Contemporary Kinetic Theory of Matter [1 ed.] 0521895472, 9780521895477

Table of contents :
Dedication
Contents
List of Figures
List of Tables
Acknowledgments
Nomenclature
1 Introduction
2 The Boltzmann Equation 1: Fundamentals
3 The Boltzmann Equation 2: Fluid Dynamics
4 Transport in Dilute Gas Mixtures
5 The Dilute Lorentz Gas
6 Basic Tools of Nonequilibrium Statistical Mechanics
7 Enskog Theory: Dense Hard-Sphere Systems
8 The Boltzmann–Langevin Equation
9 Granular Gases
10 Quantum Gases
11 Cluster Expansions
12 Divergences, Resummations, and Logarithms
13 Long-Time Tails
14 Transport in Nonequilibrium Steady States
15 What’s Next
Bibliography
Index

Citation preview

C O N T E M P O R A RY K I N E T I C T H E O RY O F M AT T E R

Kinetic theory provides a microscopic description of many observable, macroscopic processes and has a wide range of important applications in physics, astronomy, chemistry, and engineering. This powerful, theoretical framework allows a quantitative treatment of many nonequilibrium phenomena such as transport processes in classical and quantum fluids. This book describes in detail the Boltzmann equation theory, obtained in both traditional and modern ways. Applications and generalizations describing nonequilibrium processes in a variety of systems are also covered, including dilute and moderately dense gases, particles in random media, hard-sphere crystals, condensed Bose–Einstein gases, and granular materials. Fluctuation phenomena in nonequilibrium fluids and related non-analyticities in the hydrodynamic equations are also discussed in some detail. A thorough examination of many topics concerning time-dependent phenomena in material systems, this book describes both current knowledge as well as future directions of the field. j. r . d o r f m a n is Emeritus Professor at the University of Maryland. He is a fellow of the American Physical Society and the American Association for the Advancement of Science, and is a recipient of the Chancellor’s Medal for distinguished contributions to the University of Maryland. He is also the author of two books: A Course in Statistical Thermodynamics, with Joseph Kestin, and An Introduction to Chaos in Non-equilibrium Statistical Mechanics (Cambridge University Press). h e n k va n b e i j e r e n is Emeritus Professor at Utrecht University and former Scientific Director of The Journal of Statistical Mechanics: Theory and Experiment. He is a recipient of the Humboldt-Forschungs award. t. r . k i r k pat r i c k is Emeritus Professor at the University of Maryland and a fellow of the American Physical Society. He has coauthored more than 220 research papers in related areas of research.

C O N T E M P O R A RY K I N E T I C T H E O RY O F M AT T E R J. R. DORFMAN University of Maryland

H E N K VA N B E I J E R E N Utrecht Universiteit

T. R . K I R K PAT R I C K University of Maryland

University Printing House, Cambridge CB2 8BS, United Kingdom One Liberty Plaza, 20th Floor, New York, NY 10006, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia 314–321, 3rd Floor, Plot 3, Splendor Forum, Jasola District Centre, New Delhi – 110025, India 79 Anson Road, #06–04/06, Singapore 079906 Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning, and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9780521895477 DOI: 10.1017/9781139025942 © J. R. Dorfman, Henk van Beijeren, and T. R. Kirkpatrick 2021 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2021 A catalogue record for this publication is available from the British Library. Library of Congress Cataloging-in-Publication Data Names: Dorfman, J. Robert (Jay Robert), 1937– author. | Beijeren, H. van (Henk), author. | Kirkpatrick, T. R. (Theodore Ross), author. Title: Contemporary kinetic theory of matter / J.R, Dorfman, Henk van Beijeren, and T.R. Kirkpatrick. Description: Cambridge ; New York, NY : Cambridge University Press, [2021] | Includes bibliographical references and index. Identifiers: LCCN 2021002392 (print) | LCCN 2021002393 (ebook) | ISBN 9780521895477 (hardback) | ISBN 9781139025942 (epub) Subjects: LCSH: Kinetic theory of matter. Classification: LCC QC174.9 .D67 2021 (print) | LCC QC174.9 (ebook) | DDC 530.13/6–dc23 LC record available at https://lccn.loc.gov/2021002392 LC ebook record available at https://lccn.loc.gov/2021002393 ISBN 978-0-521-89547-7 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

To Matthieu Ernst, in honor of our friendship and collaborations and his many contributions to kinetic theory

Contents

List of Figures List of Tables Acknowledgments Nomenclature

page xi xxii xxiii xxv

1

Introduction 1.1 What Is Kinetic Theory? 1.2 The Kinetic Theory of Gases 1.3 Further Applications of Kinetic Theory 1.4 Outline of This Book

1 1 2 12 15

2

The Boltzmann Equation 1: Fundamentals 2.1 The Boltzmann Equation 2.2 The H -theorem of Boltzmann 2.3 The Objections of Zermelo and of Loschmidt 2.4 The Kac Ring Model

19 19 40 50 53

3

The Boltzmann Equation 2: Fluid Dynamics 3.1 The Chapman–Enskog Solution 3.2 General Properties of the Chapman–Enskog Solution 3.3 Solving the Boltzmann Equation for the Hydrodynamic Regime 3.4 The Distribution Function to First Order in μ: The Navier–Stokes Equations 3.5 The Rate of Entropy Production 3.6 Boundary Conditions on the Hydrodynamic Densities 3.7 Comparison of the Results of the Normal Solution Method with Experiment 3.8 Projection Operator Methods for the Linearized Boltzmann Equation

60 61 71 72 77 88 91 95 99 vii

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Contents

3.9 Other Solutions of the Boltzmann Equation and Models for the Collision Operator 3.10 Moment Expansions and Variational Methods for the Boltzmann Equation 3.11 Model Boltzmann Collision Operators 3.12 Other Models

110 112 114 119

4

Transport in Dilute Gas Mixtures 4.1 Introduction 4.2 The Boltzmann Equation for Dilute Gas Mixtures 4.3 The Chapman–Enskog Solution 4.4 Transport Coefficients for Binary Mixtures 4.5 The Rate of Entropy Production

123 123 124 126 144 146

5

The Dilute Lorentz Gas 5.1 Introduction 5.2 The Lorentz–Boltzmann Equation 5.3 Diffusion in the Lorentz Gas 5.4 Hard-Sphere Systems in Three Dimensions 5.5 Lorentz Gas in External Fields 5.6 Transport of Particles in a Uniform Magnetic Field 5.7 Chaos in the Lorentz Gas

149 149 151 155 159 164 171 180

6

Basic Tools of Nonequilibrium Statistical Mechanics 6.1 The Liouville Equation 6.2 Time-Displacement Operators 6.3 Hard-Sphere Systems: The Pseudo-Liouville Equation 6.4 The Pseudo-Liouville Equations and the Streaming Operators for N-particle Hard Sphere Systems 6.5 The BBGKY Hierarchy Equations 6.6 Extensions to More General Potentials 6.7 Important Relations and Identities Satisfied by T± and T¯ ± 6.8 Proof That the Binary Collision Expansions Provide a Correct Representation of the Dynamics of N Hard-Sphere Particles 6.9 The Green–Kubo Formulae

205 206 207 214

239 242

Enskog Theory: Dense Hard-Sphere Systems 7.1 Introduction 7.2 The Enskog Truncation of the BBGKY Hierarchy 7.3 The Revised Enskog Equation 7.4 The H -theorem for the Revised Enskog Equation 7.5 The Linearized Equation and Spatiotemporal Fluctuations

255 255 256 260 264 271

7

224 229 233 235

Contents

7.6 The Revised Enskog Equation for Mixtures and the Onsager Reciprocal Relations 7.7 Enskog Theory of Transport in a Hard-Sphere Crystal 7.8 The Two-Particle Distribution Function in Equilibrium 7.9 Enskog Values for Transport Coefficients in Two and Three Dimensions 8

9

ix

290 303 310 313

The Boltzmann–Langevin Equation 8.1 Introduction 8.2 The Boltzmann–Langevin Equation 8.3 Linear Hydrodynamic Equations with Fluctuations 8.4 Detection of Fluctuations about Equilibrium and Nonequilibrium Stationary States by Light Scattering 8.5 Fluctuations in Nonequilibrium Steady States 8.6 Puzzles 8.7 Other Approaches to the Linearized Boltzmann Equation with Fluctuations

349

Granular Gases 9.1 Introduction to Granular Gases 9.2 Inelastic Collisions 9.3 The Boltzmann Equation 9.4 The Homogeneous Cooling State 9.5 Driven Systems 9.6 Planetary Rings

351 351 352 354 357 377 382

10 Quantum Gases 10.1 Introduction 10.2 Density Matrices and the Wigner Function 10.3 The Uehling–Uhlenbeck Equation 10.4 Transport in a Condensed, Dilute Bose Gas 10.5 The Spatially Inhomogeneous Bose Gas at Low Temperatures 10.6 The Two-Fluid Hydrodynamic Equations for the Very-Low-Temperature Region 11 Cluster Expansions 11.1 Introduction 11.2 Generalizing the Boltzmann Equation 11.3 Difficulties in the Collision Operators

317 317 318 324 331 338 347

387 387 389 391 398 416 422 437 437 441 463

x

Contents

12 Divergences, Resummations, and Logarithms 12.1 Singular Terms, for Long Times, in the Virial Expansions of the Collision Operator 12.2 Divergences in the Nonequilibrium Virial Expansion 12.3 Ring Kinetic Equations 12.4 Applications to Green–Kubo Correlation Functions 12.5 Logarithms in the Density Expansions of Transport Coefficients 12.6 The Diffusion Coefficients for the Classical and Quantum Lorentz Gases 12.7 Final Remarks

467 467 469 476 486 490 499 503

13 Long-Time Tails 13.1 Introduction 13.2 Mode-Coupling Contributions to Uij (k,z) and to Transport Coefficients 13.3 Implications and Experimental Consequences of the Long-Time Tails 13.4 Conclusion

509

14 Transport in Nonequilibrium Steady States 14.1 Introduction 14.2 Ring Kinetic Equations for Stationary Nonequilibrium Gases 14.3 Stationary-State Couette Flow 14.4 Stationary-State Heat Flow 14.5 Nonequilibrium Is Different

541 541 545 549 558 569

15 What’s Next 15.1 Where Are We? 15.2 Kinetic Theory of Gas Flows and of Brownian Motion 15.3 Other Applications of Kinetic Theory 15.4 A Common Theme

577 577 581 583 587

Bibliography Index

507 508

526 539

589 620

Figures

2.1.1

2.1.2

2.1.3

2.1.4

2.1.5

2.1.6

2.1.7

Typical pair potentials. Illustrated here are the Lennard–Jones pair potential, φLJ , and the associated Weeks–Chandler–Anderson potential, φWCA , which gives the same repulsive force as the Lennard–Jones potential. The relative separation coordinate is scaled by the distance σ , the point at which φLJ first passes through zero, and the energy axis is scaled by the well depth, ε. This figure is courtesy of J. D. Weeks The (v1,v2 )-collision cylinder. The sphere has a radius a, which is the range of the forces. For hard-sphere molecules, a is the diameter of the molecules. Direct and restituting collisions in the relative coordinate frame. The corresponding collision cylinders, as well as the scattering angle, θ , are illustrated. Schematic illustration of the direct collisions, on the right, and the restituting collisions, on the left. The corresponding unit vectors indicating the direction of the apse lines are also shown. Schematic illustration of particle–wall collisions. In (a) the number of particles with velocity v1 is increased due to collisions of particles with the wall. In (b), the number of particles with velocity v1 is diminished when one of them collides with the wall. The effective potential energy for a two-body interaction as a function of separation for a Lennard–Jones pair potential, φLJ , at various values of the angular momentum. The energy axis is scaled by the well depth, ε. Here the reduced spatial separation is given by r∗ = r/σ , where r is the spatial separation of the particles, and σ is the distance to the first zero of the pair potential. The reduced angular momentum g∗ is given by g∗ = gb/(2mε)1/2 . The scattering angle, θ , as a function of impact parameter, b, for three different relative energies, 1 > 2 > 3 , for two particles interacting with a potential with a repulsive core and an attractive region at larger separations. Note that three different impact parameters can lead to

page 20

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2.4.1 3.7.1

3.7.2

3.7.3

3.8.1

3.8.2

3.11.1

4.4.1

5.2.1

List of Figures the same value of |θ |, which is the quantity of importance for the scattering cross-section. Rainbows occur at the minima of these curves where dθ/db = 0. The Kac ring model. The sites on the ring are indicated, and the markers between some of the sites are indicated by the check marks. The reduced second virial coefficient, B ∗ (T ∗ ), as a function of the reduced temperature [357, 356]. This figure is taken from the paper of J. Kestin, S. T. Ro, and W. A Wakeham [357] The reduced inverse coefficient of viscosity for the noble gases and some binary mixtures of noble gases in Figure 3.7.1 [357, 356]. This figure is taken from the paper of J. Kestin, S. T. Ro, and W. A. Wakeham [357] The Eucken factor for the noble gases. The dashed line is the theoretical result obtained using the 11–6–8 potential, Eq. (3.7.6), of Klein and Hanley [394, 295]. This figure is taken from the paper of B. Najafi, E. A. Mason, and J. Kestin [500] The dispersion of and absorption of sound in neon. In these figures, U0 = c, the velocity of sound, and α = kI , the sound damping coefficient. The solid curves are obtained from kinetic theory, J. D. Foch, G. W. Ford, and G. E. Uhlenbeck assuming that the particles are Maxwell molecules [224, 222]. The data points are due to to the experiments of M. Greenspan [276] Comparison of theoretical values for sound dispersion with the experimental results of M. Greenspan for a range of wave numbers and frequencies and for the noble gases [276]. The upper curve is a plot of U0 /U, as a function of the dimensionless sound frequency, ξ, where U0 = c, the ideal gas velocity of sound, and U is the phase velocity of the sound wave. The lower curve is a plot of the absorption coefficient, αU0 /ω, where α ≡ kI , as a function of ξ . The solid curves are results for Maxwell molecules. This figure is taken from the paper of J. D. Foch and M. F. Losa [223] BKW modes. The plot represents the ratio of the BKW solution to the equilibrium solution as a function of velocity for various times. The approach to equilibrium is not uniform in velocity and approaches the equilibrium from below. This figure is taken from the paper of M. H. Ernst [177] The inverse of the unlike-interaction contribution to the viscosities of binary mixtures of Xe with other noble gases. Experimental data are given by open circles for He–Xe, bottom-filled circles for Ne–Xe, side-filled circles for Ar–Xe, and filled circles for Kr–Xe. The vertical axis is defined by Eq. (4.4.3). This figure is taken from the paper of J. Kestin, H. E. Khalifa, and W. A. Wakeham [354] Fixed scatterers are placed at random in space. Moving particles interact with the scatterers but not with each other.

40 53

97

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99

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146 152

List of Figures 5.2.2

5.6.1

5.6.2

5.6.3

5.7.1

5.7.2

5.7.3

5.7.4

5.7.5

The direct collision with apse line vector, σˆ , and the restituting collision with apse line vector, −σˆ , for moving particles colliding with hard-sphere scatterers in three dimensions. Here b is the scattering impact parameter, and  denotes the azimuthal angle for the collision plane. Some possible trajectories or fragments for the cyclotron motion of charged moving particles in a two-dimensional Lorentz gas with magnetic field perpendicular to the plane of the system. Figure adapted from A. V. Bobylev, F. A. Maaø, A. Hansen, and E. H. Hauge [48] Successive collisions of a moving particle with a fixed scatterer of radius a. The successive collisions are labeled 1 → 2 → 3. The quantity denotes the distance from the center of the scatterer to the centers of the cyclotron orbits. The angle subtended by two adjacent incidence points is denoted by 2β. Figure taken from [48] The dynamics of the collision of a moving particle with a scatterer in a magnetic field perpendicular to the plane of the system. Here b = a sin α is the impact parameter. The scattering angle is denoted by ψ and α = (π − ψ)/2. This figure is taken from [48] The defocusing effect of the convex scatterers on a small pencil of trajectories of the moving particles. Here ϕ is the angle of incidence of the infinitesimally small pencil of trajectories. The change in the radius of curvature for the collision illustrated in Fig. 5.7.1. Here ρ− and ρ+ denote the radius of curvature before and after the collision, respectively. Lyapunov exponents for a dilute, two-dimensional, random Lorentz gas. The solid line are the theoretical predictions [639, 640, 649], while the data points are obtained by molecular dynamics. The dotted line represents the positive Lyapunov exponent for a Lorentz gas where the scatterers are placed on the sites of a triangular lattice. The Lyapunov √ exponents are given in units of v/a. The scaling density is ρ0 = [2a 2 3]−1 . This figure is taken from the paper of C. Dellago and H. A Posch [141] Theoretical and computer results for the two positive Lyapunov exponents for an equilibrium, dilute, three-dimensional, random Lorentz gas as a function √ of the reduced density are shown. Here the scaling density is ρ0 = 2[8a 3 ]−1 . This figure is taken from the paper of C. Dellago and H. A Posch [141] The positive and negative Lyapunov exponents, λ+ (ns a 2 ),λ− (ns a 2 ), respectively, at two different reduced densities, plotted as a function of the square of the strength of the electric field. The solid lines represent the predicted, low density, values given by Eqs. (5.7.42,48) [642, 417], and the points are the results of H. Posch and C. Dellago, using molecular dynamics for two densities [142, 141]. Here  = E/vm with E the applied field, , the mean free path, and v,m

xiii

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xiv

5.7.6

6.3.1

6.3.2

6.3.3

6.7.1

7.2.1

7.3.1

7.6.1

List of Figures are the speed and mass of the moving particle. Figure taken from the paper of H. van Beijeren, et al. [642] Theoretical and computed values for diffusion coefficients as a function of the applied electric field for a thermostatted, dilute random Lorentz gas at two densities. Figure taken from the paper of van Beijeren et al. [642] The collision cylinder for the (1,2) binary collision. The z-axis is aligned along the relative velocity before collision, and the various collision parameters used in constructing the binary collision operator are illustrated. The dynamics contributing to the binary collision operators, T+ and T− , included in the forward-streaming and backward-streaming operators, respectively, for two particles. All collisions are specular. The dashed vectors indicate unchanged relative velocity vectors that appear in the virtual parts of these operators, where the particles move as if there were no interaction and their centers can be within the collision radius, a. The dynamics contributing to the binary collision operators, T¯ + and T¯ − , included in the adjoints of the streaming and backward-streaming operators, for two particles. All collisions are specular. Detailed action of the the four binary collision operators, T¯ ±, and T± showing the action of the lifting operators by means of small circles on the appropriate velocity vectors. Figures (a) and (b) correspond to the T± operators, and (c) and (d) correspond to the T¯ ± operators. All collisions are specular. Figure (a) illustrates a simple example of the excluded volume effects incorporated in the Enskog collision integral. These effects are due to the fact that two particles in contact leave more room for the remaining ones than two particles at large separation. This figure corresponds to the function V (r 1,r 2 |r 3 ) in Eq. (7.3.2), Figure (b) illustrates a simple collisional transfer effect whereby momentum and energy are instantaneously transferred from one of the colliding particles to the other over a distance a at a collision. The figure on the left illustrates a situation immediately before a (1,2) collision. The figure on the right corresponds to the situation immediately after the collision, momentum, and energy have hopped from particle 1 to particle 2. The Mayer graphs corresponding to the three- and four-particle contributions to the χ2 (r 1,r 2,t) in the revised Enskog equation. The factors of 2 and 4 in the first two graphs for V (r 1,r 2 |r 3,r 4 ) arise from permutations of 3 and 4 in both graphs and additional permutations of 1 and 2 in the second graph. Comparisons of transport coefficients obtained from computer-simulated molecular dynamics for hard-sphere systems with the results of the Enskog theory for the coefficients of shear

201

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225

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List of Figures

8.4.1

8.4.2 8.5.1

8.5.2 9.4.1

9.4.2

9.4.3

viscosity, ηs /ηs,E , and thermal conductivity, λ/λE . Here the various data sets are the results of simulations of the hard-sphere system with differing numbers of particles. The EW data is that of Wood and Erpenbeck [199]. The curves labeled “fit” are obtained using approximate forms for the pair correlation function needed to determine values for Enskog’s χ function. Here ξ = π na 3 /6 is the packing fraction. For comparison, ξ = 0.55 is the random close packing density in these units. This corresponds to na 3 ≈ 1.05. These figures are taken from the paper of H. Sigurgeisson and D. M. Heyes [589] A very schematic representation of a light-scattering experiment where an incoming beam of light with wave vector ki and frequency ωi is scattered by the fluid. One adjusts the detector to observe light scattered into angle θ with wave vector kf and frequency ωf . The scattering volume is the shaded region where the incoming and outgoing beams intersect. For the case of the NESS with a temperature gradient, boundary plates controlling the temperature gradient are maintained at constant temperatures T1 and T2 . Rayleigh–Brillouin spectrum of light scattered by liquid argon in equilibrium at 84.97 K. Figure taken from P. A. Fluery and J. P. Boon [219] The coefficients, AT ,Aν , expressing the enhancement of Rayleigh scattering by a fluid with a stationary temperature gradient. Here the fluid is n-hexane at 25 C. The solid lines are the theoretical predictions of Kirkpatrick et al., and the data points are the results of the small-angle light-scattering experiments. Figure taken from Li et al. [432] A simple correlated collision sequence. This is a three-body recollision event, as described in the text. The velocity distribution, ρ(c), of inelastic, hard-sphere particles in a homogeneous cooling state, for different values of the restitution coefficient, n . Here c = v/vT is the velocity scaled by the thermal velocity. This figure is taken from the paper of Huthmann, Orza, and Brito [317] Experimental results for the scaled velocity distribution in a homogeneous cooling state of ferromagnetic spheres in microgravity produced by arranging magnets around the sample cell. The data clearly shows the overpopulation of high-energy particles, with a velocity distribution that closely fits an exponential decay as the first power of the velocity. Figure taken from the paper of Yu, Schröter and Sperl [693] Vortex patterns and clustering in a gas composed of identical of inelastic hard disks, as obtained from molecular dynamics. The gas is prepared in an initially homogeneous state. At about 80 collision times per particle, vortex patterns and spatial inhomogeneities appear in the gas, as illustrated on the left. At about 160 collision times, clustering of the particles is observed, as shown on the right. The

xv

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344 346

363

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9.4.4

10.3.1

10.5.1

11.2.1

11.2.2

11.2.3

List of Figures system was composed of 50,000 inelastic hard disks with  = 0.9 and at density π nσ 2 /4 = 0.4. Here σ is the diameter of the disks. This figure is taken from the paper of T. C. P. van Noije and M. H. Ernst [658] The numerical solution of Eq. (9.4.66) for the exponent, a, that determines the high-energy tail as a function of the coefficient of restitution, . This figure is taken from the paper of M. H. Ernst and R. Brito [181] The scattering of two identical particles, as seen in the collision plane in the center-of-mass frame. The detector cannot distinguish between detecting particle 1, having been scattered by an angle θ, or particle 2, having been scattered by an angle of π − θ . Representative Feynman graphs for the interactions of quasiparticles with each other or with the condensate that are accounted for in the collision term C12 , Eq. (10.5.10). Figures (a) and (d) represent processes in which a quasiparticle with wave vector k is produced, while figures (b) and (c) represent processes in which a quasiparticle with wave vector k is removed from the system. Incoming lines with wave vector ki are associated with factors f (r,ki ,t) in the collision term C12 , while outgoing lines are associated with factors of (1 + f (r,ki ,t)) in this collision term. Dynamical event with phase-space volumes growing algebraically in time. Figure (a) is a binary collision event that contributes to U −t (1|2). Figure (b) is a sequence of two collisions that contributes to U−t ( 1,2| 3). Figure (a) illustrates the action of the two-particle operator, St (1,2), on the phase points of the two particles in collision that is needed for the evaluation of the integrand in Eq. (11.2.51). Figure (b) illustrates the the coordinate system used in the evaluation of the integrals appearing in Eq. (11.2.57). Examples of dynamical events that contribute to the integrand in Eq. (11.2.58) for particles interacting with central, repulsive forces. Figure (a) illustrates a genuine three-body collision. Figures (b), (c), and (d) represent sequences of three correlated collisions between the three particles, They are called a recollision, a cyclic collision, and a hypothetical collision, respectively. In the hypothetical collision illustrated by Figure (d), one sees that particle 3 would have collided with particle 1 had it not collided with particle 2 before it could hit 1. The dashed lines and 1,3 represent the trajectories of particles 1 and 3 had the (2,3) not taken place. This corrects for the circumstance that the two-body collision integral counts the hypothetical (1,3) collision, as if it had actually taken place. Sequences (b), (c), and (d) are examples of ring events, here involving three collisions among three particles. Not illustrated are sequences of four collisions and sequences where particles 1 and 2, say, collide while 3 is

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List of Figures

11.2.4

11.2.5

11.2.6

12.2.1

12.2.2

12.2.3 12.3.1 12.5.1

overlapping 2, preceded by a collision between 1 and 3. An example is illustrated in Fig. 11.2.6. This illustrates the important difference between the T¯ − and the T− operators. In the left figure, the virtual part of the collision operator T¯ − describes the relative configuration when the two particles are just starting to overlap. In the right figure, the virtual part of the T− operator describes the relative configuration when the two particles have finished overlapping. Schematic illustration of the double-overlapping configuration contained in the product of three binary collision operators v v T¯ − (1,2)S (0) ∗ T¯ − (1,3)S (0) ∗ T¯ − (2,3)S (0) . At time τ2 , particles 2 and 3 are in contact on their way to overlap. By time τ1 , particle 1 is in contact with particle 3 while particle 3 is overlapping particle 2. At time t, particle 3 overlaps both of them. Figure (a) represents the three-body Enskog contribution f13 f23 T¯ − (1,2). Figure (b) represents the two-collision, single-overlap contribution T¯ − (1,2)f23 S (0) T− (1,3). The geometry for the construction of the binary collision operator, Ta . The actual trajectory of the collision within the action sphere is illustrated by the curved line. The apse line is the line of symmetry for the collision, with unit vector κ. ˆ The distance a(κ,g) ˆ from the center-of-action sphere along the apse line denotes the point where the incoming and outgoing relative velocity asymptotes intersect with the apse line. Sketch of recollision dynamics and times, for the recollision sequence (1,2)(1,3)(1,2). The first (1,2) collision takes place at time t2 , the (1,3) collision takes place at time t1 , and the final (1,2) collision takes place at time t. One of the many four-body ring events. These are correlated sequences of four collisions among the four particles. A repeated ring event with five collisions among four particles. Such events have phase volumes that are less divergent than the ring events. The coefficients of self-diffusion, D/D0 ; viscosity, η/η0 ; and thermal conductivity, λ/λ0 , for hard spheres, reduced by their Boltzmann values as obtained from molecular dynamics, are plotted as functions of density nσ 3, where σ is the diameter of the spheres. The data points are the results of simulations by W. W. Wood and J. E. Erpenbeck [206] for the coefficient of self-diffusion and B. J. Alder, D. M. Gass, and T. E. Wainwright for all three transport coefficients [8]. The solid curves represent the expansion given by Eq. (12.5.26) for each transport coefficient using the values for the coefficients given in Table 12.1, including the Enskog theory approximation for (3) bμ,E . The dashed lines represent the first two terms in the expansion, (3)

1 + aμ na 3 . This figure is courtesy of J. V. Sengers.

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xviii 12.6.1

12.6.2

12.6.3

13.2.1

13.2.2

List of Figures The figure on the left is a recollision event of the ring type with two scatterers for a particle in a random Lorentz gas. On the right are two events involving the moving particle and three scatterers that are equally divergent. Both diagrams labeled (a) are ring recollision events, while diagram (b) is a non-ring recollision event in which the moving particle traverses the middle scatterer twice without interacting with it. Both ring and non-ring events must be taken into account when the most divergent terms in the density expansion are summed. These figures show the inverse of the coefficient of diffusion for a moving particle in a hard-disk Lorentz gas, as obtained from computer simulations. The upper figure shows the density dependence of vσ/Dn∗ compared with the sum of the theoretical values for the first four terms in Eq. (12.6.1). Here σ is the radius of a scatterer, v is the speed of the moving particle, and n∗ = nσ 2 is the reduced density. The solid line is the theoretical result for this quantity given by the first four terms in Eq. (12.6.1). The lower figure shows the same quantity with the low-density value subtracted from it as expressed in Eq. (12.6.3). The solid line shows the value of the first logarithmic term appearing in Eq. (12.6.3). The agreement of the theory with the simulation results at low densities is evident. These figures are taken from the papers of C. Bruin [69, 70]. √ The functions f (χ ) = μ2 ln χ + μ2 ± 2χ π are plotted as functions of χ . Here χ = λq /(π ), proportional the ratio of the de Broglie wavelength of the electrons to the classical mean free path length. The six measured values of these quantities are indicated, and the two solid curves correspond to the two possible bounds on including the next term, μ3 χ . The two dashed curves represent these bounds without the logarithmic term. This figure is taken from the paper of K. I. Wysokinski, W. Park, D. Belitz, and T. R. Kirkpatrick [689, 690]. The right and left figures represent the results of B. J. Alder and T. E. Wainwright for the normalized velocity autocorrelation, ρ(s), where s = (t/t ), for a tagged particle in a gas of hard disks (left) and hard spheres (right). The velocity autocorrelation function for disks is measured at three densities characterized by A/A0 = 2,3, and 5 – where A0 is the close packing area, and for 986 (closed triangles) and 504 (open triangles) particles. The results for three dimensions for V /V0 = 3, where V0 is the volume at close packing, are plotted on a log scale, and the line has a slope of −3/2. These figures are taken from the paper of Alder and Wainwright [10]. The normalized velocity autocorrelation function for hard spheres for V /V0 = 5, as a function of the reduced time s = t/t, obtained by W. W. Wood and J. E. Erpenbeck [205] from computer simulations for different numbers of particles, N . Here as before, V0 is the volume of N spheres at close packing. The dotted curve labeled DC is the

500

502

503

513

List of Figures

13.2.3

13.2.4

13.2.5

13.2.6

13.2.7

infinite system result of Dorfman and Cohen [157, 156, 155]. The dash-dot curves are theoretical results with finite size and sound mode contributions included. This figure is taken from the paper of W. W. Wood and J. E. Erpenbeck [205] The velocity autocorrelation function as a function of time for a tagged particle in a cellular automata lattice gas placed on a Frisch–Hasslacher–Pomeau two-dimensional lattice (left figure) at a density of 0.75 per lattice site or on a three-dimensional face-centered lattice (FCHC, right figure) at a density of 0.10 per lattice site. The agreement of the computer results with the theoretical expressions for the decay of the autocorrelation function using mode-coupling theory is excellent and clearly exhibits the long-time-tail effects after about ten collision times. The lower curves indicate the estimated errors in the computations. The left figure is taken from the paper of D. Frenkel and M. H. Ernst [233], and the right figure is from the paper of M. A. van der Hoef and D. Frenkel [651] The velocity autocorrelation function multiplied by (t/t ) for a gas of hard-disk particles at various densities from computer-simulated molecular dynamics by M. Isobe. The packing fraction is defined by v = nπ a 2 /4. In this figure, the horizontal dotted line represents the inverse time decay. This decay extends to times of order of a few hundred mean free times at the lowest density, v = 0.05. The decay is more rapid for higher densities. This figure is taken from the paper of M. Isobe [321] Theoretical curves for the coefficients of the (t/tE, )−d/2 long-time tails in the velocity autocorrelation function for a particle in a hard-disk or hard-sphere gas, obtained by J. R. Dorfman and E. G. D. Cohen, using Enskog theory values for the transport coefficients, as given by Eq. (13.2.20) [157, 156, 155]. The crosses in the lower curve, for hard disks, are the results of the computer simulation by B. Alder and T. Wainwright [9, 10]. Here V0 is the close packing volume (d) and αD,E = (t/tE, )d/2 ρD (t). The normalized velocity autocorrelation function, ρD (t), is defined before Eq. (13.2.10). This figure is taken from the paper of J. R. Dorfman and E. G. D. Cohen [157]. The figure shows one-half of the averaged flow of the particles around the tagged central disk, illustrating clearly the spatial and velocity correlations in the flow pattern in the neighborhood of the tagged particle. This figure is taken from B. J. Alder and T. E. Wainwright [10] This figure shows the normalized velocity autocorrelation function, φ(t) =< vx (0)vx (t) >, for tagged particle diffusion in a cellular automata lattice gas plotted as a function of time for times up to about 600 mean free times. The solid line is the result of the self-consistent mode-coupling theory as given by Eq. (13.2.34), which at these times is an improvement over the simple mode-coupling theory. This figure is taken from the paper of C. P. Lowe and D. Frenkel [443].

xix

514

516

517

520

521

525

xx 13.2.8

13.3.1

13.3.2

13.3.3

13.3.4

List of Figures This figure shows the results obtained by M. Isobe for the long-time behavior of the velocity autocorrelation functions for hard-disk gases over similar time scales. These results show that the self-consistent expression improves upon the simple mode-coupling result over these time scales. This figure is taken from the paper of M. Isobe [321] A plot of the absolute value of the sound mode frequency in liquid argon. The solid line is ωs = ck, where c is the speed of sound. The dashed line represents the result of mode-coupling theory with the k 5/2 term in Eq. (13.3.9) included. This figure is taken from the paper of I. M. de Schepper, P. Verkerk, A. A. van Well, and L. A. de Graaf [130] The figure on the top left is a plot of the normalized peak height of the scattering function, (Q), and the one on the top right is the normalized half-width, γ (Q), as functions of the wave number Q, as shown in the paper on the neutron-scattering experiments on liquid sodium by C. Morkel, C. Gronemeyer, W. Glaser, and J. Bosse [491]. The solid lines are the predictions of mode-coupling theory given by I. de Schepper and M. H. Ernst [126]. There are no adjustable parameters in fits of the data to the theory. The dashed line in the left figure is the result of hydrodynamics. The lower figure shows the data for the Fourier transform of the velocity autocorrelation function plotted as a function of the square root of the frequency expressed as an energy [490]. The solid line is the result of mode-coupling theory [491]. Extended hydrodynamic eigenvalues of the linearized revised Enskog operator as a function of a dimensionless wave number, kσ, where σ is the diameter of the spheres, denoted √ by a in the text, as calculated for a dense hard-sphere gas (nσ 3 / 2 = 0.625). Here D labels the self-diffusive mode; H, the heat mode; ν, the viscous mode; and ±, the two sound modes. The real part of the eigenvalues appear in the negative ordinate, while the absolute values of the imaginary part of the sound modes appear in the positive region. Also, zi ≡ ωi , and the superscript s refers to the self-, or tagged particle, diffusion mode. Also appearing are viscous-like and sound-like modes whose eigenvalues do not vanish as k → 0. The wave number scale at the bottom of the figure expresses the same wave number but in a different dimensionless form, kE , using the mean free path length, E . For the density here, E = 0.052σ . The mean free time between collisions at this density is denoted in the figure by tE . This figure is taken from the paper of I. de Schepper and E. G. D. Cohen [124] Molasses tails. Theoretical and computer results for the stress-stress (left) and velocity autocorrelation (right) functions for a dense system of hard spheres at intermediate, not asymptotic, times. Mode-coupling theory accounts for the behavior of the time correlation functions even at these times including the negative, back-scattering, region in the velocity autocorrelation function at relatively short times. The softening of the heat mode eigenvalue at intermediate wave numbers

525

530

532

534

List of Figures

13.3.5

studied by I. de Schepper and E. G. D. Cohen [124] is crucial for these calculations. This figure is taken from the paper of T. R. Kirkpatrick and J. C. Nieuwoudt [387, 388]. The data points in both plots are computer simulation results obtained by W. W. Wood and J. E. Erpenbeck, Ref. [199] Negative regions in the velocity autocorrelation function. The figure on the left shows the time correlation function, denoted here by Z(τ ), on a logarithmic time scale, for different values of the packing density (the ratio of the volumes occupied by the spheres to the total volume of the system). The back-scatter region appears at higher values of the density. The figure on the right shows the time correlation function, ˜ C(t), as a function of time for a hard-sphere crystal. The left figure is taken from the paper of S. R. Williams, G. Bryant, I. K. Snook, and W. van Megen [682], and the right, from the paper of T. R. Kirkpatrick [367]

xxi

536

537

Tables

2.1

3.1 12.1

xxii

The recurrence time, tr , for 1 percent density fluctuations in a spherical volume of radius a in air at standard temperature and pressure. This table is taken from the paper of S. Chandrasekhar [86] and included in the collection of papers [674]. page 52 The Eucken factor for three noble gases. Data taken from W. G. Kannuluik and E. H. Carman [339]. 96 Coefficients in the density expansion for the transport coefficients for a gas of hard spheres (left table) and for a gas of hard disks (right table). The data are taken from the papers of Sengers and co-workers (3) [580, 581, 582, 308, 583, 337, 585]. Similar results for aμ have also been obtained by G. B. Brinser and D. W. Condiff [67]. 498

Acknowledgments

The three of us have been working in the field of kinetic theory for many years. Our approaches to research and our emphasis on clarity of presentation were strongly influenced by one person in particular – the late E. G. D. Cohen (Eddie), with whom we worked as students, postdocs, and long-time collaborators. We are also very fortunate to have a very close friend and collaborator, Matthieu Ernst, to whom we dedicate this book as a token of our appreciation. We would like to thank Yevgeny Bar Lev for his considerable help, over many years, with the preparation of this manuscript. His generous help with the LYX program enabled us to overcome many troublesome issues. There are many people who assisted us in the preparation of the book or were influential for our understanding of kinetic theory, and in one way or another contributed to the development of this subject. In addition to Eddie Cohen and Matthieu Ernst, these include Cécile Appert-Rolland, Dietrich Belitz, Mordechai Bixon, Jerzy Blawzdziewicz, Lydéric Bocquet, Javier Brey, Ricardo Brito, Cok Bruin, Leonid Bunimovich, Bogdan Cichocki, Ignatz de Schepper, Astrid de Wijn, Christoph Dellago, Carl Dettmann, Jim Dufty, Denis Evans Ubbo Felderhof, Thomas Franosch, Giovanni Gallavotti, Pierre Gaspard, Thomas Gilbert, Bob Goldman, Eivind Hauge, Walter Hoegy, Bill Hoover, Sudhir Jain, Betty Johnson, Behzad Kamgar-Parsi, Yuen Han Kan, Ray Kapral, John Karkheck, Kyozi Kawasaki, John Kinkaid, Rainer Klages, Hubert Knops, Herman Kruis, Oscar Lanford, Arnulf Latz, Joel Lebowitz, Jan Adriaan Leegwater, David Levermore, Mariano Lopez de Haro, Jon Machta, Christian Maes, Christina Marchetti, Charles McClure, Jan Michels, Gary Morriss, Oliver Mülken, Luis Nasser, Alfredo NavaTudela, Johan Nieuwoudt, Debabrata Panja, Oliver Penrose, Jarek Piasecki, Jacek Polewczak, Yves Pomeau, Harald Posch, Itamar Proccia, Linda Reichl, Lamberto Rondoni, David Ronis, Rudi Schmitz, Debra Searles, Jan Sengers, Jagdish Sharma, Yasha Sinai, Herbert Spohn, Wokyung Sung, Grzegorz Szamel, Dave Thirumalai, Urbaan Titulaer, Minh-Binh Tran, Hans van Leeuwen, Frédéric van Wijland, xxiii

xxiv

Acknowledgments

Ramses van Zon, John Weeks, Ab Weijland, Harald Wilbertz, Stephen Williams, Peter Wolynes, and Victor Yakovenko. We wish also to express our indebtedness to many friends and teachers who are no longer with us, including Berni Alder, Ted Berlin, Jan Burgers, Nicolai Chernov, Jerry Erpenbeck, Shmuel Fishman, Leo Garcia Colin, Isaac Goldhirsch, Harold Grad, Mel Green, John Tjon, Marc Kac, Nico van Kampen, Les Karlowitz, Joseph Kestin, Ed Mason, Al McLennan, Irwin Oppenheim, Jose Ortiz de Zárate, Pierre Résibois, George Stell, George Uhlenbeck, Bill Wood, and Bob Zwanzig. JRD would like to acknowledge with appreciation the hospitality of the Institute for Theoretical Physics of the University of Utrecht and the Lewiner Institute for Theoretical Physics of the Department of Physics, The Technion, Haifa, Israel, and its former director, the late Shmuel Fishman. JRD and TRK are indebted to their home departments, the Institute for Physical Science and Technology and the Department of Physics of the University of Maryland, for support for many years. They gratefully acknowledge research support over this time from the United States National Science Foundation. HvB is much indebted for the ongoing support of his home institution, the Institute for Theoretical Physics of Utrecht University. In addition he acknowledges the hospitality and support of the University of Maryland, the Université Libre de Bruxelles, the Homi Bhabha National Institute and the TATA Institute in Mumbai, the Erwin Schrödinger Institute and the University in Vienna, the Humboldt Stiftung and the Technische Universität Münich, the Australian National University and the Australian Defense Force Academy in Canberra, the Institut Henri Poincaré in Paris, and the Korea Institute for Advanced Study in Seoul. We wish to thank the Hamburger Kunsthalle, Hamburg Germany, for permission to use the painting on the cover, Geographers at Work by Cornelis de Man (1621– 1706), oil on canvas, 81×68 cm, Inv. 239. Photo credit: Elke Walford.

Nomenclature

E(k) [D (α),D (γ ) ] α αT T¯ 0 T¯ W T± (1,2) Tr± (1,2) Tv± (1,2) T¯ ± (1,2) L¯ 0,W − (ps) L¯ ± (N) (ps) L¯ W − (n) bkˆ (1,2)

β F F ext k ρ(t)  ρS σij D

Energy of Bogoliubov excitations Bracket integral Accomodation coefficient for a boundary Coefficient of thermal expansion Binary collision operator for binary collisions, when the duration of the collision and the spatial separations of the colliding particles are ignored Binary collision operator for wall–particle collisions Hard-sphere binary collision operators Real part of a hard-sphere binary collision operator Virtual part of a hard-sphere binary collision operator Barred, or adjoint, of a hard-sphere binary collision operator, T∓ (1,2) Free streaming part of Liouville operator including particle–wall interactions Barred pseudo-Liouville operator for N hard spheres Pseudo-Liouville operator including particle-wall interactions Binary collision velocity exchange operator that replaces velocities by their restituting values Inverse temperature parameter External force per unit mass in the Boltzmann equation External force in Langevin equation Wave vector Density matrix Angular velocity vector Location of a point on the boundary surface of a system Elements of the stress tensor Velocity gradient tensor xxv

xxvi

Nomenclature

g Relative velocity of two particles JK (r,t) Local energy current vector P(r,t) Local pressure tensor q(r,t) Local energy current u(r,t) Local average velocity in a gas at point r at time t c = v − u(r,t) Peculiar velocity of a particle (v) Boltzmann collision operator linearized about a total equilibrium distribution function αβ Linearized Boltzmann collision operator for binary collisions of particles of species α and β loc (v1 ) Boltzmann collision operator linearized about a local equilibrium distribution V N-particle 2dN-dimensional velocity vector in phase space u Scaled velocity for granular gas un Velocity of the normal fluid in a condensed boson gas χT Isothermal compressibility  Mean free path length n Coefficient of restitution – normal t Coefficient of restitution – tangential  The azimuthal angle η Coefficient of shear viscosity η(r,t) External source in the η-ensemble ηE Enskog theory coefficient of shear viscosity ηi (t),i (t) Descriptors for presence or absence of white or black beads at point i at time t in the Kac ring model γ Drag coefficient in Langevin equation Rate at which the number of particles with prescribed velocities + drdv increase due to binary collisions in a very small 2d-dimensional oneparticle position and velocity phase space − drdv Rate at which the number of particles with prescribed velocities decrease due to binary collisions in a very small 2d-dimensional oneparticle position and velocity phase space Sound damping coefficient s S,E Enskog theory value of the sound damping coefficient s Parameter descibing the cooling rate in a granular gas W drdv Rate of change of the single particle distribution function due to collisions of particles with a boundary wall in a small 2d-dimensional, one-particle phase space

Nomenclature

kˆ

xxvii

Unit vector in the direction of the vector from the origin to the point of closest approach in binary collision as described in the relative coordinate system centered on one of the colliding particles ˆ V Particle–particle Interaction contribution to the Hamiltonian operator σˆ Unit vector along apse line for hard-sphere collisions λ Coefficient of thermal conductivity λ Partial coefficient of thermal conductivity Eucken factor λ/(ηcv ) Enskog theory value of the coefficient of thermal conductivity λE i(±) (r,p,t) Positive and negative stretching factors Linear Boltzmann propagator acting on deviation of the single partiLk cle distribution from its equilibrium value, χ (R) Linear single particle ring propagator Lk (z) WN (x1,x2, . . . ,xN ) General N-particle function of positions and momenta symmetric under particle interchanges G0 (1,2, . . . ,s,z) Laplace transform of time displacement operator; also called a propagator L() N-particle Liouville operator Kinetic part of the N-particle Liouville operator L0 () (ps) Pseudo-Liouville operator for N hard spheres L± (N) Interaction potential part of the N-particle Liouville operator LI () S() Time displacement operator in phase space (0) St (1,2, . . . ,s) s-particle free streaming operator (eq) Husimi cluster functions for s-particles Vs μ Ordering parameter in the Chapman-Enskog solution of the Boltzmann equation Reduced mass of two particles μ12 ν Collision frequency parameter in Bhatnagar–Gross–Krook (BGK) model Low-density, equilibrium collision frequency for a particle with ν(vi ) velocity vi νc Collision frequency ω Thermal creep coefficient (±) Leading order term in sound mode eigenvalue ω Hydrodynamic eigenvalues ωi (k) ij Non-dissipative terms in the matrix form of the linearized Navier– Stokes equations Zwanzig–Mori projection operators P,P⊥

xxviii

Nomenclature

Phase of the condensate wave function Two-particle interaction potential Maxwell–Boltzmann velocity distribution function for a gas at equilibrium Maxwell–Boltzmann distribution function for particles with velocity φW (v,ρS ) v at the temperature TW (ρS ) appropriate for a point ρS on a boundary surface Potential energy for particle–wall interactions W (r i ) H Heat mode eigenfunction k (v) Condensate wave function ψ0 (r,t) (ηi ) Shear mode eigenfunctions k (v) (±) Sound mode eigenfunctions k (v) Orthonormal combinations of conserved quantities in binary i(0) collisions Elements of the vector function in composition space describing the ψi(T ) (r,t) local temperature deviation for species i in the revised Enskog theory for mixtures (p) Elements of the local momentum vector in composition space in ψij (r,t) direction j for particles of species i in the revised Enskog theory for mixtures (n) Elements of the local density vector in composition space for ψij (r,t) particles of species i in the revised Enskog theory for mixtures (R) (L) ψi,k (p 1 ),ψi,k (p 1 ) Right and left eigenfunctions of the revised Enskog collision operator ρ(r,t) Local mass density ρ(,t) N-particle phase space distribution function at phase point  at time t ρ(t) Radius of curvature One of the d + 2 hydrodynamic densities in a gas, consisting of ρα number, momentum, and energy densities ρN,me (,t) Distribution function in the maximum entropy ensemble Superfluid density ρs St (1,2, . . . ,s) s–particle streaming operator σ The distance parameter in the Lennard–Jones potential σ (r,t) Local rate of entropy production Drude’s value for the electrical conductivity in a charged Lorentz gas σD Quantum differential cross-section σqm (,g) τ Temperature jump distance f˜α (r 1,v1,t) Distribution function for particles of species α φ(r,t) φ(rij ) φ0 (v)

Nomenclature

f (r,v,t) f˜(r,v,t) f¯(r,c,t) feq (r,v) floc (r,v,t) fss (r,p) ˜ (r,v,t) ¯ (r,c,t) H (x) θij ρ˜ξ,K,E (z) ˜ k.l,z G ˜ P(k,t) ˜ q(k,t) F˜ ξ ξ02 ζ ζ (r,t) ζ1, . . . ,ζ4 ζE ζS a b ˆ B(g, k) B(T ) B(t),W (t) Bs

xxix

Single particle distribution function for a gas with boundaries or when boundary effects are ignored Single particle distribution function in the interior of a container Single particle distribution function expressed in terms of the peculiar velocity, c = v − u Single particle equilibrium distribution function, possibly in an external potential φext (r) Local Maxwell-Boltzmann equilibrium distribution function Single particle distribution function of position and momentum for a gas in a stationary state Relative deviation of the single particle distribution function from the local equilibrium distribution Deviation of the single particle distribution function from the local equilibrium distribution, expressed in terms of the peculiar velocity Heaviside function Two-particle interaction operator in the Liouville equation Laplace transform of the Enskog extension of kinetic part of the current–current time correlation function Two-particle propagator with Boltzmann-like linear collision operators Fluctuating stress tensor Fluctuating energy current Fluctuation term in Boltzmann-Langevin equation Dimensionless sound frequency Strength of fluctuations in the accelerations of particles in a granular gas Coefficient of bulk viscosity Cooling coefficient for inelastic particles Coefficients of bulk viscosity appearing in the two-fluid equations for a condensed boson gas Enskog theory value of the coefficient of bulk viscosity Slip coefficient Range of a two-particle interaction potential Impact parameter for a binary collision Quantity proportional to the differential scattering cross-section that appears in Boltzmann collision integrals Second virial coefficient Number of black beads and white beads, respectively, in the Kac ring model Bernoulli numbers

xxx

Nomenclature

Direct correlation function Current–current time correlation function Adiabatic speed of sound The spatial dimension of a system Coefficient of thermal diffusivity Coefficient of mutual diffusion for mixtures Diffusion tensor for motion in a magnetic field Higher-order diffusion coefficients for Lorentz gas Enskog theory value for the coefficient of self-diffusion Lorentz gas diffusion coefficient Enskog theory value of the coefficient of thermal diffusivity Coefficient of thermal diffusion for mixtures Local internal energy densitiy in a gas at point r at time t Local Helmholtz free energy density Single-particle distribution function Scattering amplitude Single partice distribution function for particles of species α Local Maxwell–Boltzmann equilibrium distribution function expressed as a function of the peculiar velocity c. Local equilibrium distribution for a boson gas fl (r,k,t) fn (x1,x2, . . . ,xn,t) n-particle distribution appearing in BBGKY equations Single-particle density function for a gas in a stationary state fSS (r,p) N N fW (x ,p ,t) Wigner distribution function Green’s function for diffusion equation G(r,r ,t) g(r,t) Local Gibbs free energy density Equilibrium two particle spatial correlation function g2 (r 1,r 2 ) Gλ (,t) Helfand moment for thermal conductivity Helfand moment for shear viscosity Gxy (,t) h(r,t) Local density of Boltzmann’s H -function H (t) Boltzmann’s H -function Classical Hamiltonian for a system of N-particles HN Integrated intensity of the Rayleigh peak IR (k) J (f ,f ) Short-hand notation for the nonlinear, binary collision term in the Boltzmann equation Choh–Uhlenbeck collision integral J3 (f1 |t) Jαβ (fα,fβ ) Binary collision term for collisions between one particle of species α and one of species β Collision term in Bhatnagar–Gross–Krook (BGK) model Jbgk (f ) Elements of the local momentum current tensor Jij (r,t) Ji Current conjugate to thermodynamic force Xi

C(r 1,r 2 ) C (α) (t) c d DT Dαγ Dij (j ) Dd DE DL DT ,E DT α e(r,t) f (r,t) f (r 1,v1,t) f (θ) fα (r,v,t) f¯loc (r,c,t)

Nomenclature

JK (f ,f ) k K(g ,g⊥ ) k∗ kB kT α KW (v,v ) L Lij N N0 nc ns n˜ = na d p(r,t) P (v,v ) p(r) rred S(k) S(k,ω) SE (r) Sl+ 1 2

xxxi

Kac-Boltzmann collision operator Wave number Collision kernel for inelastic, binary collisions Wave number where the static structure factor has a minimum Boltzmann’s constant Thermal diffusion ratio The collision kernel for particle–wall collisions Macroscopic length Onsager transport coefficient Number of particles in a system contained in a volume V Number of particles in boson ground state Number density of condensate in a weakly interacting boson gas Density of scatterers in a Lorentz gas Reduced number density Local pressure Probability per unit velocity that a particle with velocity v results when a particle with velocity v collides with a wall Form factor for region in which light scattering takes place Reduced radius of colliding particles Static structure function Dynamic structure factor Multiplicative constant appearing in the dynamic structure factor Sonine polynomial

T Temperature Temperature at a point ρS on the wall T (ρ S ) T (r,t) Local temperature Mean free time t Condensation temperature for an ideal Bose–Einstein gas T0 Tg Granular temperature Recurrence time tr U (r 1,r 2, . . . |r j ,r j +1, . . .) Ursell cluster functions Coefficient of the delta function in the pseudo-potential for U0 boson–boson interactions Dissipative matrix elements in linearized Navier–Stokes equations Uij V Volume of a system V (r,L) Effective potential for radial motion in a binary collision with angular momentum L Close packing volume for hard disks or spheres V0 vT (t) Thermal speed of particles in an inelastic gas

xxxii

Nomenclature

N-particle hard-sphere overlap function Function that vanishes when r is outside the boundary and is unity for r inside the system Thermodynamic force appearing in expression for entropy Xi production m Yl Spherical harmonics z Laplace transform variable Two particle kinetic operator with binary collision operators Ta a (1,2|t) describing collisions of particles 1 or 2 with a field particle 3. Collision operator appearing in revised Enskog theory for a hard C (x1 ) sphere crystal Lorentz-Boltzmann collision operator λL Boltzmann collision operator linearized about a stationary state ss (δ f˜) distribution function  Product of magnetic field vector and electric charge G Reciprocal lattice vector Linearized collision operator in kinetic equation for the distribution L12 function in a condensed boson gas (E) Linear, wavenumber dependent, propagator obtained from the Lk (p 1 ) revised Enskog equation Ring collision operator Rk,z (v1 ) T(1,2,z) General binary collision operator u(r,t) Displacement vector in a hard sphere crystal Superfluid velocity Vs Linearized revised Enskog collision operator for mixtures k,αβ Enskog collision operator for tagged particle diffusion D,E Revised Enskog collision operator, linearized about a total E equilibrium distribution function C Euler’s constant One-particle Liouville operator for particle in a magnetic field L1 Lorentz gas projection operator PL Boson field operators aˆ k, aˆ †k ˆbk, bˆ † Bogoliubov excitation field operators k ˆ0 Kinetic contribution to Hamiltonian operator H ˆj(r) Current operator ˆ n(r) Density operator ˆ tr Trapping potential contribution to the Hamiltonian operator V ˆ ψ(r), ψˆ † (r) Boson field operators ˆ ! Condensate part of field operators Unitary operators used in derivation of the two-fluid equations Uˆ i WN () WW (r)

Nomenclature

˜f(t)  H loc (v),

xxxiii

Fluctuating force in Langevin equation Energy parameter in Lennard-Jones potential Hamiltonian operator Boltzmann collision operator linearized about a local equilibrium distribution χ Density parameter for electrons in helium  Mean free path length (0) (2) ≡ ψξ(0) (c2, − k) Shortened notation for zeroth order γk(0) (1) ≡ ψγ(0) (c1,k),ξ−k in wave number hydrodynamic mode eigenfunctions T Time ordering operator μ Local equilibrium chemical potential for a condensed boson gas Maxwell-Boltzmann distribution as a function of momentum φ0 (p) (x ,t|v ) One-particle distribution function appearing in expressions for (d) 1 0 K Green–Kubo time correlation functions Maxwell-Boltzmann distribution as a function of velocity φo (v) One of the d + 2 conserved quantities in a binary collision ψα τ Dimensionless time scale for a granular gas Optical depth τo a S-wave scattering length Mode coupling matrix element appearing in the equations for Cμν (k) stationary heat flow Collision operators in the kinetic equations for the distribution Cij function in a condensed boson gas Specific heat at constant volume cv gs (x1,x2, . . . ,xs ,t) s-particle correlation function appearing in cluster expansion of nonequilibrium distribution functions h(r,t), Chemical potential per unit mass n(r,t) Local number densitiy in a gas at point r at time t Ring operator for self-diffusion RD,k,z (v1 ) s(r,t) Local entropy density S(t) Nonequilibrium entropy Incoherent dynamic structure factor SI (k,ω) χ (r,v,t) Deviation of the single-particle distribution from its equilibrium value Pair correlation function χ2 χs (x1, . . . ,xs ) s-particle cluster function appearing in linearization of deviations of nonequilibrium distribution functions from their equilibrium values χ (r,v,t) Deviation of single particle distribution function from its equilibrium value

xxxiv

Nomenclature

Pair correlation function for non-overlapping hard spheres in a one component gas χ2,αβ(r1,r2) Pair correlation function for non-overlapping hard spheres of species α and β in a mixture χ Pair correlation for hard spheres in contact in a one component gas Pair correlation function for hard spheres of species α and β in χαβ contact in a gas mixture χs (x1, . . . ,xs ,t) Linearized form of the s-particle cluster function Coherent dynamic structure factor Snn (k,ω) k (v) Wavenumber dependent linearized binary collision operator I (vi ) Interacting part of the linearized Boltzmann collision operator Adjoint of the linear propagator obtained from the revised Enskog L†(E) k equation Linearized Enskog propagator L(E) k ¯ Effective instantaneous binary collision operators for particles Ta, Ta interacting with short ranged, repulsive forces, with distance of closest approach a Boltzmann collision operator obtained by using the binary collision a (v) operators Ta PD (k) wave-number-dependent projection operator for tagged particle diffusion (bc) (1,2, . . . n) Binary collision expansion of n-particle streaming operator S−t χ2 (r1,r2 )

1 Introduction

1.1 What Is Kinetic Theory? Kinetic theory is a branch of statistical mechanics that aims to derive expressions for the macroscopic properties of fluids in terms of the microscopic properties of the constituent particles. These properties include nonequilibrium quantities, such as diffusion coefficients and viscosities, which do not follow in a straightforward way from the standard techniques based on Boltzmann–Gibbs ensembles. The microscopic properties include single-particle properties, such as particle masses and sizes as well as the interactions between particles, and their interactions through external forces with the outside world, including boundaries. The methods of kinetic theory are based upon the laws of mechanics, either classical or quantum, that describe the motion of the system of particles.1 However, kinetic theory is not based on solutions of these equations for the full system but rather on estimates of the average behavior of ensembles of mechanically identical systems that differ only in their initial conditions. In this respect, kinetic theory is properly thought of as a branch of statistical mechanics, since it uses statistical methods in order to determine the typical, or most likely, behaviors of systems of many particles. It had its origins almost three centuries ago in the work of D. Bernoulli (1738), who used a simple kinetic theory to derive the perfect gas equation of state. The first application of kinetic theory to transport phenomena may have been accomplished by J. J. Herapath (1847), who used arguments based on collision dynamics to explain experiments by T. Graham on the diffusion of gases through small holes. By the time of J. C. Maxwell’s prediction that the viscosity of a gas is independent of density (1860), the use of kinetic theory to explain transport phenomena in gases was accepted by many workers. The aim of nonequilibrium statistical mechanics, including kinetic theory, to characterize macroscopic systems that are out of equilibrium requires it to focus

1

2

Introduction

on dynamical processes taking place in the system. First of all, for particles that interact with short-range forces, these are collisions between the constituent particles or between the particles and the boundaries. In the case of particles that interact with long-range forces, such as in the interactions of charged particles, collective effects can dominate the dynamics. The particles of interest can be electrons, nuclei, atoms, molecules, excitations such as phonons, photons, colloidal particles, bits of dust, planets, galaxies, etc. In many cases, the boundaries may also be considered as a type of particle and treated as such in the basic equations. The possibility to treat such a wide variety of systems accounts for many modern applications of kinetic theory, many of which we will explore in this book.2

1.2 The Kinetic Theory of Gases 1.2.1 Dilute Gases The development of kinetic theory began in the eighteenth century with the work of Daniel Bernoulli, who was able to show3 in 1738 that the pressure of a dilute gas on its container is proportional to the mean square molecular velocity. The theory was further advanced in the first part of the nineteenth century by John Herapath (1820) and by John J. Waterston (1845). The work of these pioneers eventually led to the recognition of the connection between the thermodynamic temperature of a gas and the average kinetic energy of its molecules, which resulted in the well-known kinetic theory derivation of the perfect gas equation of state. It should be noted that this work was carried out at a time when the existence of atoms and molecules as individual particles with empty space between them was not at all obvious, and many prominent scientists at the time did not believe in them. In fact, the atomic picture of matter was not universally accepted until the early part of the twentieth century. The general acceptance of the atomic picture was due, among other things, to A. Einstein’s explanation of Brownian motion in terms of fluctuations in a fluid composed of individual particles, followed by J. B. Perrin’s quantitative confirmations of Einstein’s predictions, and to the successful explanations of thermodynamic and transport properties of solids and fluids based upon statistical thermodynamics and the kinetic theory of gases. The lack of universal acceptance of the atomic picture notwithstanding, the kinetic theory of gases became a central pillar of physics with the work of Maxwell and L. Boltzmann in the later part of the nineteenth century. Today, the methods developed by Herapath and Waterston, based on collisions of particles with each other and with the walls of the container, are often used to provide elementary derivations of the expressions

1.2 The Kinetic Theory of Gases

3

for thermodynamic and transport properties of rarefied and dilute gases [71, 72].4 They form the subject matter of elementary books on the kinetic theory of gases. Maxwell and Boltzmann used statistical methods to compute the properties of gases, recognizing that the random motion of gas molecules could be best described by distribution functions. In addition to giving the equilibrium form of the velocity distribution function for gases, Maxwell derived equations for the transport of mass, energy, and momentum for a dilute gas. For a fictitious gas of molecules that interact with central, two-body repulsive forces proportional to the inverse fifth power of the distance between the centers of a pair of particles, the so-called Maxwell molecules, Maxwell was able to derive explicit expressions for the transport coefficients appearing in the Navier–Stokes equations [465]. In particular, he was able to show that for dilute gases, the coefficients of shear viscosity and thermal conductivity would be independent of the gas density, in agreement with experimental results. The most important advance in the nineteenth century, and one that set the stage for almost all of the later developments in kinetic theory and its many applications, was made by Boltzmann in 1872 [57]. He used mechanical and statistical arguments to obtain an equation – the Boltzmann transport equation – that describes the irreversible time evolution of the single-particle velocity and position distribution function for a dilute gas not in equilibrium. The Boltzmann equation is extremely successful as a tool for calculating the transport properties of dilute gases and gas mixtures, so much so that the results obtained are often used to test different models of pair potentials by comparing theoretical values for given potentials with experimental results. The first systematic calculations of transport properties of dilute gases and their mixtures based on the Boltzmann equation were carried out independently by S. Chapman [88] and by D. Enskog [175, 176] in 1917.5 They used this equation as a starting point for a derivation of the Navier–Stokes equations of fluid dynamics and obtained explicit expressions for the transport coefficients appearing in these equations. On the basis of his theoretical work on gas mixtures, Enskog discovered the phenomenon of thermal diffusion, which – years later – provided a method for separation of isotopes of uranium for use in atomic bombs and nuclear reactors [558]. As we will discuss in greater detail in Chapter 2, the Boltzmann equation is not invariant under a time-reversal transformation where all velocities and the direction of time change sign. In fact, Boltzmann constructed a function out of the single-particle distribution function, called the H -function, which decreases monotonically in time unless the gas is at equilibrium. Of course, this means that the Boltzmann equation is not a consequence of only the applications of the basic

4

Introduction

equations of mechanics to behavior of dilute gases. Instead, there is embedded in Boltzmann’s derivation a stochastic assumption, the Stosszahlansatz, that breaks the time-reversal symmetry of the description of the dynamics of the gas. Although the Stosszahlansatz is not a purely mechanical statement, it reflects the expected and typical behavior of a dilute gas that is not in equilibrium. It can also be thought of as statement about the initial state of the gas that is assumed to be propagated forward in time.

1.2.2 Enskog’s Extension of the Boltzmann Equation to Dense, Hard Sphere Gases The first important extension of the Boltzmann equation to higher densities was made by D. Enskog [175, 176] in 1922. He considered only hard-sphere gases and modified Boltzmann’s arguments in a way that includes some but not all of the effects of higher densities. First of all, Enskog took into account excluded volume effects. At finite densities, the presence of other particles reducing the available free volume for a given pair enhances the probability for a collision between this pair. In addition, Enskog realized that the finite size of the particles, coupled with the hard-sphere potential, allows an instantaneous transfer of momentum and energy over a distance of the diameter of a sphere at each binary collision. The inclusion of excluded volume and collisional transfer effects allows the Enskog equation to apply to very dense hard-sphere fluids. The Enskog theory is only an approximation to a kinetic equation for dense gases, but it provides useful estimates for nonequilibrium properties of such gases. As we will see further on, Enskog’s methods need to be refined when applied to mixtures of hard spheres; otherwise, one obtains expressions for transport coefficients that are not consistent with the Onsager reciprocal relations [508, 509, 22]. This refinement, now called the revised Enskog equation, was obtained by H. van Beijeren and M. H. Ernst in 1973 [644, 645]. It leads to the same transport coefficients as the usual Enskog equation for pure hard-sphere gases and to transport coefficients that satisfy the Onsager relations for mixtures of hard spheres. The Enskog and revised Enskog equations can be applied to other types of monatomic particles besides hard spheres if one replaces the correct interaction potentials by “effective” hard-sphere interactions using effective hard-sphere radii obtained by finding the best fit of some thermodynamic property of the system to a hard-sphere model. In many cases, the results so obtained are in good agreement with experimental results. As we discuss later in this book, the revised Enskog theory is also capable of describing some properties of hardsphere solids, since the equations of an elastic solid can be obtained from it, with approximate expressions for the elastic coefficients, the heat conduction coefficient, and the sound attenuation constants.

1.2 The Kinetic Theory of Gases

5

1.2.3 Derivations of the Boltzmann Equation Using the Liouville Equation In order to derive the Boltzmann equation and to justify the Stosszahlansatz, one must start from some more fundamental equation for the distribution functions characterizing the gas and then obtain the Boltzmann equation by making some plausible [406] and physically motivated assumptions. Such a derivation is also important and even necessary for extending the Boltzmann equation in a systematic way to dense gases. The natural starting point is the Liouville equation [549], which describes the time development of the complete N-particle distribution function for a gas of N particles. This equation is reversible and follows directly from the mechanical equations of motion. While it was long recognized that a derivation of the Boltzmann equation should be based upon the Liouville equation, it was not until the work of N. N. Bogoliubov [55] in 1947, completed later by M. S. Green [270, 271] and by E. G. D. Cohen [98, 99, 100, 101, 102], that satisfactory derivations of the Boltzmann equation were given and progress was made on the rather intricate problem of generalizing it to higher densities [274, 153, 154].6

1.2.4 Green–Kubo Formulae Concurrently with the developments in the work to generalize the Boltzmann equation to higher densities, it was realized that the same methods are extremely useful for the evaluation of the Green–Kubo expressions for transport coefficients, for systems of moderately dense gases. These expressions, derived by M. S. Green [268, 269] and by R. Kubo [405, 408] in the 1950s, express the hydrodynamic transport coefficients as time integrals of equilibrium time correlations of microscopic currents. The Green–Kubo formulae are quite general [709, 608, 705], not restricted to dilute or moderately dense gases, and ideally suited for computersimulated molecular dynamics [312, 548]. Methods have been developed for the evaluation of the time correlation functions, and results so obtained have been very important for the development and testing of detailed theories of transport phenomena in fluids. The time correlation function method and the Boltzmann equation lead to identical results when the Green–Kubo formulae are applied to dilute gases.7

1.2.5 Divergences in the Virial Expansions of Transport Coefficients The efforts to generalize the Boltzmann equation to higher densities in a systematic way produced a number of discoveries that led to deeper understanding of nonequilibrium processes in gases, and in fluids in general. These results, in essence, opened a new era in the development of kinetic theory.8 In contrast to

6

Introduction

the essentially instantaneous binary collisions that are taken into account in the Boltzmann equation, the higher-density corrections are determined by dynamical events that take place among groups of more than two particles [274, 273, 153, 154]. These can involve much longer time scales, typically on the order of a few mean free times. As a result of these dynamical events taking place in a gas – or, more generally, a fluid – memory effects make their appearance in nonequilibrium processes, producing striking results that have no counterparts in equilibrium statistical mechanics. For example, the thermodynamic properties of dense gases over a wide range of gas densities are well represented by virial expansions [549], which are series expansions in powers of the gas density. However, no such virialtype expansions – that is, expansions in powers of the density – exist for transport coefficients. If one assumes that such a density expansion is possible, then one finds that only the first few powers have finite coefficients, while all further coefficients are divergent [677, 580, 581, 154, 256]. The coefficient of each power in the density is determined by the dynamics of a fixed number of particles in an infinite space. The first term in such a virial-like expansion is determined by the dynamics of two particles, the next power is determined by the dynamics of three particles, and the number of particles in the group increases successively for the coefficients of successively higher powers of the density. For two-dimensional systems, only the first term, the Boltzmann contribution from binary collisions, is finite. The coefficients of all higher powers of the density diverge. For three-dimensional systems, the first two terms, namely the two-particle and three-particle terms, are finite. The coefficients of all higher powers diverge.9 The divergences are due to sequences of correlated collisions that allow for correlations among the particles to extend over large distances and long times.10 These correlations do not exist when the gas is in equilibrium, but for systems not in equilibrium, they are responsible for the memory effects mentioned earlier. That long-range correlations are present in nonequilibrium fluids has been well confirmed by light scattering experiments [323, 432, 418]. 1.2.6 Mode-Coupling Theory and Long-Time Tails The memory effects lead directly to what are often called mode-coupling effects [218, 333, 346, 59, 251]. That is to say, the dominant parts of the memory effects at long times, as will be made more precise later, give rise to a special structure in the kinetic and hydrodynamic equations, in which combinations of what will be called microscopic hydrodynamic modes appear.11 Much of this book will be devoted to the properties of these modes and their effects on nonequilibrium processes. The microscopic hydrodynamical modes are best thought of as the slowestdecaying normal modes of the operator that describes the time dependence of a

1.2 The Kinetic Theory of Gases

7

spatially nonuniform, nonequilibrium distribution function. Typically these modes describe the time decays of microscopic fluctuations in an equilibrium fluid, or a fluid in a nonequilibrium stationary state, via long-wavelength microscopic collective excitations of the fluid. These collective modes can be classified as shear, sound, thermal, and diffusive modes, and combinations of two or more of them arise naturally in the equations of kinetic theory beyond the Boltzmann, binary collision approximation. Mode-coupling effects first made their appearance in the context of a theory for the anomalous behavior of transport coefficients near the critical point of gas– liquid phase transitions developed by M. Fixman [218] and by L. P. Kadanoff and J. Swift [333]. In a different direction, people working in kinetic theory – including R. Goldman and E. A. Frieman [256], Y. Pomeau [533, 534, 556, 536], and J. R. Dorfman and E. G. D. Cohen [157, 156, 155] – obtained mode-coupling contributions to the time correlation functions as a consequence of the resummation techniques needed for the renormalization of the divergences in the density expansions of transport coefficients carried out by K. Kawasaki and I. Oppenheim [350]. Kinetic theory is only one approach to obtain expressions of the mode-coupling form. Mode-coupling theories can also be obtained using more general arguments based on the assumption that the large-distance or small-wave-number parts of the microscopic densities of particles, momentum, and energy obey macroscopic hydrodynamic equations. The main observation from this work is that long-range, long-time, dynamically produced correlations exist in a nonequilibrium fluid and are manifested in the existence of algebraic decays in time correlation functions for long times and can be observed in light scattering by a fluid with a stationary temperature gradient, for example. Dorfman and Cohen [157, 156, 155] showed that these mode-coupling contributions to the time correlation functions, as obtained from kinetic theory, could explain some remarkable results obtained by B. J. Alder and T. E. Wainwright using computer-simulated molecular dynamics. Alder and Wainwright showed that the velocity autocorrelation function that determines the coefficient of self-diffusion via the Green–Kubo relations decays algebraically with time t as (t/t0 )−d/2 , where d is the number of spatial dimensions of the system and t0 is some characteristic microscopic time such as the mean free time between collisions for a particle in the gas [9, 10, 673]. These algebraic decays, generally called long-time tails, have many important consequences, both experimental and theoretical, which will be discussed in this book.12 Among other things, the slow inverse time decay of the time correlation functions for two-dimensional systems implies that the transport coefficients appearing in the linearized Navier–Stokes equations for twodimensional systems are divergent. Although for three-dimensional systems the structure of Navier–Stokes equations is unaffected by the long-time tails, since

8

Introduction

the Green–Kubo expressions are convergent, a very similar divergence difficulty appears in the higher-order hydrodynamic equations, such as the Burnett and superBurnett equations, and leads to a complex description of higher-order hydrodynamics. As a consequence, we learn that for both two- and three-dimensional systems, the equations of fluid dynamics are intrinsically nonanalytic in the gradients of the hydrodynamic fields and may contain effects of the boundaries in the equations themselves. Experimental measurements have confirmed the presence of nonanalytic terms in hydrodynamic equations. For example, the presence of fractional powers of the wave number and related mode-coupling effects have been observed in neutron scattering experiments on fluids [130, 129, 491]. In other cases, such as heat flow in a nonequilibrium stationary state with a temperature gradient, mode-coupling effects closely related to the long-time tails produce longrange spatial correlations that have a very strong influence on the properties of fluctuations of the hydrodynamic fields in nonequilibrium steady states. These effects can be orders of magnitude larger than the effects of static correlations in the fluid. The theory for these fluctuations has been confirmed by very careful light scattering experiments, as discussed in the next subsection [371, 374, 323]. Finally, it should be mentioned that for some complex fluids and for certain magnetic systems, the effects of the long-time tails are even more dramatic. In both smectic and cholesteric liquid crystals [470, 120] and in the hydrodynamic description of helimagnets [369], there are long-range static correlations due to a continuous broken symmetry (Goldstone’s theorem) that effectively multiply and amplify the long-time tails. In particular, for these systems, the lower critical dimension of two for ordinary fluids is replaced by four. This, in turn, leads to transport coefficients that diverge in three dimensions for low frequencies as 1/ω. That is, for these systems, even in bulk systems, the leading order hydrodynamic description breaks down.

1.2.7 Mode-Coupling Theory and Hydrodynamic Fluctuations Mode-coupling theories can be applied to many physical situations. We will have occasion in this book to discuss their application to light scattering by fluids in nonequilibrium stationary states, which led to predictions of, among other effects, a striking modification of the strength of the central Rayleigh peak [364, 371, 372, 373, 374, 160, 562, 627], since verified experimentally by J. V. Sengers and co-workers [418, 323].13 It is also worth mentioning that the application of kinetic theory of moderately dense gases leading to the mode-coupling equations is closely related to the derivation of the Balescu–Lenard–Guernsey (BLG) equation used in the theory of moderately dense plasmas [578].

1.2 The Kinetic Theory of Gases

9

An important example of the application of mode-coupling theory is to the theory for the behavior of glassy materials and to the theory of the liquid–glass transition [251]. Here the central idea is to construct a self-consistent formulation of modecoupling theory whereby the hydrodynamic properties of the system under study are determined by integrals that depend upon the same hydrodynamic properties, so that solutions can only be obtained in some self-consistent way. A remarkable feature of such solutions is that, while based upon ad hoc approximations, these theories provide a useful description of the transition of a liquid to an amorphous solid, or glass, and of aspects of the behavior of such a glass [251]. The fluid–glass transition seems to be a dynamical phase transition in the sense that the behavior of the system is sensitive to the time scales associated with external forces applied to the system. In such a case, the system may exhibit some form of hysteresis. Thus, the memory effects incorporated in mode-coupling theory may play an important role in establishing the correlations responsible for the phase transition. It is interesting to note that an exact solution of the dynamics of a system of hard spheres in infinite dimensions has been shown to have a mode-coupling-like glass transition [90, 451]. Further, the complicated features of the many possible metastable states in this system have also been determined. These results are consistent with the RFOT (random first-order transition) theory of the glass transition that was introduced some time ago [389, 519]. It is important to mention that mode-coupling theory can also be obtained in a variety of ways that do not require kinetic theory. The initial work of L. P. Kadanoff and J. Swift [333] was based upon an insightful and clever ansatz for the hydrodynamic modes of an N-particle Liouville operator.14 These authors took advantage of the fact that the Liouville operator is a linear differential operator. As a result, the product of two or more eigenfunctions is also an eigenfunction. This provides the mechanism for the coupling of two or more hydrodynamic modes. Perhaps the method most widely used to develop mode-coupling theory is the method of fluctuating hydrodynamics [412, 576, 323]. One assumes that the microscopic densities of particles, momentum, and energy in a fluid obey nonlinear Navier–Stokes-type equations in which white noise fluctuations are added to the dissipative fluxes. The main results of mode-coupling theory are obtained as additional terms in the hydrodynamic equations by averaging over the fluctuations. This approach to mode-coupling theory is algebraically simpler than kinetic theory and produces the same leading terms as one obtains by kinetic theory. However, it is not a systematic approach to nonequilibrium processes in fluids, and kinetic theory provides additional terms, however small, that are not obtained by the use of fluctuating hydrodynamics.

10

Introduction

1.2.8 Model Systems As with most, if not all, issues in statistical physics, a great deal can be learned by considering simplified models. Such models are constructed so as to exhibit some central features of more complex and realistic systems and yet are simple enough to allow a more detailed treatment than might be possible for more complicated systems. Kinetic theory utilizes many such model systems. One prominent model already mentioned is the hard-sphere gas. This model can be studied in any number of dimensions and has the advantage that the collisions between particles are all instantaneous, binary collisions [158, 190]. Mixtures of hard spheres are also convenient models for study. The first important extension of the Boltzmann equation to higher densities, the Enskog theory, was specifically formulated for hard-sphere particles for which both excluded volume effects and collisional transport of momentum and energy can be accounted for in a straightforward way [175, 176, 89, 644, 645]. While the Enskog theory does not provide a systematic extension of the Boltzmann equation to higher densities, and it does not account for many dynamical processes taking place in the gas, it nevertheless gives a good representation of experimental data for dense fluids, provided one uses an appropriate value of an effective hard-sphere radius for the particles of the real gas under study.15 Another version of the Boltzmann equation is the lattice Boltzmann equation [37, 684, 615]. This is a discretized version of the Boltzmann equation appropriate for a system consisting of a collection of particles moving on a lattice. The particles may jump from one site on the lattice to another at discrete times, and particles approaching the same lattice point from different directions collide with each other according to a set of collision rules, which may be either deterministic or probabilistic. All possible particle velocities are directed along the various bonds of the lattice and remain so after each collision. Such a gas is called a cellular automata lattice gas (CALG) [235, 332, 58, 684, 565]. A CALG is easily simulated on a computer since the dynamics can be reduced to the simple integer operations of a Boolean algebra. It was thought at one time that CALGs would provide an efficient way to simulate complicated hydrodynamic flows, and they attracted a great deal of attention for this reason. However, it became clear that some of the early hopes were too optimistic. Nevertheless, CALGs are indeed useful for studying such flows on a computer, and the kinetic theory for them has been studied in some detail. A mixture of particular interest for the development of kinetic theory is the Lorentz gas. This is a binary mixture of two gases, one very heavy and one very light. This model was used by H. A. Lorentz [440, 89] as a way to understand the motion of electrons in metals and to develop a theoretical expression for the resistance of a wire. One limiting case of the model that has been studied in great depth, both theoretically and by means of computer simulations, is the case where the heavy particles are fixed in space (that is, infinitely heavy), and the density

1.2 The Kinetic Theory of Gases

11

of the light particles is so low that they can be considered as independent and not interacting with each other16 [69, 70, 299, 48, 617]. One particular version of Lorentz-type models was invented by P. and T. Ehrenfest to clarify and illustrate the assumptions made by Boltzmann in his derivation of the Boltzmann transport equation. In the Ehrenfest model, also known as the wind-tree model, the fixed particles (or “trees”) are squares in a plane with their diagonals oriented along the x and y coordinate axes in the plane [173]. The moving particles (or “wind particles”) are restricted to have velocities only in the four directions along the coordinate axes (±x and ±y). At each collision of a wind particle with a tree, the particle’s speed remains constant, but its direction is rotated by ±π/2 radians, depending upon the face of the tree with which it collides. It was discovered by E. H. Hauge and E. G. D. Cohen that, if the trees are allowed to overlap each other in the plane, the model becomes sub-diffusive even at very low densities, and the Boltzmann equation does not apply to it [300, 301]. For all the versions of the Lorentz gas that have been studied, including the wind-tree model, there are interesting long-time tail phenomena taking place that differ in some crucial ways from their counterparts in a pure gas of interacting particles [70, 69]. The quantum Lorentz gas [416, 557, 381, 380, 29] exhibits the phenomenon of localization, which means that, under certain circumstances, particles do not diffuse through the system, even when the classical limit of the motion is diffusive. Localization in quantum Lorentz gases is closely related to the long-time tail effects in classical gases [422, 29]. Quantum Lorentz gases have an important application in condensed matter physics, where they are used to understand the behavior of electron transport in disordered systems, for which there is an abundance of experimental data on transport properties. Another example of a binary mixture is the Rayleigh model. Here there is one heavy particle moving in a gas of much lighter particles. The light particles can be taken as either not interacting with each other or with mutual interactions, but, in any case, the light particles do interact with the heavy particle [614]. The Rayleigh model with interactions between the gas particles is a useful model for Brownian motion. In all of these models, a further simplification is possible by making all interactions hard-sphere-like interactions – that is, only instantaneous, binary collisions take place. A version of the Rayleigh model that has attracted considerable attention is the piston model, in which the heavy particle is a piston in a tube with gases on either side of it in the tube [617, 433, 93].

1.2.9 Dilute Quantum Gases A model system of considerable interest is a weakly interacting quantum Bose– Einstein gas. Since such a system undergoes an equilibrium phase transition to

12

Introduction

a superfluid phase, the description of the nonequilibrium behavior of the gas near the Bose–Einstein phase transition is an interesting challenge for kinetic theory [385, 382, 384, 529, 530, 431, 524, 278, 537, 626]. The kinetic theory of these weakly interacting quantum gases became particularly important over recent decades because of the experimental realization of such gases and the experimental verification of their Bose–Einstein phase transition. Here we will describe some of the main features and show that below the Bose–Einstein phase transition, the hydrodynamic behavior of such a gas is described by the Landau–Khalatnikov two-fluid equations [438, 545], the two fluids being the normal and superfluid components of the gas. 1.3 Further Applications of Kinetic Theory By now, it is well known that the methods of kinetic theory, including modecoupling theory, can be applied to a wide range of topics in fluid, plasma, and colloid physics. Among the topics to which kinetic theory has been applied are the closely related phenomena of Stokes flow around large solid bodies such as spheres and cylinders [637, 638, 574, 80, 575, 478, 586], the motion of Brownian [52, 50] particles in fluids, the motion of ions in solution [340, 341], and the motion of colloidal particles in fluid suspensions [104, 108, 316]. All of these systems have a fundamentally hydrodynamic structure whose source is the interactions of macroscopically large particles with the much smaller particles that make up the surrounding fluid or with each other. The flow of a gas around a large macroscopic object such as a sphere or a cylinder, for example, can be described as a function of a dimensionless parameter – in this case, the Knudsen number, which is the ratio of the mean free path of the surrounding gas molecules to some characteristic size of the object. Kinetic theory has been used with great success to describe the dynamics, for large Knudsen numbers, of rarefied gases [264, 598, 281, 84] and has been extended, as mentioned before, to describe hydrodynamic flow, at small Knudsen numbers, where one recovers the classic results for the flow based upon the applications of the Navier–Stokes equations. Kinetic theory for gases in the small Knudsen number regime has been used to describe the diffusion of a large Brownian particle in a dilute gas. The treatment of Brownian motion often begins with the Langevin equation, where the equation of motion for the particle includes the effects of hydrodynamic friction due to the surrounding fluid. In the Langevin theory, a fluctuating term is added to the equation of motion, which is supposed to account for the fact that the collisions of the Brownian particle with the gas molecules are random, and only the average motion can be described by a frictional force. L. Landau and E. M. Lifshitz had the idea of adding fluctuating terms to

1.3 Further Applications of Kinetic Theory

13

the linearized hydrodynamic equations [412], an idea that was later generalized by R. Zwanzig and M. Bixon [42] as well as by R. Fox and G. E. Uhlenbeck [230, 231], who explored the consequences of adding noise terms to the linearized Boltzmann equation. The addition of such noise terms makes it possible to construct a theory for the macroscopic fluctuations of density in a fluid, the results of which can be used to make predictions about the scattering of light by a fluid under various circumstances [38]. This is of special interest because light scattering experiments are important methods for the study of fluid properties, and provide experimental ways of testing and checking some of the predictions of kinetic theory. The motions of ions and colloidal particles are often affected by electrodynamic interactions produced by the charges on the particles. Treatment of such interactions leads naturally to a treatment of plasmas where nonequilibrium behavior is almost entirely determined by the Coulomb interactions of the charged particles. The basic kinetic equations of nonequilibrium plasmas are the Vlasov equation and the Balescu–Lenard–Guernsey (BLG) equations [656, 578]. The Vlasov equation, unlike the Boltzmann equation, is time reversible. Nevertheless, it exhibits solutions that decay in time, associated with the phenomenon of Landau damping. This apparently paradoxical situation – a time-reversible equation with time-irreversible solutions – is resolved by a careful consideration of the function spaces in which these solutions are embedded. In addition to its importance for plasma physics, the BLG equations are also interesting because they are the analog in plasma physics of the generalized Boltzmann equation with mode-coupling contributions, valid for neutral particles interacting with short-ranged potentials. Due to space limitations, we will not cover colloids and plasmas in this book, but we refer the interested reader to some basic literature [18, 19, 455]. 1.3.1 Granular Materials Some time ago, it was realized that the Boltzmann equation could be applied to obtain a theory for the behavior of granular gases – that is, gases whose particles make inelastic collisions with each other – and the system loses energy to the environment. It is not difficult to obtain the Boltzmann equation for particles that suffer inelastic collisions when one can specify the dynamics of an inelastic binary collision. In general, this is not a simple requirement, since binary collisions – both elastic and inelastic – can have a very complex dynamics, but considerable progress can be made by using tractable and highly simplified models for inelastic collisions [66, 25, 401]. One interesting application of the kinetic theory for granular, inelastic media is to the theory for planetary rings [262, 208].

14

Introduction

1.3.2 Chaotic Dynamics Kinetic theory has recently been applied in a new context – dynamical systems theory. Dynamical systems theory attempts to understand the very complicated chaotic motions typical of classical, nonlinear mechanical systems. Such systems, although mechanically reversible and deterministic, have the property that the phase-space trajectories of two systems that start from infinitesimally nearby points in phase space, separate exponentially with time. One consequence of this exponential separation is that a small uncertainty in the specification of the initial state of such a system will lead to an exponential growth of the uncertainty in the location of the phase point with time. The rate of the growth is characterized by a set of exponents called Lyapunov exponents. Usually, a system is called chaotic if it has at least one positive Lyapunov exponent [569, 513, 243, 152]. A group of mathematicians including Ya. G. Sinai, L. A. Bunimovich, N. Chernov [592, 593], D. Szasz [616], and N. Simanyi [591, 590] have given proofs of the ergodicity of systems of hard-sphere Lorentz gases with d-dimensional spheres as fixed scatterers with d ≥ 2, and, under certain general conditions, of hard-sphere gases as well [487]. While ergodicity is generally not considered to be essential for the applications of statistical mechanics to such systems, these proofs do suggest that these hard-ball systems have interesting dynamical properties and are, in fact, chaotic. Ya. G. Sinai was the first to establish this line of approach to ergodicity of hard-sphere systems [593, 592]. As we mention later, kinetic theory can be used to calculate Lyapunov exponents and Kolmogorov–Sinai entropies of Lorentz and hard-sphere gases. These studies, while not rigorous, provide evidence that such systems are chaotic and their chaotic properties are accessible to calculations. Computer simulations by P. Gaspard, F. Baras [244, 245], H. Posch, C. Dellago, C. Forster [142], and others have verified, as much as is possible by computer studies, the chaotic nature of these systems and given values for Lyapunov exponents, Kolmogorov–Sinai entropies, topological pressures, and related quantities that characterize their dynamical behavior. On the other hand, V. Rom-Kedar, D. Turaev [629, 560], and also V. Donnay[148] have shown that if the hard-sphere potential is softened, elliptic islands with positive measure appear in the phase space of these systems. These islands spoil the ergodicity, and their appearance means that not all starting points will exhibit chaotic behavior. For non-integrable systems at high enough energy, it is expected that the fraction of the energy shell in phase space taken up by the elliptic islands is an exponentially decaying function of particle number (at fixed particle and energy density), but to our knowledge, no proofs of this are known. It is not known what conditions a non-hard-sphere potential must satisfy in order that the motion of a system of particles with interactions specified by this potential will be chaotic for

1.4 Outline of This Book

15

almost all initial points on an energy shell. It is known that not all such potentials lead to elliptic islands in phase space. For instance, A. Knauf [395] has shown that a Lorentz gas where the scatterers are placed on a square lattice and interact with the moving particles via repulsive Yukawa potentials is also a chaotic system, provided the moving particles have sufficiently high energies. Here we will show how kinetic theory can be used to calculate the dynamical properties of dilute, random hard-sphere Lorentz gases systems. Similar kinetic theory methods, such as the Boltzmann equation and others based on collision dynamics, can be used to extend the work presented here to calculate dynamical properties of dilute hard-sphere gases [135, 132, 134, 133, 663]. We shall be able to obtain expressions for Lyapunov exponents for Lorentz gases with an equilibrium distribution of moving particles as well as for Lorentz gases in a thermostatted electric field [639, 640, 165, 642, 417, 641]. In those cases where comparisons with computer simulations are possible, the agreement between kinetic theory and computer simulation results is excellent. Such results indicate that the systems we study are indeed chaotic but do not constitute mathematical proofs.

1.4 Outline of This Book Chapters 2, 3, and 4 are devoted to the Boltzmann equation for dilute gases and dilute gas mixtures. We begin in Chapter 2 with Boltzmann’s heuristic derivation of this equation. One feature of our presentation is that we incorporate boundary effects directly into the Boltzmann equation by modeling the boundary as a kind of “super-particle” with which the gas particles collide, in addition to their collisions with other particles. Then we prove that the Boltzmann entropy, the negative of the Boltzmann H -function, is monotonically non-decreasing with time, provided there is no heat flowing out of the system. Heat flow will arise naturally in this proof as a result of the way we incorporate the boundaries in the Boltzmann equation. To make this result somewhat less mysterious, especially for those readers encountering the H -theorem for the first time, we will explain the various assumptions that are made in the course of this derivation and discuss the time-symmetrybreaking assumption responsible for the irreversibility of this equation. Simple model systems have been used to clarify and illustrate the ideas, among them the Ehrenfest wind-tree model [173] and the Kac ring model [329]. Here we will discuss the Kac ring model in some detail. In Chapter 3, we present the Chapman– Enskog method for constructing solutions to the Boltzmann equation that describe hydrodynamic flows. We derive the Navier–Stokes equations and obtain expressions for transport coefficients, in particular the coefficients of shear viscosity and thermal conductivity, appropriate for monatomic gases at low densities. At this

16

Introduction

point, we summarize the experimental support for the Boltzmann equation and describe various applications of it for hydrodynamic processes. We also discuss the boundary conditions appropriate for various flows and explain how they may be obtained from the Boltzmann equation coupled with models of gas– surface interactions. In Chapter 4, the Chapman–Enskog method is applied to the Boltzmann equation for dilute gas mixtures, and the Onsager reciprocal relations are discussed. Next, in Chapter 5, we consider a very particular and very useful mixture, the dilute Lorentz gas. As described earlier, this consists of non-interacting particles that move among and collide with a collection of fixed scatterers placed randomly in space, with their average separation distance large compared to their size. This model is often used to study the motion of classical electrons in an amorphous solid. In Chapter 6, we discuss the Liouville equation for both general repulsive potentials and for hard-sphere systems. There we present the BBGKY hierarchy equations, which – in later chapters – will provide the statistical mechanical foundations for the extension of the Boltzmann equation to higher densities. In Chapter 6, we will also give a careful analysis of the dynamics of hard-sphere particles, which will be used as a model system for our later work on dense gases. An earlier and widely used approximate theory for transport in dense hard-sphere fluids, due to Enskog, with the necessary modifications due to H. van Beijeren and Ernst [644, 645], is presented in Chapter 7. Chapter 8 is concerned with the effects of adding noise terms to the linearized Boltzmann equation, a la the Langevin equation, as a way of correcting for the averaging used when deriving the Boltzmann equation. This allows for a treatment of density and other fluctuations in a gas and provides the basis for a description of the scattering of light by the gas. Two chapters are devoted to the nonequilibrium properties of systems that have been the focus of much activity, namely granular gases and quantum gases including weakly interacting, condensed Bose–Einstein gases. Chapter 9 is devoted to classical dilute gases of particles that suffer inelastic collisions and lose energy to the environment. The latter systems, a part of the general field of granular matter, also exhibit interesting behavior, and the general topic has important applications – among others, to systems of interest for astrophysics, such as planetary rings. Dilute quantum gases, Bose–Einstein gases, below the superfluid phase transition are discussed in Chapter 10. The next three chapters are devoted to the kinetic theory of moderately dense gases, in particular the microscopic foundations, and applications of mode-coupling theory. In Chapter 11, we describe nonequilibrium cluster expansions that provide the foundations for extending the Boltzmann equation in a systematic way to higher densities. These methods allow for a virial expansion of the generalized Boltzmann equation where the successive powers of the density take into

Notes

17

account dynamical events involving groups of two, three, and so on, particles. As discussed in Chapter 12, virial expansion methods fail for nonequilibrium gases because all but the first few terms in the virial expansion contain secular divergences in time. Resummations designed to cure these divergence problems due to K. Kawasaki and I. Oppenheim [350] are presented in this chapter. When applied to the density dependence of the various transport coefficients, the resummation leads to the appearance of logarithms in the density expansions of transport coefficients. In Chapter 13, we show that the resummations provide a microscopic basis for mode-coupling theory, which, in turn, leads to a kinetic theory description of long-time tails in the Green–Kubo time correlation functions. The presence of the long-time tails shows that there are serious problems in providing a successful foundation for the equations of hydrodynamics, especially, but not exclusively, for two dimensional gases. The following chapter, Chapter 14, is the final technical chapter. It is devoted to the efforts to reformulate the equations of hydrodynamics for both two- and threedimensional systems in the light of the difficulties due to the existence of longrange spatial and long-time temporal correlations in the gas produced by sequences of correlated collisions and that are ignored in both the Boltzmann and Enskog equations. This chapter concludes with a discussion of one of the central results in the theory of nonequilibrium processes in fluids, the theory leading to the major enhancement of the central, Rayleigh peak, in the small-angle scattering of light by a fluid maintained in a nonequilibrium stationary state. We conclude the book in Chapter 15 with brief discussions of some of the many applications, some very recent, of kinetic theory that are not covered in the book and with final thoughts on some related developments in cosmology. The mathematical methods used here are not rigorous. Rigorous results have been obtained for certain areas covered in this book. In particular, there is a large mathematical literature on the existence of solutions of the Boltzmann equation under various conditions, as well as for the ergodic properties of various types of hard-sphere gases. When appropriate, we provide references to the literature so that the interested reader can explore the mathematical discussions at greater depth.

Notes 1 Standard references for the kinetic theory of gases, apart from textbooks, include S. Chapman and T. G. Cowling [89]; H. Grad [265]; E G. D. Cohen [103]; H. Spohn [606]; J. H. Ferziger and H. G. Kaper [215]; C. Cercignani [79, 81, 83, 85]; P. Résibois and M. de Leener [555]; P. Schram [578]; and P. L. Krapivsky, S. Redner, and A. Ben-Naim [401]; among many others. 2 This book is a greatly expanded version of a review article that two of us published in 1976, several years before a number of topics discussed here had been developed [164]. In view of the many advances and applications of kinetic theory, it seemed to us that a more lengthy review of this subject would be warranted.

18

Introduction

3 This discussion is based upon historical studies by Steven Brush, as described, for example, in his book The Kind of Motion We Call Heat [72]. 4 The books by S. Brush [71, 72], to which we refer, contain a bibliography of important papers for the history of kinetic theory as well as a thorough description of this history. We can do no better than to refer to these books rather than to the individual papers. 5 The standard reference for the kinetic theory of dilute gases and gas mixtures is the book by S. Chapman and T. G. Cowling [89]. While their notation is cumbersome, the discussion is complete and clear. 6 There were a number of other efforts to obtain the Boltzmann equation using the Liouville equation as a starting point, in addition to those of N. N. Bogoliubov, M. S. Green, and E. G. D. Cohen. The paper by Cohen [98] provides a comparison of different methods, and the paper by E. G. D. Cohen and T. Berlin [107] exposes an important facet of the Bogoliubov method and gives an essential clue as to the origin of irreversibility in this and other approaches to the derivation of the Boltzmann equation. 7 A proof of the general equivalence of kinetic theory and time correlation function formalisms for evaluating transport coefficients is complicated by the fact that nonanalytic and even divergent terms appear when one looks at either of these methods for gases at moderate or high density. Instead, one can verify the equivalence by explicit calculations as will be discussed in later chapters. For low-density gases, there are several discussions of the equivalence of the two methods. Here we mention only one of them [189]. 8 A brief characterization of the various periods in the development of kinetic theory can be found as a footnote in the Introduction to J. Stat. Phys. 109, Numbers 3/4, by M. H. Ernst, H. van Beijeren, and E. G. D. Cohen [194]. See also the paper by Ernst [180]. 9 These remarks apply to gases in which the particles interact with short-range forces, which are repulsive for small inter-particle distances. 10 Several authors almost simultaneously realized that the virial expansions of transport coefficients were plagued by divergence difficulties. The history of this discovery is summarized in volume 3 of Brush’s three-volume set of books on kinetic theory [71]. 11 If one thinks, loosely speaking, of the hydrodynamic modes as eigenfunctions of the Liouville operator, which is a linear differential operator, it follows that products of these modes are also eigenfunctions. 12 Kinetic theory is one of several approaches for studying the long-time tails. Identical results have been obtained at the same time using mode-coupling or related hydrodynamic arguments by M. H. Ernst, E. H. Hauge, and J. M. J. van Leeuwen [191, 192, 193, 195, 196] and by R. Zwanzig and M. Bixon [706]. 13 Closely related work on long-range correlations in fluids maintained in nonequilibrium stationary states was carried out by a number of groups. See, for example, [544, 543, 564, 628, 627]. 14 To fully appreciate this method, one has to realize that besides proper eigenfunctions under L2 norm with purely imaginary eigenvalues, there exist generalized eigenfunctions with real or complex eigenvalues. These are known as Pollicott–Ruelle resonances [567, 532]. 15 Values for the effective hard-sphere radius of particles in a real gas are generally obtained by fitting results from equilibrium measurements, such as virial coefficients, to hard-sphere models for those properties. 16 This model is also usually referred to as the Lorentz model. We will indicate whenever necessary which version is under discussion.

2 The Boltzmann Equation 1: Fundamentals

2.1 The Boltzmann Equation 2.1.1 Dilute, One-Component, Monatomic Gases We begin our discussion of the kinetic theory of fluids with a somewhat modified version of Boltzmann’s derivation of the transport equation bearing his name [57, 631]. As mentioned in Chapter 1, the Boltzmann transport equation provided the fundamental equation for the time evolution of the position and velocity distribution of particles in a dilute gas, allowing for treatments of a wide variety of transport phenomena in dilute gases from a single point of view. Here we derive this equation in terms of (a) free motion, or motion under the influence of external forces; (b) collisions of the gas particles with the walls of the vessel containing them; and (c) binary collisions between the particles themselves. With respect to the collisions of the particles with the wall, it is necessary to specify the details of the interactions between the particles and the wall. These may be of at least three types: (a) interactions via conservative forces with an interaction potential that is fixed in the inertial frame of the containing vessel; (b) interactions, e.g. with particles colliding with oscillating walls, where the velocity of the gas particle after collision is determined according to some specified rule whose possible outcomes depend upon its velocity before collision; or (c) a stochastic-type interaction where the final velocity is sampled from a distribution of final velocities and does not depend upon the velocity before collision. The collisions of the gas particles with each other will always be taken to be governed by a conservative, central, two-body interaction potential, φ(rij ), which generally is repulsive at very short distances and may be attractive for somewhat larger distances. Here rij = |r i −r j | denotes the magnitude of the distance between the center of particle i at r i and that of particle j at r j . For the current discussion, we will suppose that the potential vanishes for distances larger than some value, denoted by a. An example of such a potential, a Lennard–Jones potential with a 19

20

The Boltzmann Equation 1: Fundamentals 3

2 f WCA

1

f (r) e

0

f LJ

–1 1

1.2

1.4 1.6 r/ s

1.8

2

Figure 2.1.1 Typical pair potentials. Illustrated here are the Lennard–Jones pair potential, φLJ , and the associated Weeks–Chandler–Anderson potential, φWCA , which gives the same repulsive force as the Lennard–Jones potential. The relative separation coordinate is scaled by the distance σ , the point at which φLJ first passes through zero, and the energy axis is scaled by the well depth, ε. This figure is courtesy of J. D. Weeks

“cutoff” at a [306], is illustrated in Fig. 2.1.1. In this figure, we also illustrate a well-known, totally repulsive potential often used in computer simulations. This is the Weeks–Chandler–Anderson (WCA) potential [675], which is constructed to have the same repulsive force as the Lennard–Jones potential. Another important potential for our discussions is the hard-sphere potential.1 This potential is infinite for inter-particle distances less than some value, a, and zero otherwise. For a hard-sphere gas of identical particles, we may imagine that the particles are spheres of radius a/2. The interaction potential of the gas particles with a velocityindependent external field is denoted by φext (r i ), and for the moment, we consider that the gas particles interact with the wall via a wall–particle potential energy, W (r i ). In this case, the complete Hamiltonian, HN , for a system of N particles interacting with each other via central forces, and interacting with an external field and with the walls through conservative forces is given by 2 HN =

 mv2 i

i

2

+

 i a, no collision can take place since the center of particle 2 never enters the action sphere. Suppose now that the impact parameter b is such that b < a. By looking at Fig. 2.1.2, we can see that if the center of particle 2 is

24

The Boltzmann Equation 1: Fundamentals

inside the collision cylinder with volume πa 2 |v2 − v1 |δt, a (v1,v2 ) collision will take place within time interval δt, if no other particles intervene earlier, which we assume to be the case. We also ignore any collisions that may be already in progress at time t. Such events lead to higher-order density contributions, and vanish if the particles are hard spheres with instantaneous binary collisions. The Stosszahlansatz We now have all the geometry that we need to begin assembling the components needed to obtain Boltzmann’s expression for − . The argument proceeds as follows: • The number of particles with velocity v1 in the volume δr 1 δv 1 at time t is f (r 1,v1,t)δr 1 δv 1 . • Each collision cylinder attached to a particle with velocity v1 for a collision with a particle with velocity v2 has volume πa 2 |v2 − v1 |δt, so that the total volume of these collision cylinders in δr 1 δv 1 is f (r 1,v1,t)δr 1 δv 1 × πa 2 |v2 − v1 |δt. • In principle, we now would have to examine each of these collision cylinders to see if they are occupied by one or more particles with velocity v2 . If so, we then would need to see if the v1 particle would collide with one of the v2 particles in this collision cylinder in time δt rather than with a particle that is contained in another cylinder also “attached” to the v1 particle. Since we are unable to do this without following the dynamical history of each and every particle in the system,6 we will need to use some probabilistic reasoning to calculate the number of (v1,v2 ) collisions. • The stochastic argument is based upon Boltzmann’s Stosszahlansatz [173]: The number of (v1,v2 ) collisions in the volume δr 1 δv 1 δv 2 in time δt is given by the product of the total volume of the collision cylinders under consideration and the number of particles per unit volume with velocity v2 in the range δv 2 , namely f (r 1,v2,t)δv 2 . This product is f (r 1,v1,t)f (r 1,v2,t)δr 1 δv 1 δv 2 πa 2 |v2 − v1 |δt.

(2.1.4)

Now we have assembled all of the pieces needed for the calculation of − . It is obtained by integrating the number of (v1,v2 ) collisions just obtained over all possible velocities v2 . That is,  (2.1.5) − δr 1 δv 1 δt = πa 2 dv2 |v1 − v2 | f (r 1,v1,t)f (r 1,v2,t)δr 1 δv 1 δt.

2.1 The Boltzmann Equation

The expression for − is then given by  2 dv2 |v1 − v2 | f (r 1,v1,t)f (r 1,v2,t). − = πa

25

(2.1.6)

Some additional assumptions were made in the final steps. First, by setting the number of v1,v2 collisions equal to the number of particles with velocity in the volume δv 2 around v2 in the (v1,v2 ) collision cylinder, we assumed that each particle with velocity v2 in this cylinder indeed collides with the particle with velocity v1 to which the cylinder is attached. It was also assumed that the distribution functions do not vary significantly between r 1 and the actual position r 2 of the other colliding particle, so that both distribution functions can be evaluated at the same spatial point, r 1 .7 We emphasize that the Stosszahlansatz is a probabilistic statement, not a mechanical one. It is a statement about the chance of finding a particle with velocity v2 in a collision cylinder. That is, the Stosszahlansatz is the assumption that, before a collision, particles about to collide are completely uncorrelated, so that, irrespective of the knowledge that there is a particle with velocity v1 at position r 1 , the number of particles with velocity in the volume δv 2 around v2 contained in the collision cylinders is taken to be equal to product of the number of particles with velocity, v2 , in the volume δv 2 around v2 , per unit volume with the volumes of the (v1,v2 ) collision cylinders.8 The Stosszahlansatz assumes that the colliding particles are uncorrelated before their collision, but they will certainly be correlated after collision. If this were not the case, we could use the Stosszahlansatz to describe the time-reversed motion, and the Boltzmann equation would then necessarily be time reversible, which it is not. We will discuss this point further after we complete Boltzmann’s derivation of the H -theorem. In realistic systems, correlations between colliding particles are indeed important because, among other things, they are responsible for the mode-coupling effects mentioned in Chapter 1, and these correlations must be included in any systematic extension of the Boltzmann equation to gases at higher densities. We will discuss this point at greater length in the chapters devoted to the generalized Boltzmann equation. Although we will discuss this point in greater detail later in this chapter, the utility and success of the Boltzmann equation in providing a quantitative description of transport in dilute gases is due to the fact that the Stosszahlansatz describes the typical behavior of members of an ensemble of copies of the gas. That is, it should be thought of as describing the average behavior of the ensemble and not as a correct description of any individual member. Instead, individual systems will

26

The Boltzmann Equation 1: Fundamentals

exhibit fluctuations about the average behavior. The deviations from the average behavior should be small for large systems, and a more correct description of a gas can be obtained by including a fluctuation term in the equation itself. This issue will be considered in Chapter 8. 2.1.3 The Gain Term:  + Next, we turn our attention to the calculation of + , the rate at which particles with velocity v1 are created in the small volume δr 1 δv 1 about (r 1,v1 ). The task at hand is to find the velocities of two colliding particles such that after collision, one of them has velocity v1 . We will then use arguments similar to those used to obtain − to obtain an expression for + . However, the calculation of + requires a more careful discussion of the collision dynamics since this term depends on a determination of restituting velocities. These are the velocities that a pair of colliding particles must have before a collision so that one of them will have velocity v1 after the collision. First, we examine the direct collision – that is, the v1,v2 collision discussed earlier. Since we assume that the particles interact with central, conservative forces, there are three conservation laws that must be satisfied – namely, the conservation of momentum, that of energy, and that of angular momentum. The conservation of energy and of momentum may be stated in terms of the initial velocities v1,v2 , and the velocities after the collision, v1,v2 as v1 + v2 = v1 + v2,

(2.1.7)

2 v21 + v22 = v2 1 + v2 ,

(2.1.8)

where we used the fact that the two colliding particles are mechanically identical to eliminate the mass in these conservation equations. An immediate consequence of these equations is the conservation of the magnitude of the relative velocity, g = v2 − v1 , g = |v2 − v1 | = |v2 − v1 | = g  .

(2.1.9)

This follows from the fact that the energies before and after collision are purely kinetic energies and may be expressed as the sum of the kinetic energy of the center of mass and the relative kinetic energy, as 1 (2.1.10) v21 + v22 = 2V 2 + g 2, 2 where V = (v1 + v2 )/2 is the velocity of the center of mass. Since the kinetic energy of the center of mass is conserved separately as a result of the conservation of momentum, the relative kinetic energy is also conserved, from which Eq. (2.1.9) follows immediately.

2.1 The Boltzmann Equation

27

Restituting collision cylinder

−k v2’ − v’1 dt

q

bdbde ’ b g’

k b

v2 − v1 d t

Direct collision cylinder g

bdbde

Figure 2.1.3 Direct and restituting collisions in the relative coordinate frame. The corresponding collision cylinders, as well as the scattering angle, θ , are illustrated.

In addition, the conservation of angular momentum will have a somewhat more delicate role to play in the determination of the restituting velocities, as we will see later. First, we set up the geometry of the direct collision in the relative coordinate system with origin fixed at the center of particle 1. We consider the relative motion, which takes place in the plane through the origin normal to the relative angular momentum mr 21 × g. Before the collision, the center of particle 2 moves toward particle 1 in a straight line with velocity g. When it reaches the action sphere of radius a about the origin, the collision begins. The distance of the trajectory of particle 2 from a line through the origin in the direction of g is the impact parameter of the binary collision. After the collision is completed, particle 2 will again move along a straight line, but with velocity g  . Let us denote the distance of this line from a parallel line through the origin by b . The conservation of angular momentum requires that gb = g  b , but the fact that g = g  implies that b = b, as illustrated in Fig. 2.1.3. If we follow the collision through its completion, we can see that the spatial trajectory of particle 2 has a mirror symmetry about a line, called the apse line, from the origin through the point of closest approach of the trajectory to the origin. There are a number of ways to prove this symmetry, but the simplest is to imagine two particles, the one we have been discussing and a “ghost particle” that travels toward the origin on the other side of the apse line, along the trajectory, but in the opposite direction. If the two particles start at equal distances from the action sphere

28

The Boltzmann Equation 1: Fundamentals

with the same impact parameters and the same relative speed, g = g , they will each experience central forces of equal magnitude at equal distances from the origin and will arrive at the apse line at the same time. At that point, the two trajectories will be mirror images of each other through the apse line. Based on this observation, we see that the relative velocities g and g  have equal components in the direction perpendicular to the apse line and opposite components in the direction of the apse line. If we denote by kˆ a unit vector in the direction to the point of closest approach from the origin along the apse line, it follows from this discussion that ˆ k. ˆ g  = g − 2(g · k)

(2.1.11)

This result allows us to express the final velocities of both particles in terms of the initial relative velocity and the unit vector kˆ along the apse line as ˆ k, ˆ and v1 = v1 + (g · k) ˆ k. ˆ v2 = v2 − (g · k)

(2.1.12)

Of course, the crucial quantity that determines the outcome of the collision is the ˆ which, in turn, depends upon the impact parameter b and both the unit vector k, direction and the magnitude of g. The actual form of this relationship is determined by the interaction potential between the two particles. It is at this point, the deterˆ that one needs more information about the collision mination of the direction of k, dynamics than is provided by the conservation laws. Assuming we can solve the problem of determining kˆ as function of b and g, we are now in a position to determine the restituting velocities. We imagine two particles that are aimed to collide with the same impact parameter b as in the direct collision and with pre-collision velocities given by v1 and v2 (but with opposite relative angular momentum). This collision is illustrated in Fig. 2.1.4, and it can easily be visualized by a simple rearrangement of the particles in space. Under these circumstances, the apse line of this new collision will be in the direction of ˆ opposite to that of the direct collision. Notice now that the relative velocity −k, after this collision will be the original relative velocity, and we have succeeded in determining the restituting velocities, in terms of the parameters and results of the direct collision. To see this, we denote, for the moment, the relative velocity after this second type of collision by g  and use Eq. (2.1.11) to obtain ˆ kˆ = g − 2(g · k) ˆ kˆ + 2(g · k) ˆ kˆ = g. g  = g  − 2(g  · k)

(2.1.13)

It follows from Eq. (2.1.13) that one can determine the restituting velocities simply by finding the final velocities for the direct collisions. These velocities depend,

2.1 The Boltzmann Equation v2

v1

v’1

29 v’2 k

−k v’2

v’1

v1

v2

Figure 2.1.4 Schematic illustration of the direct collisions, on the right, and the restituting collisions, on the left. The corresponding unit vectors indicating the direction of the apse lines are also shown.

as we have seen, on the collision parameters and on the interaction potential. The observation that the final velocities of the direct collisions are the initial velocities of the restituting collisions and vice versa will play an essential role when we turn our attention to the Boltzmann H -theorem later in this chapter. We now have enough information to determine + . We use arguments similar to those used to obtain − , but with a more delicate construction of the collision ˆ along the apse line depends upon cylinders. Since the direction of the unit vector, k, the parameters of the collision, we consider collision cylinders for all possible values of the collision parameters. We illustrate the calculation for three-dimensional systems, and the extensions to other spatial dimensions will be immediate. For three dimensions, the collision parameters of the direct collisions are the impact parameter, b, and the azimuthal angle  of the collision plane. The latter is a polar angle in a plane perpendicular to g measured with respect to a fixed plane in the relative coordinate system that contains the z-axis, taken to be in the direction of g. For the restituting collisions, the collision parameters are the impact parameters, taken, for reasons given before, to be the same as in the direct collision for each pair of final velocities v1,v2 , but the azimuthal angle,   , is a polar angle in a plane perpendicular to g  measured with respect to a fixed plane containing the z -axis, which is along the direction of g  . Let us now consider restituting collisions that take place with impact parameters in the range b to b + db, and azimuthal angles in the range between   and   + d  . We follow the method used before to obtain − : 1. The volume of a small collision cylinder about particle 1 for (v1,v2 ) collisions is b|g  |δt db d  . 2. The number of such collision cylinders in the six-dimensional volume δr 1 δv 1 is f (r 1,v1,t)δr 1 δv 1 , which is simply the number of particles with velocity v1 in this volume. 3. We now apply the Stosszahlansatz for the restituting collisions: The number of particles with velocity in the volume δv 2 around v2 in these collision cylinders is

30

The Boltzmann Equation 1: Fundamentals

the product of the total volume of all the (v1,v2 ) collision cylinders considered before with the number of particles per unit volume with velocities in the range δv 2 about velocity v2 , at time t, given by f (r 1,v2,t)δv 2 . 4. We assume that all of the particles with velocity v2 in the collision cylinders actually collide with the particles with velocity v1 . Putting all of these pieces together, we find that the number of particles produced by (v1,v2 ) collisions with velocity v1 in the region δr 1 δv 1 in time δt is given by f (r 1,v1,t)f (r 1,v2,t)|g  |bdbd  δr 1 δv 1 δv 2 δt.

(2.1.14)

In order to put this expression into a form that can be used to obtain + , we make use of some transformations of the volume elements in velocity space and of the collision parameters. The expression for − as well as expression (2.1.14) can be written in terms of ˆk, the unit vector in the direction of the apse line, by using the fact that, in the collision plane, kˆ makes an angle of (π − θ)/2 with the z-axis, and −kˆ makes the same angle with respect to the z -axis, as illustrated in Fig. 2.1.3. For the direct collisions, we may write     1 1 ˆ d k = sin (π − θ) d (π − θ) d 2 2 1 = cos(θ/2)dθd 2    dθ  1 (2.1.15) = cos(θ/2)   dbd. 2 db Therefore, we may write

where

ˆ k, ˆ |v2 − v1 |bdbd = B(g, k)d

(2.1.16)

   db  ˆ B(g, k) = 2gb   [cos(θ/2)]−1 . dθ

(2.1.17)

ˆ depends upon the intermolecular potential since that deterThe quantity B(g, k) mines the scattering angle, θ, for a given value of b and g, but B does not depend upon the azimuthal angle . In a similar way, for the restituting collisions, we can write ˆ kˆ = gbdbd , B(g , − k)d

(2.1.18)

where we have used the fact that the apse line points in the direction of −kˆ for the ˆ = d k, ˆ and restituting collisions. It is important to note that we have used d(−k)

2.1 The Boltzmann Equation

31

the fact that the same unit vector kˆ can be described in a coordinate system with z-axis in the direction of g or in a coordinate with a z-axis in the direction of g  , as in the restituting collisions. Furthermore, since |g| = |g  |, and −kˆ makes and angle of (π − θ)/2 with g  , it follows that ˆ = B(g, k). ˆ B(g , − k)

(2.1.19)

Equation (2.1.19) must be regarded with some caution. It is correct as it stands if b is a single-valued function of θ. As we will discuss later in this chapter (see Section 2.7), typical potentials that are repulsive at small inter-particle separations and attractive at larger separations may have more than one value of the impact ˆ kˆ parameter b for a given scattering angle θ. In that case, one must redefine B(g, k)d to be the sum of all possible values of gbdbd for which the impact parameters b ˆ and a similar sum for B(g , − k)d ˆ Then, with these definitions, ˆ k. lead to the same k, the equality Eq. (2.1.19) is correct. Next we note that + δr 1 δv 1 δt is given by    ˆ (r 1,v ,t)f (r 1,v ,t), (2.1.20) dv1 dv2 B(g.k)f + δr 1 δv 1 δt = δr 1 δt d kˆ 1 2 R

where R  denotes the region in the space of the variables v1,v2, kˆ determined by the requirement that particle 1 has velocity in the region δv 1 about velocity v1 after the (v1,v2 ) collision. This integration is most easily performed if we note that the Jacobian for the transformation from restituting collision velocities to direct collision velocities is unity, ˆ dv1 dv2 = dv1 dv2, for fixed k.

(2.1.21)

This is most easily seen by imagining a transformation from laboratory velocities to center of mass and relative velocities for the velocities before and after a collision. The center-of-mass velocity is unaffected by the collision, and the relative velocity ˆ as described by Eq. (2.1.11). is just reflected with respect to the plane normal to k, Since the Jacobian of such a reflection is unity, Eq. (2.1.21) follows immediately. With this result, we can write Eq. (2.1.20) as    ˆ (r 1,v ,t)f (r 1,v ,t), (2.1.22) ˆ + δr 1 δv 1 δt = δr 1 δt d k dv1 dv2 B(g. k)f 1 2 R

Where R denotes the region of v1,v2, kˆ space where v1,v2 are the final velocities ˆ under the condition after the (v1,v2 ) collision with apse line in the direction of −k, that the velocity of particle 1 is allowed to range over the small region, δv 1 . Note that we have set the final integration of the right-hand side of Eq. (2.1.22) to be

32

The Boltzmann Equation 1: Fundamentals

ˆ However, we can now the kˆ integration since Eq. (2.1.21) holds only for fixed k. rearrange the integrals on the right-hand side of Eq. (2.1.22) to obtain    ˆ ˆ (r 1,v1,t)f (r 1,v2,t), k)f dv1 dv2 d kB(g, + δr 1 δv 1 δt = δr 1 δt δv 1



= δr 1 δv 1 δt

 dv2

ˆ ˆ (r 1,v1,t)f (r 1,v2,t) d kB(g, k)f

(2.1.23)

for sufficiently small δv 1 . The expression for + follows immediately. We have now obtained Boltzmann’s collision integral, which – by referring to Eqs. (2.1.3), (2.1.6) and (2.1.23) – we can write as9     ˆ ˆ f (r 1,v1,t)f (r 1,v2,t) − f (r 1,v1,t)f (r 1,v2,t) . k) + − − = dv2 d kB(g, (2.1.24) It is often convenient to use a shorthand notation whereby f1 ≡ f (r 1,v1,t), f2 ≡ f (r 1,v2,t) f1 ≡ f (r 1,v1,t), f2 ≡ f (r 1,v2,t), which enables us to write + − − =



 dv2

ˆ ˆ 1 f2 − f1 f2 ]. d kB(g, k)[f

(2.1.25)

(2.1.26)

It is equally convenient to introduce a somewhat less cumbersome notation for the binary collision term in the Boltzmann equation; namely, we use the notation J (f ,f ) for this term. That is,   ˆ ˆ 1 f2 − f1 f2 ]. J (f ,f ) ≡ dv2 d kB(g, k)[f (2.1.27) It is worth repeating at this point that our derivation of the gain and loss terms has made some essential assumptions: 1. The gas consists of mechanically identical, monatomic particles that interact with central, short-ranged forces. 2. The gas is sufficiently dilute that only binary collisions need be taken into account. 3. The number of direct and restituting collisions taking place in a short time interval δt can be calculated using the Stosszahlansatz. 4. The distribution functions vary slowly enough with position that the difference between the positions of the centers of the colliding particles immediately before collision can be neglected when the rates of direct and restituting collisions are calculated.

2.1 The Boltzmann Equation

33

5. The distribution functions vary slowly enough with time that their changes over a time interval on the order of the duration of a binary collision also be neglected. Very often, the Boltzmann equation is used to calculate quantities such as the viscosity or thermal conductivity of a gas of particles interacting with infinite ranged potentials, such as the Lennard–Jones 6–12 potential, or “Maxwell” molecules [89] which interact with repulsive r −4 potentials. Such applications require the potential function to decrease sufficiently rapidly with distance that the right-hand side of Eq. (2.1.26) exists, and that the distributions do not vary much over distances that characterize, in some way, the range of the potential. Potentials such as the Lennard–Jones potential, with both repulsive and attractive regions, have the additional feature that long-lasting, orbiting collisions are possible where the pair of colliding particles can form a quasi-bound state. We will comment on the effect of these collisions later in this chapter, in Section 2.1.7. 2.1.4 Interactions with the Walls:  W The final term in the form of the Boltzmann equation used here is somewhat unusual. In the usual derivations of this equation, the boundaries are not included in the equation but are accounted for by specifying boundary conditions that the distribution functions must satisfy for positions on the boundaries of the vessel [83]. Here we treat the boundaries as if they were just another kind of particle with which the gas particles interact. While this may seem artificial for a gas in a container, the same method can be used to describe the motion of a large particle in a gas, such as would occur in a description of Brownian motion or of Stokes or rarefied gas flow around a sphere or other macroscopic object. Thus, our method here can be seen as setting the stage for later applications of the Boltzmann equation, and of kinetic theory in general. As in the previous discussion, we focus on particles with some velocity, v1 , and write (+) (−) − W , W = W

(2.1.28)

(+) is the rate at which particles in region δr 1 acquire velocity v1 through where W (−) collisions with the boundaries, and W is the rate at which particles in region δr 1 with velocity v1 collide with the boundary and thereby change their velocity. We (−) begin, as before, with the loss term, W . To compute this term, we fix our attention on a small section of the surface located with respect to some fixed coordinate system by ρ S , as illustrated in Fig. 2.1.5. Let dS be the area of the small section, and let nˆ be a unit vector normal to the surface and pointing into the region occupied by the gas. Only particles with

34

The Boltzmann Equation 1: Fundamentals v’1 v1

n

n

v r

S

v’’ 1

1

r

S

(a)

(b)

Figure 2.1.5 Schematic illustration of particle–wall collisions. In (a) the number of particles with velocity v1 is increased due to collisions of particles with the wall. In (b), the number of particles with velocity v1 is diminished when one of them collides with the wall.

velocities v1 satisfying the condition v1 · nˆ < 0 can collide with the wall. The probability that a particle at position r 1 in the range δr 1 is on the boundary at point ˆ Here δ(r 1 − ρ S ) is ρ S and heading toward the wall is δ(ρ S − r 1 )δr 1 H (−v1 · n). a Dirac delta function, and H (x) is the Heaviside function taking the value unity when x > 0 and zero otherwise. By constructing a small collision cylinder with ˆ we see that the number base dS on the small surface element and height |v1 · n|δt, of particles that are removed from the velocity range δv 1 about velocity v1 in the volume element δr 1 about r 1 in time δt due to collisions with the boundary at ρ S is given by  (−) ˆ ˆ (r 1,v1,t)δtδr 1 δv 1, W δtδv 1 δr 1 = dS|v1 · n|δ(r 1 − ρ S )H (−v1 · n)f ∂V

(2.1.29) where we have carried out an integration over the entire surface bounding the gas, (−) as which we denote by ∂V . This expression leads directly to one for W  (−) ˆ ˆ (r 1,v1,t). = dS|v1 · n|δ(r (2.1.30) W 1 − ρ S )H (−v1 · n)f ∂V

(+) is very similar to that just given for The determination of an expression for W the loss term. However, we do need to make some further assumptions in order to obtain an expression for the gain term. These assumptions are as follows.

1. If a particle collides with the surface, it leaves the surface instantly at the point where the impact takes place. This is equivalent to the assumption that the collision duration is small compared to δt and that the range of motion of the particle during the collision is small compared to δr 1 .

2.1 The Boltzmann Equation

35

2. When a particle with velocity v1 collides with the surface, the probability that it leaves the surface with velocity v1 in the range δv 1 is given by P (v1,v1 )δv 1 , where P is taken to be independent of the time, t, and to satisfy the following conditions: (a) P (v1,v1 ) vanishes unless v1 ·nˆ < 0 and v1 ·nˆ > 0, so that P can be expressed in terms of a collision kernel KW (v1,v1 ) as ˆ H (v1 · n). ˆ P (v1,v1 ) = KW (v1,v1 )H (−v1 · n)

(2.1.31)

(b) P (v1,v1 ) is properly normalized, so that  dv1 P (v1,v1 ) = 1 for v1 · nˆ < 0.

(2.1.32)

ˆ v1 ·n>0

(c) Since we have assumed that particles striking the boundaries are instantly reflected from the point of impact in a way that is described by the collision kernel KW , we must impose the condition that the rate at which particles leave the surface at the point of impact should be equal to the rate at which particles arrive at that point on the surface. This condition is expressed by the relation  ˆ H (v1 · n)f ˆ (ρ S ,v1,t) = ˆ W (v1,v1 )f (ρ S ,v1,t). dv1 |v1 · n|K |v1 · n| ˆ v1 ·n 2 > 3 , for two particles interacting with a potential with a repulsive core and an attractive region at larger separations. Note that three different impact parameters can lead to the same value of |θ |, which is the quantity of importance for the scattering cross-section. Rainbows occur at the minima of these curves where dθ/db = 0.

the interacting pair will be disturbed due to collisions with other particles in the gas. While the effects of such long-lived interactions are not properly treated by the Boltzmann equation, estimates show that orbiting collisions do make some contribution to the transport properties of the gas, but, for all but the lowest temperatures, measured in units of the Lennard–Jones parameter, , their contributions are very small. We refer to the literature for further details on classical orbiting processes in binary collisions [306, 215, 392]. It is worth mentioning that at low temperatures, a correct description of binary collision processes requires a quantum treatment, or possibly a semiclassical treatment. A standard approximation is to replace the classical differential cross section in the Boltzmann collision integrals by quantum or semiclassical cross sections. In the quantum treatment, singularities in the classical cross sections are smoothed out by quantum effects, and the effects of orbiting collisions are reduced due to tunneling through the potential barrier [306, 493, 94]. 2.2 The H -theorem of Boltzmann It follows from experiment and from equilibrium statistical mechanics that for a dilute gas confined to a container at rest and in thermodynamic equilibrium in an external field φext (r), the single-particle distribution function has the MaxwellBoltzmann form

βm feq (r,v) = nWW (r) 2π

d2 Ce

−β

1 2 2 mv +φext (r)



,

(2.2.1)

2.2 The H -theorem of Boltzmann

41

where n is the number density of the gas, n = N/V , β = (kB T )−1 , where kB is Boltzmann’s constant, T is the thermodynamic temperature, d is the spatial dimension of the system, and WW (r) is the step function defined after Eq. (2.1.36), and C is a normalization constant given by −1  −βφext (r) drWW (r)e . (2.2.2) C=V allspace

It is therefore natural to ask whether or not this result is in fact predicted by the Boltzmann equation. That is to say: 1. Can one show, using the Boltzmann equation, that a dilute gas not in equilibrium at some time will, as time progresses, approach an equilibrium state with distribution function given by Eq. (2.2.1), and that once in this state, the gas will remain in this state? 2. Can one describe in detail how the approach to this equilibrium state proceeds? The answer to the second question requires a construction of the solution to the Boltzmann equation corresponding to some initial nonequilibrium distribution, which allows us to follow the development of the distribution function with time. This is typically a very difficult problem since the Boltzmann equation is a nonlinear integro-differential equation with very few known solutions. We shall describe some features of the solutions, in so far as we can identify these features, in the next chapter. Here we consider the first question and outline its solution as given by Boltzmann, in what is certainly one of the great accomplishments of nineteenth century physics. Boltzmann showed that one can give an affirmative answer to the first question without having to construct a solution to the equation, assuming, of course, that a solution to the equation exists and that it satisfies some smoothness and integrability conditions. To do this, he proved a theorem, the Boltzmann H -theorem [57], which states that the function H (t) defined by  drdvf (r,v,t)(ln f (r,v,t) − 1), (2.2.3) H (t) = V

is a monotonically decreasing function of time, provided that f (r,v,t) is a solution and there is no energy exchange between the gas and the outside world, including the walls of the vessel.12 The form of the H function was suggested by the fact that if one replaces f in Eq. (2.2.3) by feq given by Eq. (2.2.1), then H = −S/kB + constant, where S is the equilibrium entropy of an ideal gas of N particles in equilibrium in volume V at temperature T . The H function ceases to change with time when the distribution function takes the equilibrium, Maxwell–Boltzmann form, as given by Eq. (2.2.1). In order to check this, let us substitute this form into

42

The Boltzmann Equation 1: Fundamentals

the Boltzmann equation (2.1.41). One easily checks that, for points in the interior of the container, the second and third term on the left-hand side of this equation cancel each other. Furthermore, the first term on the right-hand side, J (f ,f ), also vanishes as a consequence of the energy conservation during a collision, as expressed in Eq. (2.1.10). So the Maxwell–Boltzmann distribution will be stationary indeed provided the action of T¯ W − upon it will vanish if the wall is in equilibrium with the gas. A sufficient condition for this is the so-called thermostat condition, which is of the form ˆ ˆ W (v,ρ S ) · n|φ H (v · n)|v  ˆ ˆ  · n|K ˆ W (v,v )φW (v,ρ S ), = H (v · n) dv H (−v · n)|v

(2.2.4)

where φW (v,ρ S ) is a Maxwell–Boltzmann distribution with temperature TW (ρ S ) given by13

3/2  2 m − 2k mv T (ρ ) B W S . φW (v,ρ S ) = e (2.2.5) 2πkB TW (ρ S ) In the sequel, we will assume throughout that this thermostat condition is satisfied. The H -theorem naturally provoked a considerable amount of controversy at the time of its publication, in part because of the inherent conflict between the monotonic decrease of Boltzmann’s H with time and the time reversibility of Newton’s equations of motion and in part because the atomic theory of matter had not at the time (1872) been firmly established. Here we present the H -theorem in a form appropriate for a gas in a bounded system that allows an exchange of energy between the gas and the boundaries. We will then discuss in some detail the source of irreversibility in the solutions of the Boltzmann equation, explain how one can resolve the two famous paradoxes [72, 631, 152] posed by E. F. F. Zermelo and J. Loschmidt, and argue why the Boltzmann equation cannot be a direct consequence of Newton’s equations of mechanics. Nevertheless, we will argue that the H -theorem provides, for dilute gases at least, a statistical mechanical framework for understanding the molecular origins of the second law of thermodynamics.14 2.2.1 Some Models for the Interactions of the Gas Particles with the Boundaries The interactions between gas particles and the physical boundaries of the vessel containing them can be quite complicated. There is a considerable body of literature concerning the exchange of energy between the gas and the boundaries and many theoretical treatments of it [83, 85, 116, 15]. Since our focus is not on the details

2.2 The H -theorem of Boltzmann

43

of the physical processes responsible for this exchange, we will use here simplified models that capture some of the important features necessary for understanding Boltzmann’s H -theorem. The simplest type of gas–surface interaction is described by specular, elastic collisions of the gas molecules with a surface at rest,15 as would result from a hardcore wall potential that is infinite outside the vessel and vanishes inside.16 In such a case, the collision kernel of Eq. (2.1.31) is given by   sp ˆ nˆ · v1 ) (2.2.6) KW (v1,v1 ) = δ v1 − v1 + 2n( Here nˆ is a unit vector normal to the boundary and chosen to point into the interior of the gas. Notice that this reflection law trivially satisfies the thermostat condition. Although this is a simple boundary condition, which will be used often in the coming chapters, it is by no means the only boundary condition that one might naturally think of. A very natural and useful boundary condition is constructed by supposing that particles striking the boundary at a point ρ S are re-emitted with a Maxwell–Boltzmann velocity distribution with a temperature, TW (ρ S ), characteristic of the wall at the point of incidence. From the thermostat condition (2.2.4) together with the normalization condition (2.1.33), one readily finds that the thermal reflection kernel is given by

mkB Tw (ρ S ) −1/2 th  ˆ ˆ W (v,ρ S ). H (v · n)|v · n|φ (2.2.7) KW (v,v ) = 2π We have by no means exhausted all of the possible boundary conditions that will be used in this book for the distribution function. However, we postpone a discussion of other conditions until we use them in later chapters. 2.2.2 Proof of the H -theorem We now turn to the proof of the H -theorem in a form that is appropriate under the thermostat condition discussed earlier. That is, we will prove that if the kernel KW (v,v ) satisfies the thermostat condition, then the time rate of change of the H function satisfies  dQ(ρ S ) 1 1 dH dS ≤− , (2.2.8) dt kB ∂V TW (ρ S ) dt where the integration is over the entire bounding surface of the gas, dQ(ρ S )/dt is the energy flux from the walls to the gas at the point ρ S on the boundary, and T (ρ S ) is the temperature at that point [116, 215, 83]. We hold the boundary fixed so that there is no work done on the gas, and the energy flow is equal to the heat flow. The net heat flow into the gas will be given by

44

The Boltzmann Equation 1: Fundamentals

dQ(ρ S ) = dt

 ˆ dv(v · n)

mv2 f (ρ S ,v,t). 2

(2.2.9)

Since the integration is over all values of v, the heat flow includes both the energy transferred to the walls and that transferred to the gas. Using the definition of the H function given by Eq. (2.2.3), we define a density of H (t), h(r,t) by    h(r,t) = dvf (r,v,t) ln (f (r,v,t)/A) − 1 (2.2.10) with

 H (t) =

drh(r,t).

(2.2.11)

Here A is a constant inserted to make the argument of the logarithm dimensionless. It plays no role in the following, and we drop it here. It now follows from Eq. (2.1.26) that  ∂f (r,v,t) dh(r,t) = dv ln f (r,v,t) dt ∂t  = −∇r · j h − dvf (r,v,t) (∇v · v˙ )       ˆ ˆ f  f1 − ff1 dv1 d kB(g, k) + dv ln f (r,v,t)  (2.2.12) + dv ln f (r,v,t)T¯ W f (r,v,t). We have used the notation introduced in Eq. (2.1.27) and simplified the notation so that (r 1,v1 ) → (r,v) and (r 2,v2 ) → (r 1,v1 ). Here j h is the flux in h due to the free motion of the particles. The second term on the right-hand side typically vanishes for external fields that depend on position alone or have the form of a Lorentz force on a charged particle or a Coriolis force in a rotating coordinate frame. We retained this term for the moment since it plays an important role in the physics of systems maintained in steady states by what is called a Gaussian thermostat [312, 212]. This will be discussed in Chapter 5. In the present chapter, and in most of the chapters to follow, we will drop this term, to use it again when we come to a discussion of Gaussian thermostats. To obtain Eq. (2.2.12), we have assumed that f (r,v,t) vanishes sufficiently rapidly at large velocities so that    (2.2.13) dv∇v · v˙ f (ln f − 1) → 0. This will certainly be the case for an isolated system with a fixed number of particles and a fixed total energy.17

2.2 The H -theorem of Boltzmann

45

Next we consider the change in h due to binary collisions, the third term on the right-hand side of Eq. (2.2.12). To analyze this term, we first establish a more general and very useful identity:     1   ˆ ˆ dvψ(v)J (f ,f ) = f1 − ff1 ) dv dv1 d kB(g, k)(f 4   × ψ(v) + ψ(v1 ) − ψ(v ) − ψ(v1 ) . (2.2.14) We start by noting that     1 ˆ ˆ [ψ(v) + ψ(v1 )] dv dv1 d kB(g, dvψ(v)J (f ,f ) = k) 2 × (f  f1 − ff1 ).

(2.2.15)

Here we just exchanged the variables v and v1 , summed, and divided the sum by ˆ by B(g , − k), ˆ kˆ by −kˆ and dvdv1 by dv dv , to two. Next we replace B(g, k) 1 write     1 ˆ [ψ(v) + ψ(v1 )] d kˆ dv dv1 B(g , − k) dvψ(v)J (f ,f ) = 2 × (f  f1 − ff1 ).

(2.2.16)

At this point, one does something interesting: We take advantage of the fact that the velocities, (v,v1 ), are the initial velocities in the restituting collision in which (v,v1 ) are the final velocities. This allows us to remove the primes in the integration variables on the right-hand side of Eq. (2.2.16). Next we can exchange primed and unprimed variables in this equation to obtain       1 ˆ ψ(v ) + ψ(v1 ) ˆ dvψ(v)J (f ,f ) = d k dv dv1 B(g, k) 2 × (ff1 − f  f1 ).

(2.2.17)

If we now combine Eqs. (2.2.16) and (2.2.17) and divide again by two, we obtain Eq. (2.2.14). Applying this result to Eq. (2.2.12) and dropping the ∇v · v˙ term, as mentioned before, we now obtain      1 dh(r,t) ˆ ˆ f  f1 − ff1 ln ff1 dv dv1 d kB(g, k) = −∇ · jh + dt 4 f  f1  + dv ln f T¯ W f . (2.2.18) The second term on the right-hand side is either negative or zero, since (b − a) ln(a/b) ≤ 0 for any real positive a and b. The equality sign only holds if a = b,

46

The Boltzmann Equation 1: Fundamentals

of course. As we will see later, the equality ff1 = f  f1 leads to the Maxwell– Boltzmann distribution. For the moment, we continue with the derivation of the H -theorem, which we can now write in the form  dh(r,t) (2.2.19) ≤ −∇ · j h + dv(ln f )T¯ W f . dt To obtain an inequality for H (t), we integrate over all space, use the fact that the distribution function vanishes outside of the container and find that   dH (t) ¯ W f , or ≤ dr dv(ln f )T dt   dH (t) ≤ dS dv(ln f (ρ S ,v,t)) dt ∂V   ˆ ˆ  · n|K ˆ W (v,v )f (ρ S ,v,t) dv H (−v · n)|v × H (v · n)  ˆ . ˆ (ρ S ,v,t)H (−v · n) − |v · n|f

(2.2.20)

The last equation can be transformed to a form that will be convenient for the application of another mathematical inequality, Jensen’s inequality [566], if we use the definition of the wall collision kernel, KW , and the normalization condition given by Eqs. (2.1.31) and (2.1.32), respectively, to write    dH (t) ˆ H (−v · n)|v ˆ  · n|K ˆ W (v,v ) ≤ dS dv dv H (v · n) dt ∂V × [f (ρ S ,v,t) ln f (ρ S ,v,t) − f (ρ S ,v,t) ln f (ρ S ,v,t)].

(2.2.21)

We now wish to make use of the thermostat condition, Eq. (2.2.4), and therefore, we introduce the Maxwell–Boltzmann distribution function, φW defined by Eq. (2.2.5) into the previous equation by writing f (ρ S ,v,t) ≡ φW (ρ S ,v)f0,W (ρ S ,v,t),

(2.2.22)

so as to rewrite Eq. (2.2.21) as    dH (t) ˆ H (−v · n)|v ˆ  · n|K ˆ W (v,v )φW (ρ S ,v ) dS dv dv H (v · n) ≤ dt ∂V  × f0,W (ρ S ,v,t) ln f0,W (ρ S ,v,t) − f0.W (ρ S ,v,t) ln f0,W (ρ S ,v,t) + f0,W (ρ S ,v,t) ln φW (ρ S ,v) − f0,W (ρ S ,v,t) ln φW (ρ S ,v )]. (2.2.23) The crucial step in the argument now is to use the fact that the function x ln x is a convex function for all x ≥ 0. That is, the straight line connecting any two points

2.2 The H -theorem of Boltzmann

47

p1 and p2 on the curve always lies above the curve. Expressed mathematically, any such convex function ψ(x) satisfies the inequality,     ψ ai xi ≤ ai ψ(xi ), (2.2.24) i

i

 where the ai are real, positive numbers, satisfying i ai = 1. Here we use an integral form of this inequality, known as Jensen’s inequality, which we express in the form needed here as     dxf (x)g(x) dxf (x)g(x) dxg(x)f (x) ln f (x)    ln ≤ , (2.2.25) dxg(x) dxg(x) dxg(x) where f (x) and g(x) are real, positive functions. To apply Jensen’s inequality to Eq. (2.2.23), we set the region of integration to be over all v with v · nˆ < 0 and let ˆ H (v · n)K ˆ W (v,v )φW (ρ S ,v ), g(v ) = |v · n| f (v ) = f0,W (ρ S ,v,t), or f (v ) = φW (v ). When Jensen’s inequality is applied to these functions, we obtain    ˆ H (−v · n) ˆ v · nˆ K(v,v )φW (v )f0,W (v ) ln f0,W (v) dv H (v · n)    ˆ H (−v · n) ˆ v · nˆ K(v,v )φW (v )f0,W (v ) ln f0,W (v ). ≤ dv H (v · n) (2.2.26) Here we used Eq. (2.1.33) and the thermostat condition, Eq. (2.2.4), as well as the identity   ˆ H (v · n)f ˆ (ρ S ,v,t) |v · n| ˆ ln ˆ ln f0,W (ρ S ,v,t). (2.2.27) θH (v · n) = H (v · n) ˆ H (v · n)φ ˆ W (ρ S ,v) |v · n| The inequality given by Eq. (2.2.25) allows us to strengthen the inequality for dH /dt, Eq. (2.2.23), by dropping a negative quantity on the right-hand side of this equation. That is, we find that    dH (t) ˆ H (−v · n)|v ˆ  · n|K ˆ W (v,v )φW (ρ S ,v ) dS dv dv H (v · n) ≤ dt ∂V   × f0,W (ρ S ,v,t) ln φW (ρ S ,v) − f0,W (ρ S ,v,t) ln φW (ρ S ,v ) . (2.2.28)

48

The Boltzmann Equation 1: Fundamentals

If we now insert the expression for φW (ρ S ,v), Eq. (2.2.5), in Eq. (2.2.28), using Eqs. (2.1.32) and (2.1.33), we obtain   1 mv2 1 dH (t) ˆ ≤− dv(v · n) f (ρ S ,v,t), dS (2.2.29) dt kB ∂V TW (ρ S ) 2 where the velocity integration is over all v. This is the form of the H -theorem announced by Eqs. (2.2.8) and (2.2.9). The original form of Boltzmann’s H -theorem, dH /dt ≤ 0, is recovered if there is no net energy flow into or out of the container. This can occur if the particles make elastic, specular collisions with the walls, or if the wall temperature is uniform, and the velocity distribution function of the gas is a Maxwell–Boltzmann distribution with the temperature equal to that of the walls. In fact, one stationary solution to the Boltzmann equation, under these circumstances, has the form given by Eq. (2.2.1). Therefore, if in the course of time the distribution reaches a Maxwell–Boltzmann equilibrium distribution in this form, this distribution function will be maintained for all later times. 2.2.3 The Equilibrium Distribution Function Now we turn our attention to Boltzmann’s argument that the H -theorem can be used to show that gas will eventually reach a state of total equilibrium with temperature equal to the wall temperature, provided the particle–wall interactions satisfy the thermostat condition, Eq. (2.2.4) and the wall temperature has a uniform value TW . Our goal is to show that under these circumstances the distribution function takes the form given by Eq. (2.2.1). We use an argument of J. L. Lebowitz and P. G. Bargmann and note that we can define a time-dependent, nonequilibrium Helmholtz free energy [419, 420], F (t), by18 F (t) = E(t) − TW S(t) = E(t) + kB TW H (t), where the energy function, E(t), is given by   2   mv + φext (r) f (r,v,t). E(t) = dr dv 2

(2.2.30)

(2.2.31)

It follows from the Boltzmann equation and the H -theorem that dF (t) ≤ 0. dt

(2.2.32)

This is a well-known expression of the Clausius inequality applied to systems in contact with a heat bath, except that, for a dilute gas, the definition of the free

2.2 The H -theorem of Boltzmann

49

energy has been generalized toward arbitrary nonequilibrium states. If we assume that the total energy, E(t) is bounded from below and the total number of particles, N, is fixed we easily find that the Helmholtz free energy is also bounded from below.19 Hence, from Eq. (2.2.31), it follows that F (t) is a bounded, monotonic non-increasing function of time. It then approaches a limit as t → ∞, such that dF (t)/dt → 0 in this limit. The observation that the Helmholtz free energy eventually approaches a constant value is the essence of the argument for the approach of the distribution function to an equilibrium form. An examination of the proof of the H -theorem shows that dF (t)/dt = 0 if and only if the equality f (r,v,t)f (r,v1,t) = f (r,v,t)f (r,v1,t)

(2.2.33)

is satisfied. Taking logarithms, we find that Eq. (2.2.33) is equivalent to the statement that ln f is conserved in a binary collision. That is, ln f (r,v,t) + ln f (r,v1,t) = ln f (r,v,t) + ln f (r,v1,t).

(2.2.34)

From this result, it follows that the logarithm of the distribution function satisfying Eq. (2.2.33) must be of the form ln f (r,v,t) = a(r,t) + b(r,t) · v + c(r,t)v2,

(2.2.35)

since the quantities conserved in a binary collision are the number of particles, the total momentum, and the total energy of the system of two particles. One might also include the total angular momentum about some point, but this would be redundant. This follows from the fact that all four distribution functions in Eq. (2.2.34) are evaluated at the same point, r, so the conservation of angular momentum, r × v, where × denotes a vector product, follows from the conservation of total momentum.20 Thus, as t → ∞, the distribution function approaches f ∗ (r,v,t), where 2 f ∗ (r,v,t) = WW (r)e[a(r,t)+b(r,t)·v+c(r,t)v ],

(2.2.36)

and for this distribution, J (f ∗,f ∗ ) = 0. The condition that f ∗ be a solution of the Boltzmann equation also requires that F ∂f ∗ + v · ∇r f ∗ + · ∇v f ∗ = 0 ∂t m

(2.2.37)

inside the vessel containing the gas. If the container is at rest, and if the external force derives from a fixed potential φext (r), it follows that at long times f ∗ must have the form21 [631] ∗

f (r,v,t) = AWW (r)e

−β

1 2 2 mv +φext (r)



,

(2.2.38)

50

The Boltzmann Equation 1: Fundamentals

where A and β are constants. We can determine A from the normalization of f ∗ as

d βm 2 , (2.2.39) A = nC(β) 2π where n = N/V is the number density of the gas, we have returned to a general number of spatial dimensions, d, and  −1 . (2.2.40) C(β) = V drWW (r)e−βφext (r) Finally, if the walls are maintained at temperature TW , the thermostat condition requires that β = βW = (kB TW )−1 when the distribution function reaches a stationary solution of the form Eq. (2.2.38). We have therefore shown, by means of the H -theorem, that if the walls are maintained at a constant temperature TW and the particle–wall interaction satisfies the thermostat condition, or if there is no flux of energy at the walls, the distribution function for the gas approaches the Maxwell–Boltzmann equilibrium distribution function as t → ∞. Throughout the book, we will use a simple notation, φ0 (v), for the normalized Maxwell–Boltzmann distribution function in d dimensions, given by

  βm d/2 mv 2 . (2.2.41) exp −β φ0 (v) = 2π 2 The normalized Maxwell–Boltzmann distribution function expressed in the momentum variable is, with a slight abuse of notation, φ0 (p), given by

d/2   β exp −βp2 /2m . (2.2.42) φ0 (p) = 2πm 2.3 The Objections of Zermelo and of Loschmidt By means of the Boltzmann transport equation and its consequence, the H -theorem, Boltzmann appears to have found the connection between macroscopic thermodynamic behavior of dilute gases and the microscopic properties of the particles composing the gas and their interactions with each other and with the boundaries. Boltzmann’s solution addresses, in particular, the irreversible increase of entropy or the decrease of free energy in the approach of the gas to an equilibrium state. This is a very problematic result, however satisfactory it may be from the point of view of thermodynamics. In fact, the irreversibility of the Boltzmann equation and the H -theorem stand in apparent contradiction to the fundamental reversibility of the laws of mechanics that govern all of the interactions between the various constituent parts of the system.

2.3 The Objections of Zermelo and of Loschmidt

51

Let us consider the case where the gas particles make specular, elastic collisions with the walls, since this simplifies the argument without affecting the basic points. The laws of mechanics require that the form of the equations describing the motion of a system be invariant under the transformation t → −t,v → −v, and r → r. Inspection of the Boltzmann equation for the behavior of the gas away from the boundary ∂f F + v · ∇r f + · ∇v f = J (f ,f ) ∂t m

(2.3.1)

shows that each term on the left-hand side of this equation changes in sign under this transformation, while the right-hand side does not. Therefore, no matter what else might be correct about the Boltzmann equation, the statement that it is a direct consequence of the dynamics of the particles making up the gas and the vessel is not correct. This point was made sharply in the form of two interesting objections to the H -theorem – one by Loschmidt in 1876, shortly after the publication of the H -theorem in 1872 by Boltzmann, and the other in 1896 by Zermelo [173, 72]. Loschmidt argued that the H -theorem must be in violation of Newtonian mechanics because if the H -function were to decrease in time for a mechanical system, it would have to increase in time for the time-reversed motion. Since for any valid solution of the Newtonian equations, the time-reversed motion is an equally valid solution, there is no way that the H -theorem could be correct for both motions, unless, of course, the H -function were a time-independent constant, which it is not. Zermelo’s objection was somewhat more subtle. It relied upon a mathematical result of Poincaré called the recurrence theorem [290, 329, 152].22 Poincaré’s theorem states that any isolated mechanical system with a bounded energy surface is recurrent. This means the following: Imagine a system of a finite number of particles contained in a finite container represented by a fixed external potential and isolated from the rest of the universe. Now imagine some initial state specified by the positions and momenta of all of the particles in the system. The recurrence theorem states that the mechanical system will eventually come arbitrarily close to its initial state, and this will happen over and over again, given enough time. Thus, even if the H -function were to decrease for some time in the motion of a system, it would eventually have to increase in order to come arbitrarily close to its previous higher value. Of course, the time needed for a macroscopic system to return sufficiently close to some initial state is ridiculously long, but that in no way invalidates the mathematical argument advanced by Zermelo. To get some feeling for the enormous times associated with the Poincaré recurrence theorem, we present in Table 2.1 some results obtained by S. Chandrasekhar [86, 674] for the recurrence of a relatively mild fluctuation in density (which will occur extremely more frequently than an actual Poincaré recurrence!).

52

The Boltzmann Equation 1: Fundamentals

Table 2.1. The recurrence time, tr , for 1 percent density fluctuations in a spherical volume of radius a in air at standard temperature and pressure. This table is taken from the paper of S. Chandrasekhar [86] and included in the collection of papers [674]. a [cm]

tr [sec]

∼ 10−5 ∼ 2.5 × 10−5 ∼ 3 × 10−5 ∼ 5 × 10−5 ∼1

≈ ≈ ≈ ≈ ≈

10−11 1 106 1068 14 1010

Given the evident problems reconciling the Boltzmann equation and its consequences with Newtonian mechanics but the evident parallel between the equation and the laws of thermodynamics, we must now determine in what sense the Boltzmann equation can be understood as a correct description of the nonequilibrium behavior of a dilute gas. The answer, as we have indicated previously, is that it is a statement about the probable behavior of a gas with macroscopically, but not microscopically, specified initial conditions, rather than about the individual behavior of any particular realization of the gas. To get some physical feeling for this situation, imagine an isolated room containing a macroscopic number of molecules of a gas, and imagine an initial situation specified only by the condition that all of the molecules are confined to a small region of the room – say, a corner. We would expect to see the gas molecules spread throughout the room, and this is indeed the typical behavior. However, we can also imagine initial arrangements of the positions and velocities of the molecules such that for an interval of time, the gas becomes more compressed in the corner or even moves without too much change into the opposite corner of the room. Any mechanically possible behavior could, in principle, occur, however unlikely or bizarre. But for such behavior to occur, the requirements on the initial phases of the molecules are so special that the probability they will occur in a physical realization of the specified macroscopic initial state is effectively zero, and we never see such things. This example illustrates the statistical nature of the Boltzmann equation. In the next section, we consider a simple model system that will enable us to understand more fully the issues raised by Loschmidt and Zermelo and how they can be satisfactorily resolved.

2.4 The Kac Ring Model

N 1 N−1

53

2 3 4

Figure 2.4.1 The Kac ring model. The sites on the ring are indicated, and the markers between some of the sites are indicated by the check marks.

2.4 The Kac Ring Model 2.4.1 The Source of Irreversibility in the Boltzmann Equation As we stated in the derivation of the Boltzmann equation, a nonmechanical assumption is employed to obtain an expression for the rate of direct and of restituting collisions. This assumption was the Stosszahlansatz. It would be very illuminating to have a model system that would be simple enough to illustrate the statistical nature of the Stosszahlansatz and show how it might be understood as resulting from the application of statistical arguments for determining the probable behavior of an ensemble of identical reversible mechanical systems differing only in the initial phases of the constituent particles. Such a useful and clarifying model was invented by M. Kac, and we turn to it now. It is known as the Kac ring model [329]. Consider a circle with a large but discrete number, N, of lattice sites placed along its circumference. Suppose further that there are M < N markers placed between adjacent sites, as illustrated in Fig. 2.4.1. The markers are distributed along the circle completely randomly, the only rule being that there is at most one marker between any two adjacent sites. The markers are fixed once they are placed around the circle. Now, on each of the N lattice sites, there is exactly one bead. The beads come in two colors, black and white, and the N black or white beads are distributed along the circle in some way. The dynamics of the model take place at discrete time steps, and at each time step, all the beads move exactly one step to the next lattice site, in a clockwise direction along the circumference of the circle. If the bead passes one of the markers in a step from one site to the next, it changes its color to the other one. That is all there is to the model and its dynamics. The dynamics are clearly reversible: If we start from some initial state and run the dynamics for t steps, we can recover the initial state by simply running the dynamics counterclockwise for t steps. Furthermore, no matter what the initial arrangement of the beads might be, there is a Poincaré recurrence of 2N steps, since at the 2Nth step, each bead will have passed every marker twice

54

The Boltzmann Equation 1: Fundamentals

and be back to its initial position and color. Of course, certain arrangements of the colors and markers may have shorter recurrence times as well, but 2N is a universal recurrence time for this model.23 The macroscopic quantity of interest will be taken to be the total number of beads of either color. Let B(t) and W (t) denote the number of black beads and white beads, respectively, after t steps. We wish to determine an expression for (t) = B(t) − W (t). We will do this in two ways: first, by using an argument employing the Stosszahlansatz and, second, by solving the exact equations of motion and then applying the methods of statistical mechanics to the exact dynamics. The following two steps are mechanically correct, in any case: The total number of beads is N; B(t) + W (t) = N.

(2.4.1)

If we denote by b(t) and w(t) the number of black beads and the number of white beads, respectively, at time t, about to change color in the next step, we find that B(t + 1) = B(t) − b(t) + w(t), W (t + 1) = W (t) − w(t) + b(t), or

(t + 1) = (t) − 2 (b(t) − w(t)) .

(2.4.2)

This is as far as we can go without examining the beads around the circle to learn which ones are about to change color. This is, of course, similar to our problem of trying to count the number of collisions taking place in time δt when we derived the Boltzmann equation. There we resorted to the Stosszahlansatz to get a number. Here we do the same: We say that the number of black or white beads about to change color is equal to the total number of beads of each color multiplied by the fraction of sites that have a marker in front of them – namely, μ = M/N. This is the analog of the Stosszahlansatz for the ring model, and it introduces the irreversibility into the equation for (t). That is, b(t) = μB(t),w(t) = μW (t) and

(t + 1) = (1 − 2μ) (t).

(2.4.3)

The solution of Eq. (2.4.3) for (t) is obviously

(t) = (1 − 2μ)t (0).

(2.4.4)

Equation (2.4.4) predicts a monotonic decay of | (t)| to its “equilibrium” value, zero. There is no sign in this equation of either the reversibility of the model or the Poincaré recurrence phenomena.24 How then can it possibly be correct, in any sense whatever? To answer this question, we now turn to a more exact description of the dynamics of the model. We define some additional quantities, ηi (t) and j , where ηi (t) = 1

2.4 The Kac Ring Model

55

if the bead at site i at time t is black and ηi (t) = −1 if the bead at site i at time t is white. Also, i = 1 if there is no marker in front of site i, and i = −1 if there is a marker in front of site i. Using these two definitions, we can easily see that the microscopic equation of motion for this model is simply ηi (t + 1) = i−1 ηi−1 (t).

(2.4.5)

This simple equation embodies all of the dynamics of the model. By iteration we obtain ηi+1 (t + 1) = i i−1, . . . ,i−t ηi−t (0). By summing ηi (t) over all the sites, we obtain an expression for (t), as  i i−1, . . . ,i−t ηi−t (0).

(t + 1) =

(2.4.6)

(2.4.7)

i

Notice that the Poincaré recurrence time 2N appears as a consequence of Eq. (2.4.6) or (2.4.7). That is, ηi (2N) = ηi (0) and (2N) = (0) since, for this case, every j appears twice in each product of the  values. Now consider an ensemble of Kac rings, each with the same distribution of beads around the circle but differing in the placement of the markers. In fact, we shall prepare the ensemble in such a way that μ = M/N is the probability that there will be a marker in front of any site. This means that the average number of markers is still M, but individual members of the ensemble may have more or fewer markers than M in accordance with the prescription. Under these circumstances, the i may be considered to be identical, individually distributed, random variables with average values < i >= (1 − μ)(+1) + μ(−1) = (1 − 2μ). Using this result, we may compute all of the averages, being careful to note if a particular j appears more than once in the product. For this reason, we obtain different results if 0 < t ≤ N and if N < t ≤ 2N. For t in the interval 0 < t ≤ N, we find  i−1, . . . ,i−t  (0),  (t) = i

= (i )t (0) = (1 − 2μ)t (0).

(2.4.8)

This is exactly the result that we obtained using the Stosszahlansatz, but its validity is now restricted to a fixed range of time t. On the other hand, for N < t ≤ 2N, the counting is somewhat different. We have to ignore all markers that appear twice in the products since they give a factor of unity. Thus, suppose t = N + s; then any markers between site N and site s − 1 would be counted twice. In that case, there will be N − s = 2N − t terms in the products of the  values, and as a result,  (t) = (1 − 2μ)2N −t (0),

(2.4.9)

56

The Boltzmann Equation 1: Fundamentals

which is quite consistent with the Poincaré recurrence theorem, since the initial value of is always recovered after exactly 2N steps. The exponential decay of < (t) >, as described by Eq. (2.4.8), is the average result for our ensemble for times less than the number of sites. Of course, if N is a huge number – say, on the order of 1023 – it would seem unlikely that we would see anything else. To make this point more precise, we need to ask about the size of the fluctuations about the results just obtained. If N is very large, we would expect these results to be typical in the sense that the fluctuations about them are very small. We will calculate the mean square deviation of the actual results from those predicted by Eq. (2.4.4), for times much less than the recurrence time to show that the fluctuations are indeed very small for large N. However, there are certainly members of the ensemble where there are large regions free of, or filled with, markers or long segments with markers on even sites and none on odd sites. For these members, we would expect to see large local fluctuations away from the average Boltzmann behavior. There are occasionally circumstances where rare clusters do lead to physical effects, a phenomenon usually associated with so-called Lifshitz tails [435]. 2.4.2 Pre- and Post-collision Correlations Before turning to a more detailed discussion of the fluctuations about the average behavior in a ring model system, we make a few comments about the effect of correlations produced by collisions in this system. The Stosszahlansatz assumes, for this model, that as a bead moves from a site to an adjacent site, it may encounter a marker with probability μ and may not encounter a marker with probability 1−μ. Since the markers are fixed once they are distributed around the ring, the timereversed motion is totally determined. That is to say we can only guess what will happen in a given time step, but once that step takes place, we no longer have to guess what will happen if we time-reverse the motion. Thus, according to the Stosszahlansatz, there is no time-reversal symmetry of the motion of the system; the forward and time-reversed motion are treated in different ways due to the presence of correlations between beads and markers created by the “collision” mechanism. There are only two ways to restore the time-reversal symmetry – one can follow the exact dynamics for a given configuration of markers on the ring and not employ the ansatz to describe the effects produced by the encounters of the beads with the markers, or one can construct a new model where the markers are redistributed after every step, before the next one. In the first case, one must follow the exact dynamics in both the forward and time-reversed directions; in the second case, one can apply the Stosszahlansatz equally well for both the forward and time-reversed motions, since the redistribution of markers breaks any correlations between beads and markers produced in any step.

2.4 The Kac Ring Model

57

2.4.3 Deviations from the Stosszahlansatz in the Ring Model Since we argued before that the Stosszahlansatz is a probabilistic statement about the number of collisions taking place in a small time interval or, for the ring model, about the number of color changes that take place at each time step, we wish to examine the behavior of fluctuations about the results obtained from its use. The mathematical simplicity of the ring model allows us to calculate quantities that would be difficult, if not impossible, for an ordinary gas of colliding particles [263]. The most accessible quantity that provides a measure of the fluctuations about the Boltzmann equation result, Eq. (2.4.4), is the mean square deviation, δ(t)2 , of the actual value of (t) about its average,  (t), defined by δ(t)2 = ( (t) −  (t))2  =  (t)2  − ( (t))2 .

(2.4.10)

In order to calculate this quantity, we will need the exact expression for (t) given by Eq. (2.4.6). Using this expression, we find that

(t) = 2

N N  

i−1, . . . ,i−t j −1, . . . ,j −t ηi−t (0)ηj −t (0).

(2.4.11)

i=1 j =1

We can simplify the calculation to some extent by imagining that initially all the beads on the ring are of one color – say, black. Then, for t < N, we can write  (t) = N(1 − 2μ)t , and ( (t))2  is then given by N N   ( (t))  = i−1, . . . ,i−t j −1, . . . ,j −t . 2

(2.4.12)

i=1 j =1

The right-hand side of Eq. (2.4.12) can be simplified, under the averaging, to ( (t))2  = N

N  1 2, . . . ,t j j +1, . . . ,j +t−1 .

(2.4.13)

j =1

For sufficiently short times – namely t < N/2 – the right-hand side of Eq. (2.4.13) can be evaluated easily and is given by   ( (t))2  = N 1 + 2(1 − 2μ)2 + 2(1 − 2μ)4 + · · · + (N + 1 − 2t)(1 − 2μ)2t   1 − (1 − 2μ)t 2t − 1 + (N + 1 − 2t)(1 − 2μ) . (2.4.14) =N 2μ(1 − μ) The term of order N 2 on the right-hand side of this equation is just ( (t))2 , so the mean square fluctuation about the average, or Boltzmann equation, value of (t) is, for this case, given by   1 − (1 − 2μ)t 2 2t − 1 + (1 − 2t)(1 − 2μ) . δ (t) = N (2.4.15) 2μ(1 − μ)

58

The Boltzmann Equation 1: Fundamentals

We see that the fluctuations about the average value are small in comparison the average value itself, of order N 1/2 compared to an average value of order N. Thus, in the “thermodynamic limit” for this model, where N → ∞ and t remains finite, we find that δ 2 (t) ( (t))2

→ 0.

(2.4.16)

2.4.4 The Boltzmann–Langevin Equation The calculation presented for the mean square fluctuations within the ensemble of rings depends upon the simplicity of the motions of the beads and the collision dynamics in the ring model. A similar calculation, including the effects of correlated collisions, cannot be easily carried out to obtain expressions for the fluctuations of the numbers of collisions in a dilute gas about the number predicted by the Stosszahlansatz, due to the complexity of particle motions and collision dynamics. However, a number of authors have addressed the problem of describing fluctuations about the average behavior as described by the Boltzmann equation by extending the linearized Boltzmann equation by adding Langevin-like fluctuations to it. The resulting equation, known as the Boltzmann–Langevin equation [42, 230, 231], has a number of useful properties allowing for some estimates of fluctuations of physical quantities about the average values as described by the linearized Boltzmann equation. We will discuss the Boltzmann–Langevin equation in Chapter 8, after we have shown how the Navier–Stokes equations can be obtained from the Boltzmann equation. Notes 1 We will use the term hard sphere without reference to the number of spatial dimensions of the system. When necessary, we will be more explicit about the actual dimensionality of the system. 2 In the long tradition of kinetic theory, we use velocities rather than momenta, but, of course, all equations may be trivially reformulated in terms of the latter. In the course of this book, we will use both. 3 For air at standard temperature and pressure, n ∼ 3 × 1025 molecules per cubic meter, and a ∼ 2 × 10−10 m. Thus this number is on the order of 3 × 10−4 . 4 For later convenience we denote a particular point in space by r 1 and the velocity of a representative particle in the vicinity of r 1 by v1 5 We will see presently that the typical length scale for this is the mean free path between collisions, which indeed for sufficiently dilute gases is much larger than the average distance between particles. The ratio of the average distance between the particles to the mean free path is roughly on the order of (na d )(d−1)/d , where d is the spatial dimension of the system. 6 This process is indeed carried out in computer-simulated molecular dynamics. 7 The error made by ignoring the difference in position of the colliding particles is of order a|∇r f |, which in turn is of order na d , since the distribution function changes over distance on the order of a mean free path. This is a first-order density correction to the Boltzmann equation. We will see how to account for changes in the distribution function over distances of order a in the derivation of the Enskog equation to be presented in Chapter 7.

Notes

59

8 The presence of correlations would imply a breakdown of the assumption that the probability of finding particles with velocities v1,v2 together in some region is simply the product of the probabilities of finding particles of each velocity in the region. 9 Here we have obtained the expression for B(g,k) for three-dimensional systems, but it can be suitably redefined for general dimensions, d. 10 Here, by using the simplified notation, T¯ (1,2),bkˆ (1,2) , we are anticipating the notation used for binary collision and exchange operators in later chapters. 11 For energies between the maximum of the effective potential and the minimum located closer to the origin of the r-axis, there even exist stable periodic orbits, but these cannot be reached from an initial situation where the two particles are far apart, without the intervention of a third particle. 12 Boltzmann, in his derivation of the H -theorem, did not explicitly take into account any interactions between gas particles and the walls, but his treatment does allow for walls that can be represented by fixed external potentials. Here we will present a derivation of the H -theorem that allows for more general ways of representing the interactions with the walls. Still, we will have to impose the so-called thermostat condition on these interactions to be able to derive the H -theorem. 13 We continue to use the case of three-dimensional systems as an example. All of the expressions given here can easily be generalized to any number of spatial dimensions. 14 Boltzmann wrote in his book Lectures on Gas Theory that “this proof has, as it will appear, a not uninteresting connection with the entropy principle” [57]. 15 Reflection laws for moving surfaces are found most easily by first formulating them in the rest frame of the surface and then transforming to the rest frame of the gas. 16 For smooth wall potentials that increase from zero to infinity over a small distance, the specular collision kernel is not exact, but it is an excellent approximation, comparable to that of the neglect of the difference in position of two colliding particles. The effect of such potentials may also be accounted for by including them in the external potential determining v˙ in Eq. (2.1.41). In most cases, however, the representation by the specular collision kernel – or, equivalently, by a matching condition between the distribution functions with incoming and reflected velocities – is simpler. 17 Statistical mechanics also gives good arguments why the same will hold for systems in contact with particle and/or heat reservoirs since, in such cases, the combined system plus reservoir can be considered to be an isolated system. 18 We have assumed that the walls of the container are kept fixed. We can relax this condition; e.g. for systems with mobile walls kept under constant external pressure, we could use the Gibbs free energy instead of the Helmholtz free energy in the argument presented after Eq. (2.2.30). 19 Variational calculus applied to Eq. (2.2.31) shows that the minimum is obtained for a Maxwell–Boltzmann distribution with the temperature TW . 20 Since r × v + r × v1 = r × v + r × v1 follows from v + v1 = v + v1 . 21 Note that if the gas particles make specular collisions with the walls of the container, then the Maxwell–Boltzmann form is preserved by the interactions with the walls, and there is no energy exchange between the walls and the gas itself. The conclusions reached in the discussion to follow also apply to this case except that β is not determined by the walls but rather by the total energy of the system. 22 This theorem was anticipated by Nietzsche’s doctrine of eternal return [502]. Nietzsche’s argument was not unlike that of Poincaré’s proof but much less precise. See S. Brush [72]. 23 In contrast to the recurrence properties of a gas, the Kac model has a recurrence time that is the same for all initial configurations of beads and markers. 24 Note that according to Eq. (2.4.4) and for the special case that μ = 1/2, the number of beads of each color will be the same after one step and remain so for any number of steps. This is unlikely to be true for most distributions of markers around the circle but may be true as an ensemble average, as we discuss later.

3 The Boltzmann Equation 2: Fluid Dynamics

In this chapter, we will discuss solutions of the Boltzmann equation that are relevant for our understanding of transport phenomena in dilute gases. We first discuss the so-called normal, or Chapman–Enskog, solution of the equation that is appropriate for a gas that is close to a local equilibrium state at every point in space [631, 103, 89, 215, 83]. That is to say, the state of the gas is such that at every point, one can define a local density, mean velocity, and temperature, but these quantities may vary on some scale, large compared to the mean free path length. The Chapman–Enskog solution to the Boltzmann equation is predicated upon the existence of such a situation and then proceeds to derive, by methods to be discussed here, the macroscopic, Navier–Stokes, equations of fluid dynamics together with expressions for the transport coefficients appearing in them in terms of the dynamics of binary collisions taking place in a dilute gas. Having obtained expressions for transport coefficients and related macroscopic quantities in terms of the intermolecular potential energies, we then turn to a comparison of these predictions with experimental results. These predictions compare very favorably with experiment, thus providing strong support for Boltzmann’s arguments when deriving the equation as well as a strong motivation for trying to extend the Boltzmann equation to higher densities, as will be discussed in Chapters 7 and 11–13. After reviewing the Chapman–Enskog method, we present a more direct way to derive the linearized forms of the hydrodynamic equations based upon the application of projection operator methods to the linearized Boltzmann equation. We also discuss hydrodynamic modes of the linearized Boltzmann equation, which provide a way to describe the long time behavior of local deviations from equilibrium in the gas. Both projection operator methods and hydrodynamic modes will play important roles in the later chapters of this book. We will also consider other types of solutions of the Boltzmann equation in this chapter. These include the very useful Grad 13-moment method [264] and some special solutions due to A. V. Bobylev, M. Krook, T. T. Wu, and others 60

3.1 The Chapman–Enskog Solution

61

[44, 402, 403, 45, 177, 178]. Finally, we comment briefly on the many interesting open mathematical questions related to proving the existence of solutions to this complex and interesting equation. 3.1 The Chapman–Enskog Solution The Boltzmann equation is an integro-differential equation. As such, it is extremely difficult to solve in general, and many endeavors to do so have met with deep mathematical problems. Very few exact solutions are known. Among these are the equilibrium Maxwell–Boltzmann distribution function and some special solutions, which will be discussed at the end of the chapter. The Chapman–Enskog, or normal, solution was developed with the purpose of deriving the equations of fluid dynamics from the Boltzmann equation. It is based on some ideas that naturally arise in the context of Boltzmann’s H -theorem and the approach to an equilibrium Maxwell– Boltzmann distribution. It is not an exact solution in a strict mathematical sense. It envisages an expansion of the solution of the Boltzmann equation as a series in gradients of the hydrodynamic densities – that is, gradients of the local number, momentum, and energy densities – of all orders and in powers of their gradients. In order to keep track of the various terms in this series of the same order in the gradients, we introduce a small parameter, μ = /L, which can be specified as the ratio of the mean free path length,1 , between two subsequent collisions of the same particle and some characteristic macroscopic length, L. We suppose that each gradient introduces a power of μ, and use powers of μ to collect terms of equal order in the gradients. When the order of each term is clear we eliminate μ in favor of the gradients and set μ equal to unity. The Euler, or ideal fluid, equations arise from the solution up to the first order in this parameter, the Navier– Stokes equations, containing additional dissipative terms, result from the solution through second order, and so on [412]. It is not known whether this procedure ever converges to a complete solution with increasing order in the expansion or only generates an asymptotic expansion of a solution.2 In any case, only the first two terms, leading to the Euler and to the Navier–Stokes equations, are generally used. Some results are known about the convergence of the linearized version of the Chapman– Enskog expansion, notably McLennan’s theorem [476, 477], but not much else. The third and fourth terms in the expansion lead to the Burnett and super-Burnett corrections to the Navier–Stokes equations, which are useful for describing sound propagation and shock waves in a dilute gas but have limited utility otherwise [89, 87, 239]. They require additional boundary conditions, which tend to give rise to spurious, nonphysical solutions. As described later in this chapter, when we discuss the dispersion of sound in a dilute gas, Burnett and higher-order corrections to the Navier–Stokes results significantly improve the agreement of kinetic theory

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predictions with experimental results. A recent review of the applications of the Burnett equations to shock waves can be found in the article by L. S. Garcia-Colin et al. [239]. Before getting into the details of the Chapman–Enskog solution, we find it convenient to separate the Boltzmann equation Eq. (2.1.41), of the previous chapter, into two separate equations, one governing the behavior of the distribution function in the interior of the container and one for the distribution function at the boundaries of the system. We recall from Eq. (2.1.36) that we can express the distribution function for the gas within the system’s boundaries in the form f (r,v,t) = W (r)f˜(r,v,t),

(3.1.1)

where W (r) is a step function equal to unity whenever r is within the boundaries confining the gas, and zero otherwise.3 The function f˜(r,v,t) is taken to be continuous at the boundary. If we insert this expression for the distribution function into the Boltzmann equation, Eq. (2.1.41), we find two types of terms, those governing the distribution function in the interior of the container, which lack delta functions evaluated at the boundaries, and those governing the distribution function on the boundary, which are proportional to such delta functions. Since we suppose that f˜(r,v,t) is continuous at the boundaries, we can separate the terms with the delta functions from the other terms, and as a result, we obtain two equations given by ∂ f˜(r 1,v1,t) + ∇r1 · (v1 f˜(r 1,v1,t)) + ∇v1 · (F f˜(r 1,v1,t)) ∂t   = dr 2 dv2 T 0 (1,2)f˜(r 1,v1,t)f˜(r 2,v2,t),

(3.1.2)

where we introduced the symbol F for the external force per unit mass and an equation that determines the behavior of the distribution function at the boundaries f˜(r,v,t)v · ∇W (r) = T¯ W f˜(r,v,t).

(3.1.3)

For the time being, we will concentrate on Eq. (3.1.2) and return to the boundary equation, Eq. (3.1.3) later in Section 3.6. 3.1.1 The Equations of Fluid Dynamics There are d + 2 equations of fluid dynamics for a one-component fluid in d spatial dimensions. These equations describe the rate of change with time and space of the local densities of mass, energy, and momentum of the fluid.4 These densities are directly associated with mechanical quantities that obey microscopic conservation laws. They change on time and space scales that are large compared to the scales

3.1 The Chapman–Enskog Solution

63

on which the densities of non-conserved quantities change. The latter change drastically at every collision, whereas coarse-grained densities of conserved quantities like mass, momentum and energy (determined e.g. by the long-wavelength components of their Fourier transforms in space) change only very gradually, both in collisions and during free flight. In a purely macroscopic theory, the Navier–Stokes equations of fluid dynamics result from combining general conservation equations with a set of macroscopic relations called constitutive laws [412]. The conservation laws relate the change in time of the conserved densities to the spatial derivatives of currents of these densities. The constitutive laws are explicit expressions for the currents appearing in the conservation laws as functions of gradients in the conserved densities. The precise expressions for the constitutive laws, generally known as Fourier’s law of heat conduction, Newton’s law of friction, Fick’s law of diffusion, etc., depend on the tensorial character of the currents and upon the symmetry properties of the fluid. The requirement that the irreversible rate of entropy production be positive definite also places restrictions on the forms of the constitutive laws.5 They also contain undetermined constants such as the coefficients of thermal conductivity, shear and bulk viscosity, and so on. There are interesting and important effects in fluid mixtures that we will discuss in a later chapter on gas mixtures. To set the stage for the normal solution or Chapman–Enskog procedure, we’ll first obtain the conservation laws from the Boltzmann equation and then exhibit, without derivation, the constitutive laws mentioned previously. The derivation of these laws from the Boltzmann equation forms the core of the Chapman–Enskog procedure. The Conservation Laws To obtain the conservation laws using the Boltzmann equation, we simply multiply this equation by one of the quantities that, summed over all particles, are conserved in a binary collision. Explicitly, these are 1, corresponding to the conservation of particles or, equivalently, m, corresponding to the conservation of mass; mv, corresponding to the conservation of momentum; and mv 2 /2 + φext (r), corresponding to the conservation of energy.6 Here φext (r) is the external potential energy per particle, whose gradient in most cases can be identified with minus the external force per particle. We neglect the potential energy of interactions between the particles when determining the energy density. In fact, the Boltzmann equation approximates the actual dynamics of the gas by a sequence of instantaneous binary collisions in which pre-collisional velocities are replaced by post-collisional ones. In this approximation, the potential energies of interaction never explicitly appear. The interaction energy contribution to the energy density are included when considering transport theories for fluids at higher densities.

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We introduce some notation by organizing the conserved quantities into a vector with components denoted by ψα , where the ψα are the d + 2 quantities conserved in binary collisions, namely, ψm = m; ψi = mvi for i = 1, . . . ,d; and ψe = mv 2 /2 + φext (r). We denote the corresponding spatial and time-dependent fluid or hydrodynamic densities by ρα , where the mass density, ρ; the momentum density, ρi ; and the energy density, ρe , are given by  ρ(r,t) = m f˜(r,v,t)dv = mn(r,t); (3.1.4)  (3.1.5) ρi (r,t) = m f˜(r,v,t)vi dv = ρui (r,t),i = 1, . . . ,d;  ρe (r,t) = [mv 2 /2 + φext (r)]f˜(r,v,t)dv = ρ(r,t)u2 (r,t)/2 + n(r,t)φext (r) + n(r,t)eK (r,t), where the local kinetic energy density, eK (r,t), is given by  n(r,t)eK (r,t) ≡ (mc2 /2)f¯(r,c,t)dv.

(3.1.6)

(3.1.7)

Here n(r,t) is the local particle density. We have defined the local average velocity u(r,t) in terms of the momentum density ρi = ρui and introduced the vector c = v − u, the relative velocity of the particles with respect to the local velocity of the fluid.7 The expression for ρe (r,t) given previously treats the energy density as the sum of three contributions: First, the kinetic energy density of the local fluid motion, ρu2 /2; secondly, the density of external potential energy n(r,t)φext (r); and finally, an internal energy density n(r,t)e(r,t), which, in the case of a dilute gas, is approximated by the kinetic energy density of the gas particles in the local rest frame of the fluid. This is denoted here by n(r,t)eK (r,t). Above we introduced the notation f¯(r,c,t) ≡ f˜(r,v = c + u,t).

(3.1.8)

Now, one may obtain the local conservation laws of mass, momentum, and energy as follows: start by multiplying the Boltzmann equation, Eq. (3.1.2), without the wall collision term, by each of the functions ψα , use the symmetrization procedure of Eq. (2.2.14) together with the collisional conservation laws described by Eqs. (2.1.7, 2.1.8) to arrive at the equations ∂ρ + ∇ · (ρu) = 0; ∂t

(3.1.9)

∂(ρui ) + ∇r · J i − ρFi = 0; ∂t

(3.1.10)

3.1 The Chapman–Enskog Solution

65

∂ρe + ∇ · [J K + unφext (r)] = 0. ∂t

(3.1.11)

Here J i is the ith row of the momentum current density tensor, with elements Jij given by   ˜ (3.1.12) Jij = m vi vj f (r,v,t)dv = ρui uj + m ci cj f¯(r,c,t)dc and

 Ji = ρui u + m

ci cf¯(r,c,t)dc.

(3.1.13)

In order to simplify the equations and notation in this chapter, we will often, but not always, omit the space, r, and time, t, variables in the arguments of the local quantities, ρ, u, and T , as well as in the local pressure tensor and heat flow vector defined later. The elements of the momentum current density tensor denote the current density of the ith component of the momentum in the j th direction, in the gas. We see from Eq. (3.1.12) that Jij is a symmetric tensor. The first term on the right-hand side of Eq. (3.1.12) is a macroscopic convective contribution, which vanishes in the local rest frame. The second term represents the local pressure tensor for a dilute monatomic gas. The vector J K is the kinetic energy current defined by  2

v JK = v m f˜(r,v,t)dv 2   mc2 ¯ u2 f (r,c,t)dc. (3.1.14) = ρu + nueK + mu · ccf¯(r,c,t)dc + c 2 2 Note that the average value of c on integration over the distribution function f¯(r,c,t) is, by definition, zero. The restriction, made for convenience, that the external forces on the particles do not depend upon their velocities can easily be removed. Next, one has to consider the effects of inclusion of the wall collision term, T¯ W f , in the Boltzmann equation. This gives rise to additional terms on the right-hand sides of Eqs. (3.1.9)–(3.1.11), which vanish everywhere but on the surface of the container. They describe the contributions of the walls to the rates of change of the momentum and energy of the gas, since the walls can be sources and sinks of energy and momentum of the fluid.8 Forms of the conservation equations that apply to general fluids, not just dilute gases, can easily be obtained by considering the time rate of change of mass, momentum, and energy in a small volume in the fluid due to flows and to forces exerted by and work done by the surrounding fluid on the small volume. These general forms can be obtained using microscopic arguments based upon the Liouville equation [631, 294].

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The Boltzmann Equation 2: Fluid Dynamics

One important difference between the equations based upon the Boltzmann and Liouville equations comes from the fact that the Liouville equation method will include the contributions from the potential energies of interactions between the particles to the energy density and to the momentum and energy currents appearing in the conservation equations.9 The Constitutive Laws It is important to notice that the expressions for the two current densities, J i ,J K , contain only two quantities that cannot be immediately expressed in terms of the hydrodynamic densities, ρα (r,t), or more specifically, upon ρ(r,t),u(r,t), and eK (r,t). These quantities are the pressure tensor, P, appearing in the momentum current density and defined by  P(r,t) = mccf¯(r,c,t)dc, (3.1.15) and the internal energy flow vector, q, appearing in the energy current density and defined by  mc2 ¯ f (r,c,t)dc. (3.1.16) q(r,t) = c 2 The constitutive equations will, under the proper circumstances, allow us to express P and q in terms of spatial gradients of the hydrodynamic densities defined earlier.10 As we will see further on in the chapter, when we derive them from the Boltzmann equation, the constitutive laws for the currents in the conservation laws are an appropriate description for flows in a fluid only when the fluid is close to a state of local equilibrium where the fluid has at every point a well-defined local mass density, ρ(r,t); local mean velocity, u(r,t); and local energy density, eK (r,t) and associated local temperature, T (r,t); pressure, p(r,t); entropy density, s(r,t); and Helmholtz and Gibbs free energy densities, f (r,t) and g(r,t), respectively. One must suppose that these quantities change slowly with space and time, and on spatial and time scales that are large with respect to characteristic microscopic scales, such as the mean free time between collisions and the mean free path length of the gas particles. We remark that the requirement that the system be close to a state of local equilibrium does not necessarily imply that the system is close to a total equilibrium state. One might consider, for example, a system where the change of the temperature over the length of the container is large compared to the temperature at some point in the interior, but there is a small variation of the temperature over distances of the mean free path of the gas particles. The local equilibrium condition is a requirement that the local hydrodynamic densities are related to each other through standard thermodynamic relations,

3.1 The Chapman–Enskog Solution

67

generalized in such a way that thermodynamic quantities are replaced by their space and time-dependent analogs. That is to say, the temperature, T (r,t); pressure, p(r,t); and other thermodynamic variables are supposed to satisfy the same thermodynamic relations as do their equilibrium counterparts. For the case of a monatomic dilute gas, the local thermodynamic equations have the usual simple forms, such as the local caloric equation of state d (3.1.17) kB T (r,t), 2 where kB is Boltzmann’s constant. Eq. (3.1.17) defines the local temperature in terms of the local kinetic energy density. The local thermodynamic equation of state is eK (r,t) =

p(r,t) = n(r,t)kB T (r,t).

(3.1.18)

Also, all other local thermodynamic functions show the same relationships as for a monatomic ideal gas in equilibrium, but in a state characterized by local parameters, e.g. temperature and pressure. The Velocity Equation The constitutive laws express the currents appearing in the conservation laws in terms of the hydrodynamic densities as follows: For the conservation of momentum equation, we combine Eqs. (3.1.12) and (3.1.15) to obtain ∂ Fi ∂ρui =− [ρui uj + Pij ] + ρ(r,t), ∂t ∂xj m

(3.1.19)

where we use the usual summation convention for repeated indices. The constitutive equations for the components of the pressure tensor, P, for an isotropic fluid are of the form Pij = pδij − σij .

(3.1.20)

The two terms on the right-hand side of Eq. (3.1.20) require some explanation. The first term is the hydrostatic pressure in the fluid. If, in the absence of external forces, there is a gradient of this pressure in any direction, the hydrostatic forces will not be in balance and the fluid will be forced to flow. If only this term is retained in the expression for the pressure tensor, one obtains the Euler, or ideal fluid, equation for the momentum density. The Euler equation is invariant under the transformation t → −t,u → −u. That is to say, it is a reversible equation, and doesn’t, under usual circumstances, exhibit a decay to an equilibrium state.11 Thus, the Euler equation for the momentum density is given by ∂(ρu) F + ∇ · (ρuu) = −∇p + ρ. ∂t m

(3.1.21)

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The Boltzmann Equation 2: Fluid Dynamics

Irreversibility enters the equation for the fluid velocity when one introduces, for the second term on the right-hand side of Eq. (3.1.20), a constitutive relation known as Newton’s law of viscosity. This expresses the stress tensor σ as

∂uk ∂ui ∂uj 2 ∂uk σij = η + − δij + ζ δij . (3.1.22) ∂xj ∂xi d ∂xk ∂xk Here the summation convention is used again to denote a sum over repeated indices, and δij denotes a Kronecker delta function. Unlike the terms in the pressure tensor included in the Euler equation, Eq. (3.1.21), the stress tensor does change sign upon the time reversal transformation, and with its inclusion, the equation for the momentum density is no longer reversible. The coefficients η and ζ are called the coefficients of shear and bulk viscosity, respectively. The structure of Newton’s law, Eq. (3.1.22), is determined by the assumption that the fluid is isotropic and by the requirements that the stress tensor: (1) should contribute only when there is a relative motion between nearby parts of the fluid, such as would occur in a shear flow and (2) vanishes when the velocity is zero or when there is a uniform rotation of the fluid. In the latter case, the fluid velocity, u, is proportional to r ⊗ , for some angular velocity, , and the tensor σij vanishes. Combining the conservation equations for mass and momentum with the full expression given earlier for the pressure tensor and ignoring possible dependencies of shear and bulk viscosity on local parameters such as temperature and density, we obtain the following equation for the local velocity of the fluid at points away from the boundaries12 :



1 ∂u 2 ρ + (u · ∇)u = −∇p + η∇ u + ζ + η ∇(∇ · u). (3.1.23) ∂t 3 This is the well-known Navier–Stokes equation for the velocity field of a fluid. The Energy Equation Now we turn our attention to the energy density of the fluid. To obtain a useful equation for the energy density at a point not on the wall, we need the expression for the heat flow vector, q, defined by Eq. (3.1.16), appearing in the conservation law, Eq. (3.1.11). The constitutive equation for this current is given by Fourier’s law of heat conduction as13 q = −λ∇T ,

(3.1.24)

where the parameter λ is called the coefficient of thermal conductivity and T is the local thermodynamic temperature. Like the viscosities, the coefficient of thermal conductivity may depend upon space and time since it is, in principle, a function

3.1 The Chapman–Enskog Solution

69

of the local temperature and density. Returning to the energy conservation law, Eq. (3.1.14), and using the expressions (3.1.20) and (3.1.22) for the pressure tensor as well as Fourier’s law for the heat flow vector, we obtain an equation for the local kinetic energy density n

∂eK + nu · ∇eK + p∇ · u − σ : D + ∇ · q = 0. ∂t

Here the symmetric velocity gradient tensor, D, has elements   1 ∂ui ∂uj ; + Dij = 2 ∂xj ∂xi

(3.1.25)

(3.1.26)

p is the local hydrostatic pressure; and the stress tensor, σ , is defined in Eq. (3.1.22). If we drop the terms in Eq. (3.1.25) in which the transport coefficients η,ζ , and λ appear and use the local thermodynamic relations plus the continuity equation, we obtain the reversible, Euler equation for the local temperature, namely ∂(nT −d/2 ) + u · ∇[T −d/2 ] = 0. ∂t

(3.1.27)

This Euler equation for the temperature is clearly an equation for adiabatic, or constant entropy, motion of the fluid. The inclusion of Newton’s and Fourier’s laws makes the energy equation, Eq. (3.1.25), irreversible. The full equation, Eq. (3.1.25), is called the Navier–Stokes equation for the energy of a fluid and can be written in the form14

∂ ρT (3.1.28) + u · ∇ s = σij Dij + ∇ · (λ∇T ). ∂t In this form, one can see that the irreversible terms on the right-hand side of this equation are responsible for entropy production, both from the conduction of heat from higher to lower temperatures and through viscous friction. 3.1.2 Relevant Parameters for the Normal Solution of the Boltzmann Equation Much of the discussion in this chapter, and in the book as a whole, will focus on the identification of the relevant parameters for describing the system of interest. Often it is possible to use ratios of these parameters to construct small dimensionless parameters that may be useful for characterizing the physical state of the system as well as for obtaining series expansions of important physical quantities. For a dilute monatomic gas, the choice of the relevant physical parameters is quite clear. There are fundamentally three length scales. These are (1) the characteristic size of the gas atoms, a; (2) the mean free path length, l, between subsequent collisions of

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The Boltzmann Equation 2: Fluid Dynamics

the same gas particle; and (3) a macroscopic length, L, that may be characterized as the smallest wavelength of a non-negligible Fourier component of the distribution function. In many cases, this is of the order of the size of the bounded region in which the gas is contained. In other cases, it may be determined by the size of a macroscopic object out of equilibrium, such as a sphere dragged through or heated inside the gas. Finally, it may be the wavelength of a sound, heat, or shear wave present in the gas, but not related to the size of the system or of objects inside it. Dividing by the mean speed of the gas particles, v0 , one can turn these length scales into time scales, namely ta , the time a particle typically takes to travel a distance equal to the size of the particle; t , the mean free time between collisions; and a macroscopic time scale, tL , the typical time it would take a particle, moving without collisions, to cross a distance L.15 We can describe the density of the gas using the dimensionless parameter a/ ∼ na d . As we discussed earlier, dilute gases are characterized by a small value for this parameter – that is, a/ 1 – while for densely packed systems a/ ∼ 1 or larger. Another independent dimensionless parameter involving the size of the particles is the ratio a/L. For macroscopic systems, one has a/L 1, since the size of a macroscopic container is always many orders of magnitude greater than the size of a particle. However, the motion of atoms in nanotubes and in other nanostructures, which is of considerable importance, corresponds to situations where a/L ∼ 1. The third possible dimensionless ratio of lengths, K = /L, is called the Knudsen number, and it will be of special importance for this chapter. For a dilute gas, the region where K 1 can be considered to be the hydrodynamic regime, while the region for which K 1 is the rarefied or Knudsen regime. One can understand the reason for referring to K 1 as the hydrodynamic regime in the following way. In this regime, particles on average make many collisions with other gas particles between any two encounters with the wall of the container. These collisions allow for the eventual establishment of a state of local equilibrium throughout the system. If there are temperature, density, or velocity gradients in the system, the hydrodynamic densities vary from place to place in the container. These variations may be characterized in many cases by conditions at the walls of the container, such as different temperatures on different sides of the container, producing a temperature gradient across the system, or a non-zero relative velocity between two opposite walls, producing a shear flow, for example. Physically, a necessary condition for the hydrodynamic equations to yield a valid description of the state of the gas is that the variations of the hydrodynamic fields over a mean free path are small compared to the variations of these fields over the typical macroscopic length scale L. This condition can be formulated as  |∇ρ| 1, |ρ|

(3.1.29)

3.2 General Properties of the Chapman–Enskog Solution

71

where ρ is one of the hydrodynamic densities. Since the gradient of a hydrodynamic field is of order ρ/L, when boundary conditions are responsible for the presence of gradients, the condition expressed by Eq. (3.1.29) leads to the requirement that K 1. For systems where L corresponds to the system size, the Knudsen regime, K >1, is one in which the gas particles collide with the walls more often than they collide with each other. In cases where L corresponds to the size of an internal object, one has a Knudsen regime in a region with a size of a few mean free paths around each such object. This is the important regime for rarefied gas dynamics. Interesting questions arise when one tries to understand the properties of transport phenomena in gases over a range of values of K, say from the hydrodynamic to the Knudsen regimes. We can summarize this discussion with a few simple inequalities. • A dilute gas is characterized by the requirement that a l

or,equivalently,

(3.1.30)

na d 1. For a dense gas, the mean free path length, , and the size of a particle are comparable, and at high densities, the mean free path may even be smaller than the size of a particle. • The hydrodynamic regime for a fluid of any density is characterized by the requirement that  L.

(3.1.31)

• A Knudsen, or rarefied, gas is characterized by the requirement that L ≤ .

(3.1.32)

3.2 General Properties of the Chapman–Enskog Solution For the remainder of our discussion of the Chapman–Enskog solution, we will consider the hydrodynamic regime for a dilute gas, for which the following inequalities obtain a  L.

(3.2.1)

Nonequilibrium gases satisfying these conditions are often described by hydrodynamic equations, supplemented by appropriate boundary conditions at the walls of the system.16 Therefore, it should be possible to construct solutions of the Boltzmann equation that at each instant of time can be characterized completely by the hydrodynamic densities and whose time evolution therefore can be described by the solutions of the hydrodynamic equations. On the other hand,

72

The Boltzmann Equation 2: Fluid Dynamics

these solutions should follow from the Boltzmann equation alone and therefore the transport coefficients, which are free parameters in the hydrodynamic equations must be fixed by the Boltzmann equation. The Chapman–Enskog solutions of the Boltzmann equation are precisely these solutions. They are obtained by solving the Boltzmann equation (3.1.2) without boundary term, while the boundary conditions satisfied by the hydrodynamic densities are obtained by means of the general boundary condition, Eq. (3.1.3), applied to the Chapman–Enskog solution of Eq. (3.1.2). Central to the construction of the Chapman–Enskog solution is a picture of the evolution of the distribution function f˜(r,v,t) for a gas in the hydrodynamic regime starting from some initial distribution. In this picture, the collisions between the gas molecules produce drastic changes in the distribution function over a few mean free times. In this period, the Boltzmann collision integral leads to changes in the distribution function of typical magnitude νc |f˜(r,v,t)|, where νc = v0 / is the average collision frequency and v0 is a typical velocity of a particle. The collisions drive the gas toward a state close to a local equilibrium state, where the changes in the H -function are driven by the hydrodynamic processes in the gas and take place in time scales that are large compared to the time between collisions.17 The specific form of such a local equilibrium state is is a local Maxwell–Boltzmann distribution, f˜loc (r,v,t), with spatial and time dependent density, n(r,t); temperature, T (r,t); and mean velocity, u(r,t). In d dimensions, it is given by

 β(r,t)m d/2 − β(r,t)m (v−u(r,t))2 2 ˜ floc (r,v,t) = n(r,t) e . (3.2.2) 2π Here β(r,t) = (kB T (r,t))−1 . Once the gas has become close to a local equilibrium state, the slow hydrodynamic processes described by the equations of fluid dynamics are responsible for producing a uniform density, temperature, and mean velocity throughout the system, unless external circumstances – such as walls maintained at different temperatures – preserve the nonuniformities. The typical time scale for this is tL . Still, even during this stage, it is the collisions between gas molecules that are responsible for the further, slow decrease of the H -function.18 3.3 Solving the Boltzmann Equation for the Hydrodynamic Regime The essence of the Chapman–Enskog solution, as suggested in the previous section, is (1) to suppose that after a few mean free times, the collisions taking place in the gas produce a state that is determined by the hydrodynamic densities alone (apart from the boundary conditions); and then, (2) to express the solution of the Boltzmann equation as an expansion in powers of the gradients of the

3.3 Solving the Boltzmann Equation for the Hydrodynamic Regime

73

hydrodynamic fields, assumed to be characterized by a small parameter μ, which is roughly of order /L. As we will see, the zeroth-order term in this expansion will be a local equilibrium distribution. The hydrodynamic fields determining its precise form a priori are fully arbitrary, but for the expansion to make sense, one has to require that they vary on the length scale L only. In accordance with this approach, we write19 ˜ 1 (r,v,t) + μ2  ˜ 2 (r,v,t) + · · · ]. f˜(r,v,t) = f˜loc (r,v,t)[1 + μ

(3.3.1)

˜ n (r,v,t) are to be determined from the Boltzmann equation, Here the functions  together with and as functions of the hydrodynamic variables n(r,t),T (r,t) and u(r,t) and products of powers of gradients acting upon these. Here μ is a small parameter on the order of the Knudsen number, and we use it in what follows to order the terms in the gradient expansion of the distribution function, Eq. (3.3.1). To make this expansion more understandable, we take a closer look at the various terms in Eq. (3.1.2). The typical magnitude of the collision term, on the right-hand side, is νc |f˜|, νc the collision frequency. This is clearest from the analysis of the loss term made in Section 2.1.2. Since the normal solution is supposed to depend on position through the hydrodynamic densities alone, which vary on the length scale L, the ∇r · (vf˜) term on the left-hand side of Eq. (3.1.2) can be estimated to be of order (v0 /L)|f˜|. The same estimate can be made for the ∇v · (Ff˜/m) term by observing that macroscopic potential gradients will typically be of order kB T /L. Finally, since also the time derivative of f˜ completely results from its dependence on the hydrodynamic fields, one may anticipate on the basis of the hydrodynamic equations that ∂ f˜/∂t will be of order (c/L)|f˜|, with c being the sound velocity, which is comparable to v0 . The most straightforward way of solving the Boltzmann equation perturbatively now would be expanding both the distribution function and the hydrodynamic fields into series in powers of μ, determining the zeroth order from the equation J (f˜0, f˜0 ) = 0,

(3.3.2)

˜ n , together with subsequent and solving successively for subsequent orders  contributions to the hydrodynamic fields, starting with f˜0 as a total equilibrium distribution function This procedure is known as the Hilbert expansion.20 It works in principle, but to reproduce e.g. a solution of the nonlinear Navier–Stokes equations over a not even very long time, one already has to iterate over many orders.21 In contrast, the Chapman–Enskog solution reproduces the full Navier– Stokes equations already after two iterations. The price paid for this is that it cannot be considered a fully systematic expansion of the solution in powers of μ, but it leads to a systematic expansion of the hydrodynamic equations in powers of μ.

74

The Boltzmann Equation 2: Fluid Dynamics

The starting point of the Chapman–Enskog procedure is also Eq. (3.3.2), but here the zeroth-order term, f0 , is taken to be a local equilibrium distribution function with space- and time-dependent local density, temperature, and mean velocity.22 ˜ 1,  ˜ 2, . . . ,  ˜ j , . . ., are solved Then, as in the Hilbert solution, the functions  successively. The difference, however, is that the time and position dependence of ˜ j , are expressed the hydrodynamic fields – in terms of which all these functions,  – are determined from the full set of equations through order j in powers of the ˜ j is expressed in terms of local density, gradients. That is, each of the functions  velocity and temperature, including up to j th-order gradients of these, but the actual time and position dependence of these fields can be determined only after deriving and solving the hydrodynamic equations through order j . As a consequence of this procedure, the time derivative of the distribution function, appearing on the left-hand side of Eq. (3.1.2), is not simply of order μ because the time derivatives of the hydrodynamic fields contain terms of all orders in μ, starting with the first order in μ. In this expansion procedure, the fields themselves are treated at all times as being of order μ0 , while each spatial derivative is treated as producing an additional order of μ. The local equilibrium distribution, which is taken to be the general solution of Eq. (3.3.2), contains the full hydrodynamic fields for all times. Chapman and Enskog found out the solution method can be made unique in a simple and useful way by requiring that this zeroth-order solution alone reproduces the full hydrodynamic densities for all times and positions, viz.   (3.3.3) ψ(v)f˜(r,v,t)dv = ψ(v)f˜loc (r,v,t)dv, where the ψ is any linear combination of the conserved quantities, m, v, and mv 2 /2. defined just before Eq. (3.1.4). This condition implies that  ˜ j f˜loc (r,v,t)dv = 0, for j = 1,2, . . . . ψ(v) (3.3.4) Before turning to the actual details of the Chapman–Enskog expansion, we point out that the pressure tensor and the heat flow vector defined by Eqs. (3.1.15) and (3.1.16), respectively, will also have expansions in powers of μ when the proposed solution, Eq. (3.3.1), is inserted in the integrals appearing in Eqs. (3.1.15) and (3.1.16). One obtains ⎛ ⎞   n(r,t) ¯ j (r,c,t)⎠ dc μj  (3.3.5) P(r,t) = 1 + ccf¯loc (r,c,t) ⎝ β(r,t) j

3.3 Solving the Boltzmann Equation for the Hydrodynamic Regime

75

and  q(r,t) =

⎛ c(mc2 /2)f¯loc (r,c,t) ⎝



⎞ ¯ j (r,c,t)⎠ dc. μj 

(3.3.6)

j

Here 1 is the d × d unit tensor, and we have expressed the distribution function in terms of the peculiar velocity, c. 3.3.1 The Boltzmann Equation Expanded in Powers of μ If we insert the proposed solution, Eq. (3.3.1) into Eq. (3.1.2), we obtain23   ∂ ˜ 1 (v) + · · · ] + v · ∇r + F · ∇v f˜loc (v)[1 + μ ∂t ˜ 1 (v) = J (f˜loc, f˜loc ) + μf˜loc loc (v)   ˜ 2 (v) + J (f˜loc  ˜ 1 (v), f˜loc  ˜ 1 (v1 )) + μ2 f˜loc loc (v) + O(μ3 ),

(3.3.7)

where the linearized Boltzmann collision operator, loc (v), is defined by ˜ = J (f˜loc , f˜loc ) + J (f˜loc, f˜loc ) f˜loc (v)loc (v)(v)  ˜ ˜ 1 )] = f˜loc (v) dv1 dr 1 f˜loc (v1 )T¯ 0 (r,v,r 1,v1 )[(v) + (v = f˜loc (v)



dv1 f˜loc (v1 )



ˆ ˆ ˆ (v,v1 ) − 1) × [(v) ˜ ˜ 1 )]. d kB(g, k)(b + (v k (3.3.8)

Here we used the definition of the binary collision operator, T¯ 0, given by Eq. (2.1.39) of the previous chapter. The order in powers of the parameter μ of the terms appearing on the right-hand side of Eq. (3.3.7) is immediately clear, and we can also easily assess the ordering of terms on the left-hand side. As noted in the previous subsection, both gradient terms raise the order of terms they act upon by one power of μ. The time derivatives of the various contributions to f˜(r.v,t) all result from the time dependence of the hydrodynamic densities. These in general contain higher-order gradients and powers of gradients, so it is reasonable to suppose that the time derivatives of these densities, ρi (r,t), are given by expansions in powers of μ, which we write in the form

76

where

The Boltzmann Equation 2: Fluid Dynamics



∂ρi (r,t) ∂ρi (r,t) 2 ∂ρi (r,t) +μ + ··· , =μ ∂t ∂t ∂t 1 2 ∂ ∂t j

(3.3.9)

should be understood to represent terms of order μj .

˜ j can be deterExplicit expressions for the equations of order μj , from which  mined, must be obtained by combining the conservation equations with the explicit form of the μ expansion of the distribution function. A crucial observation in this connection is that at each order of μ, the time derivative on the left-hand side of the Boltzmann equation will depend only upon lower-order terms in the expansion of the distribution function, since the time derivative itself is at least of order μ. This feature of the Chapman–Enskog expansion makes it possible to determine all terms in the series successively. A feature of the Chapman–Enskog solution that will have a strong echo when we consider the kinetic theory of dense gases is the interesting functional dependence of the distribution function upon the hydrodynamic densities.24 As noted, the distribution function depends upon space and time only through the hydrodynamic densities. There is no explicit space or time dependence. The space and time dependence of the hydrodynamic densities themselves are given by explicit solutions of the hydrodynamic equations, once the transport coefficients and boundary conditions are determined. 3.3.2 The Distribution Function to Zeroth Order in μ and the Euler Equations We will follow this procedure order by order for the first few orders, so as to obtain the Euler and then the Navier–Stokes equations of hydrodynamics. In this subsection, we will consider the distribution function to zeroth order in the parameter μ, leading to the Euler equations for the hydrodynamic densities. In fact, we have already assembled all the pieces needed for the calculation to this order in μ. The left-hand side of the Boltzmann equation, as we argued earlier, is at least of order μ. Thus, to zeroth order in μ, the Boltzmann equation, as noted before, is simply J (f˜0, f˜0 ) = 0,

(3.3.10)

where f˜0 is the zeroth order approximation to the distribution function. We have already identified the solution for this with the local equilibrium distribution function f˜0 (r,v,t) ≡ f˜loc (r,v,t),

(3.3.11)

3.4 The Distribution Function to First Order in μ: The Navier–Stokes Equations 77

where the local equilibrium distribution has the Maxwell–Boltzmann form and is given by Eq. (3.2.2). The derivation of this result has been discussed before, in Eqs. (2.2.33–2.2.36) of subsection 2.2.3. Now to zeroth order in μ, the pressure tensor and heat flow vector are given by the zeroth-order terms on the right-hand sides of Eqs. (3.3.5) and (3.3.6), respectively, as P0 (r,t) = n(r,t)kB T (r,t)1, and

(3.3.12)

q 0 (r,t) = 0.

(3.3.13)

If we insert these expressions for the pressure tensor and heat flow vector in the conservation equations, we obtain the d + 2 Euler, or ideal fluid, equations mentioned earlier, namely

∂ρ + ∇ · (ρu) = 0; (3.3.14) ∂t 1

∂u ρ + ρu∇ · u = −∇p + Fρ; (3.3.15) ∂t 1   dnkB ∂T (3.3.16) + u · ∇T + nkB T ∇ · u = 0. 2 ∂t The preceding equation for the temperature can be rewritten in a way that one can see that it describes the adiabatic motion of the gas. That is, 

 ∂ + u · ∇ (nT −d/2 ) = 0. (3.3.17) ∂t 1 So the Euler or ideal fluid equations are the hydrodynamic equations resulting from terminating the Chapman–Enskog expansion at zeroth order. In addition, they reproduce the first-order time derivatives of the hydrodynamic densities. From Eqs. (3.3.15) and (3.3.17), one may conclude that the motion of an ideal fluid is both inviscid and adiabatic. For a gas inside a container, these equations must be supplemented by boundary conditions, which we discuss in Section 3.6. 3.4 The Distribution Function to First Order in μ: The Navier–Stokes Equations ˜ 1, which will lead us to the Navier– Now we turn our attention to determining  Stokes equations of fluid dynamics and provide us with expressions for the coefficients of viscosity and thermal conductivity in terms of the intermolecular forces that govern the binary collisions taking place in the fluid. In accordance with our previous discussions, the order μ terms in the Boltzmann equation lead directly to the equation

78

The Boltzmann Equation 2: Fluid Dynamics



 ∂ f˜loc (r,v,t) + v · ∇r f˜loc (r,v,t) + F · ∇v f˜loc (r,v,t) ∂t 1   ˆ f˜loc (r,v,t)f˜loc (r,v1,t) = d kˆ dv1 B(g, k) ˜ 1 (r,v1,t) −  ˜ 1 (r,v,t) −  ˜ 1 (r,v1,t)]. ˜ 1 (r,v,t) +  × [

(3.4.1)

The subscript on the time derivative in the preceding equation serves as a reminder that we retain only terms of order μ when calculating the time derivative. To determine these terms, we write       ∂ f˜loc (r,v,t) ∂ f˜loc (r,v,t) ∂n(r,t) ∂ f˜loc (r,v,t) ∂u(r,t) = + · ∂t ∂n(r,t) ∂t ∂u(r,t) ∂t 1 1 1   ∂ f˜loc (r,v,t) ∂β(r,t) + . (3.4.2) ∂β(r,t) ∂t 1 These time derivatives, to first order in the gradients, are obtained from the Euler equations. For our purpose, it is useful rewriting these as equations for the time derivatives of number density, local velocity, and temperature. The time derivative of the particle density is given, in fact to all orders, but also to first order by   ∂n(r,t) = −∇ · (n(r,t)u(r,t)). (3.4.3) ∂t 1 Using this in Eq. (3.3.15), one finds the time derivative of the local velocity is given by   1 ∂u(r,t) ∇p(r,t), (3.4.4) = −(u(r,t) · ∇)u(r,t) + F (r) − ∂t mn(r,t) 1 where p is the local pressure satisfying p(r,t) = n(r,t)kB T (r,t). The Euler equation for the time derivative of the temperature follows from Eq. (3.3.15) after some straightforward algebra as   ∂T (r,t) 2T (r,t) = −u(r,t) · ∇T (r,t) − ∇ · u(r,t). (3.4.5) ∂t d 1 Inserting these time derivatives in the left-hand side of Eq. (3.4.1), we obtain the following integral equation for 1 , expressed in terms of the peculiar velocity25 

 2

2 c mc d + 2 − c · ∇ ln T (r,t) f¯loc (r,c,t) β(r,t)m cc − 1 : D + β d 2 2 ¯ loc (c) ¯ 1 (c). (3.4.6) = f¯loc (r,c,t)

3.4 The Distribution Function to First Order in μ: The Navier–Stokes Equations 79

The Linearized Boltzmann Collision Operator ¯ loc (c), appearing in this equation, The linearized Boltzmann collision operator,  written in full, is given by Eq. (3.3.8), is   ˆ ¯ ¯ loc (c)(r,c,t) ≡ dc1 d kˆ f¯loc (r,c1,t)B(g, k)   ¯ ¯ ¯ ¯ ,t) + (r,c × [(r,c 1,t) − (r,c,t) − (r,c 1,t)]. (3.4.7)

This operator is symmetric and negative definite  outside of the space of conserved ¯ ¯ h(c), for any two funcquantities, with an inner product of the form dcφ¯ 0 (c)g(c) ¯ ¯ tions, g, ¯ h, with weight function, φ0 (c), a normalized Maxwell–Boltzmann distribution function. It depends implicitly on the temperature, density, and average velocity at the position r. Eigenfunctions belonging to different eigenvalues are orthogonal to each other. The proof of the negative definite property follows the lines of the proof of the H -theorem, and is simply the statement that     1 ˆ ¯ loc (c)ψ(c) ¯  ¯ =− dcf¯loc (r,c,t)ψ(c) dc dc1 d kˆ f¯loc (c)f¯loc (c1 )B(g, k) 4 ¯ ¯ 1 ) − ψ(c ¯  ) − ψ(c ¯ 1 ) 2 < 0, × ψ(c) + ψ(c (3.4.8) ¯ provided ψ(c) is not a linear combination of quantities conserved in a binary collision. We list here some important properties of the linearized collision operator since they will be useful as we proceed [76, 265, 266, 267, 149, 87, 82, 83, 477, 85]. • Since the operator is isotropic in velocity space, the eigenfunctions will take the form of a product of a function of c2 with an element of an irreducible tensor in the d-dimensional velocity space. As a result, the eigenvalues will have a degeneracy equal, apart from accidental degeneracies, to the number of spherical harmonics needed to represent all the elements of the particular irreducible tensor. For three-dimensional systems, for example, the degeneracy is 2l + 1, corresponding to the spherical harmonics. • The solutions of the eigenvalue equation for the collision operator depend upon the intermolecular potential and upon the space of allowable functions ψλ . The dependence upon the space of allowable functions arises from the fact that the eigenvalue spectrum typically has both discrete values and a continuous range of values. The eigenfunctions corresponding to the discrete spectrum typically belong to a space of square-integrable functions, but the eigenfunctions corresponding to the continuous spectrum are distributions, which do not belong to this space.

80

The Boltzmann Equation 2: Fluid Dynamics

ˆ is independent • For some intermolecular potential models, the function B(g, k) ˆ of g and is a function of k alone. Molecules that interact according to such potentials are called Maxwell molecules. The potential energy of interactions of such molecules is repulsive and an inverse power of their interparticle separation. Maxwell potentials have the form φ(r) ∼ r −s , where s = 2(d − 1) in d dimensions. The fact that the collision operator is independent of g means that polynomial functions of degree n will remain polynomials of degree n when the operator acts on them. This allows for some simplification of the eigenvalue problem, and, in fact, for the special case of three-dimensional Maxwell molecules, the complete spectrum of the linearized Boltzmann collision operator is known. It is discrete, with values ranging between zero and negative infinity. There is, of course, a gap between the zero eigenvalues and the first non-zero, negative value, which is the slowest decay rate of the decay of a velocity perturbation about a Maxwell–Boltzmann equilibrium distribution. The eigenfunctions take the form

mc2 l m (r) ψrlm (c) = Nrlm Sl+ 1 β c Yl (θ,φ), (3.4.9) 2 2 (r) where the Ylm are spherical harmonics and the Sl+ 1 are Sonine polynomials, pro2

portional to the associated Laguerre polynomials. The quantities Nrlm are normalization constants. The eigenvalues have the 2l + 1-fold degeneracy mentioned before. For completeness, we mention that the Sonine polynomials, Sm(r) (x), can be defined as the coefficients of t r in the expansions in powers of t of of a set of generating functions, viz. xt

(1 − t)−m−1 e− 1−t =

∞ 

t n Sm(r)(x) .

(3.4.10)

r=0

Explicit expressions for the polynomials are Smr (x) =

r  p=0

(−x)p

(m + 1 + r) , p! (n − 1)! (m + 1 + p)

(3.4.11)

with m being an arbitrary real number. ¯ loc (c) is composed of a continuum • For hard-sphere molecules, the spectrum of  extending from some finite, negative value to negative infinity, as well as an infinite number of discrete values within a finite negative range, in addition to the value zero [149, 477, 266, 267]. • Since both mathematically and numerically it is hard to deal with potentials of infinite range, it is customary to consider cutoff potentials where the potential is set equal to zero for intermolecular separations greater than some given value, or where the cross section is set equal to zero for grazing collisions (often referred to

3.4 The Distribution Function to First Order in μ: The Navier–Stokes Equations 81

as an “angular cutoff”). There is an extensive mathematical literature on various aspects of the spectrum of the linearized Boltzmann collision operator to which we refer the interested reader [266, 267, 226, 85, 83, 667]. • For three-dimensional systems, H. Grad proved that if the potential is repulsive and varies with the separation, r, as φ(r) = Kr −n , for all values of r, then the spectrum depends on the value of n, of course. For potentials that decrease more rapidly with r than Maxwell molecule potentials, there is a continuous part of the spectrum that is bounded away from zero. If the potential decays more slowly, the continuum extends to the value zero, and zero, in addition, is an accumulation point of the discrete spectrum [266, 267]. • In our discussions, we will assume that the potential is such that there is a gap between the zero eigenvalues and the first non-zero value of the spectrum. This assumption is sufficient to ensure that a small, spatially homogeneous perturbation away from equilibrium will decay back to equilibrium at an exponential rate. We now turn to the construction of the solutions to the integral equation, Eq. (3.4.6), and the calculations of the coefficients of shear viscosity and thermal conductivity.

3.4.1 The First Sonine Polynomial Approximation for the Solution of the Linearized Boltzmann Equation By examining Eq. (3.4.6) and separating the different tensor forms, we obtain two equations, each of which determines different tensorial contributions to the ¯ 1 as a ¯ 1 (r,c,t). We use velocity variables c = v − u and write  expression for  linear combination of two different irreducible tensors in velocity, c, space, in the form   c2 2 2 ¯ : D, (3.4.12) 1 (r,c,t) = A(c )c · ∇ ln T + B(c ) cc − 1 d where A(c2 ) and B(c2 ) are scalar functions of the peculiar velocity. Then, using the ¯ loc (c) is an isotropic operator in velocity space, we write Eq. (3.4.6) as fact that  two separate equations. The equations are   

 c2 c2 2 ¯ ¯ ¯ : D = floc (r,c,t) βm cc − 1 : D , floc (r,c,t)loc (c)B(c ) cc − 1 d d (3.4.13) which determines the response of the gas, to first order in μ, to a gradient in the local velocity field, and 

 mc2 d + 2 2 ¯ ¯ ¯ − c · ∇ ln T , floc (r,c,t)loc (c)A(c )c · ∇ ln T = floc (r,c,t) β 2 2 (3.4.14)

82

The Boltzmann Equation 2: Fluid Dynamics

which determines the first-order response of the gas to a temperature gradient. Equations (3.4.13) and (3.4.14) are linear, inhomogeneous integral equations. Only the particular solutions orthogonal to 1, c, and c2 are important because of the conditions imposed on the solutions by Eq. (3.3.4).26 A standard way of solving integral equations of this type is to assume that the solution has an expansion in terms of a complete set of functions of velocity and then to use the equation to determine the coefficients. There are many possible complete sets of functions that are appropriate for use as expansion functions. If we were to use Cartesian coordinates to describe points in d-dimensional velocity space, for example, then products of d Hermite polynomials, one for each Cartesian direction, would be a standard choice. Since we know from the simple arguments given before that the functions A and B can only be functions of c2 , it is simpler to expand these functions in complete sets of functions of the variable c2 , equivalent to expanding the solutions of Eq. (3.4.6) in a complete set of functions appropriate for d-dimensional polar coordinate systems. The most familiar functions of this type, for three-dimensional systems, are products of spherical harmonics for the angular variables with generalized Laguerre polynomials for the radial variable. In kinetic theory, it is customary to use a somewhat different, but closely related, set of functions for the radial coordinate, namely, the generalized Sonine polynomials, which we encountered already in the exact expressions for the eigenfunctions of the ¯ loc (c), for three-dimensional Maxwell molecules. linearized collision operator,  For two-dimensional systems, the spherical harmonics are replaced by angular functions, exp (imθ), where m is a positive or negative integer, and the radial functions are again Sonine polynomials, but different ones than those used for three-dimensional systems. Since the details of constructing the solutions of these equations are quite standard, we will not present them here [89, 215]. Instead, we ¯ loc (c), will express the solutions in terms of the inverse of the collision operator  which is well defined when acting on functions orthogonal to the conserved quantities, 1,c,mc2 /2. We will then be able to obtain simple approximate solutions to these equations. The approximate solution we are about to obtain corresponds to using the first term in the Sonine polynomial expansion. It is called the first Enskog approximation, and by including further polynomials in the expansion of the solution, one can obtain further Enskog approximations. For the special case of Maxwell molecules, the first Sonine polynomial approximation is exact since, as mentioned before, the eigenfunctions are, in combination with spherical harmonics, eigenfunctions for the linearized Boltzmann equation for these particles. We begin by obtaining the formal solutions to Eqs. (3.4.13) and (3.4.14). Thus, we write 

  c2 c2 −1 ¯ : D = loc βm cc − 1 : D , B(c ) cc − 1 d d 

2

(3.4.15)

3.4 The Distribution Function to First Order in μ: The Navier–Stokes Equations 83

and −1

¯ loc A(c2 )c · ∇ ln T = 

  mc2 d + 2 − c · ∇ ln T . β 2 2

(3.4.16)

3.4.2 The Coefficients of Thermal Conductivity and Shear Viscosity The heat flow vector, to first order in μ, denoted by q 1 , is now given by combining the general expression for the heat flow vector, Eq. (3.1.16), with the first-order correction to the local equilibrium distribution function. We find27  mc2 q 1 = dcf¯loc cA(c2 )c · ∇ log T 2 

    1 mc2 d + 2 ¯ −1 mc2 d + 2 ¯ dcfloc β = − cloc β − c · ∇ ln T = −λ∇T . β 2 2 2 2 (3.4.17) An expression for the coefficient of thermal conductivity, λ, can be obtained by letting the direction of the temperature gradient be in the x-direction, so that we may write 

    2 2 d + 2 d + 2 mc mc −1 ¯ loc β − cx  − cx . (3.4.18) λ = −kB dcf¯loc β 2 2 2 2 A similar expression can be given for the coefficient of shear viscosity using Eqs. (3.1.15) and (3.4.15). It is sufficient to consider only the xy component of the pressure tensor and the case where the fluid velocity u = yˆ uy (x). Then the macroscopic form of the pressure tensor is given by Newton’s law Pxy = −η

∂uy (x) , ∂x

and the Boltzmann equation result is  ∂uy (x) ¯ −1 , Pxy = m dcf¯loc cx cy  loc βmcx cy ∂x leading to a formal expression for the shear viscosity,  2 ¯ −1 dcf¯loc cx cy  η = −βm loc cx cy .

(3.4.19)

(3.4.20)

(3.4.21)

Note that the presence of the local equilibrium distribution function in the preceding expressions for the transport coefficients indicates that the transport coefficients themselves are functions of position and time. This is not surprising and this dependence must be taken into account in the most general applications of fluid dynamics. The formal expressions for the transport coefficients, Eq. (3.4.18) and (3.4.21), are examples of expressions that will appear often as we develop kinetic theory in this

84

The Boltzmann Equation 2: Fluid Dynamics

and in later chapters. This is the form that results from the projection operator solution of the linearized Boltzmann given later in this chapter, and related forms will result from evaluations of the Green–Kubo time correlation function expressions for transport coefficients for a general fluid [268, 269, 405, 408]. These will be discussed in Chapters 6 and 12–14, as we develop methods for describing transport phenomena in dense fluids. It is now important that we find a way to evaluate these formal expressions. The idea of the first Sonine approximation is to suppose that the action of the inverse linearized Boltzmann operator on some function, for which this inverse is well defined, is proportional to the function itself. Thus we would write, for example, ¯ −1  loc cx cy ≈ αcx cy ,

(3.4.22)

¯ loc cx cy . cx cy ≈ α 

(3.4.23)

or

This is, in fact, what one would obtain by using the appropriate first Sonine polynomial approximation in solving the linearized Boltzmann equation. The proportionality coefficient, α, is then obtained by multiplying both sides by f˜loc cx cy and integrating over all velocities. For a wide range of potentials, this procedure leads directly to a good approximation for the two transport coefficients given by  dcf¯loc cx2 cy2 , (3.4.24) η ≈ −βm  ¯ loc cx cy dcf˜loc cx cy  and

2  mc2 d + 2 ¯ dcfloc β cx2 − 2 2 λ ≈ −kB    .   2 2 mc mc d + 2 d + 2 ¯ loc β cx  cx dcf¯loc β − − 2 2 2 2 

(3.4.25)

The negative signs in these expressions guarantee the positivity of the transport coefficients since the collision operator is negative definite. It is important to note that η and λ given above are independent of the gas density, n. The values of integrals in the denominators depend upon the potential energy used to describe the interactions of the gas particles. For hard-sphere molecules in two or three dimensions, the coefficient of thermal conductivities are [89, 156] 1/2

λ(2) =

2kB3 T aπm

λ(3) =

75kB3 T 1/2 , 64a 2 πm

, (3.4.26)

3.4 The Distribution Function to First Order in μ: The Navier–Stokes Equations 85

where a is the diameter of the disks or spheres. For most other intermolecular potentials, the corresponding expressions for the thermal conductivity must be obtained numerically. The general form of the expression for the thermal conductivity for three-dimensional systems in this approximation is λ(3) =

75kB2 T , 32m(2,2) (T )

(3.4.27)

where (2,2) (T ) is an integral involving the intermolecular potential, through the ˆ For a number of potentials, it is tabulated in various scattering function B(g, k). places, including the books by Chapman and Cowling [89], and by Hirschfelder, Curtiss, and Bird [306]. We shall return to the results obtained here after a brief discussion of the corresponding calculations for the coefficient of shear viscosity. Expression (3.4.24) for the coefficient of shear viscosity can easily be evaluated for hard-disk and hard-sphere particles, and one finds that in this approximation,

m 1/2 1 (2) , η = 2a βπ

5 m 1/2 (3) . (3.4.28) η = 16a 2 βπ Results for the low-density values for the coefficient of shear viscosity for other potential energy functions can be evaluated numerically and are available in the literature for a wide variety of potential functions. The general expression for threedimensional systems, similar to that for the thermal conductivity, is η(3) =

5kB T , 8(2,2) (T )

(3.4.29)

where the collision integral (2,2) (T ) is the same one appearing in Eq. (3.4.27) for the thermal conductivity. Note that, in this first Sonine polynomial approximation, the thermal conductivity and shear viscosity are linearly related, and the proportionality factor, 5cv /2, is called the Eucken factor [89]. That is, λ(3) =

5cv (3) η , 2

(3.4.30)

where cv is the isochoric specific heat. We have used the dilute gas value, cv = 3kB /2 in Eq. (3.4.30). We mention in passing that since these expressions are obtained using the local equilibrium distribution functions, the temperature variable appearing in them is, in general, a function of position. This spatial dependence of the temperature is usually dropped or ignored for systems close to total equilibrium.

86

The Boltzmann Equation 2: Fluid Dynamics

3.4.3 The Navier–Stokes Equations To summarize, we have determined the terms of order μ0 and order μ1 in the μ-expansion of the distribution function. These terms can be used to evaluate the heat flow vector and the pressure tensor to the same order in μ. When these expressions are inserted in the conservation laws, we then obtain equations for the hydrodynamic densities, for a monatomic dilute gas, that consistently include all terms up to and including terms of order μ2 . These are the Navier–Stokes equations for a dilute monatomic gas, for which we have obtained explicit expressions for the transport coefficients in terms of the interatomic potential to any desired precision, in principle, using the Sonine polynomial expansion of the first-order distribution function 1 . For such dilute gases, the Navier–Stokes equations are given by dn + n∇ · u = 0, dt

  1 du = nF − ∇(nkB T ) + ∇ · 2η D − 1(∇ · u) , mn dt d   dT d 1 2 nkB + nkB T (∇ · u) = ∇ · λ∇T + 2η D : D − (∇ · u) , 2 dt d

(3.4.31) (3.4.32) (3.4.33)

where the the tensor D is defined by Eq. (3.1.26), and the total time derivative is given by d ∂ = + u · ∇. (3.4.34) dt ∂t For a more general fluid, the only changes in the forms of these equations are that the ideal gas pressure, a valid approximation for dilute gases, is replaced by the appropriate fluid pressure, and there is an additional bulk viscosity term in the stress tensor, as described earlier in this chapter. In addition, one must generalize the Boltzmann equation to higher densities or use another method, such as the Green– Kubo time correlation function method, for calculating the transport coefficients in these systems. We postpone discussions of this issue to later chapters, where we address this problem.

3.4.4 Recapitulation Before discussing the question of whether or not the normal solution satisfies the proper boundary conditions on the distribution function at the walls of the container, it is worthwhile to summarize the principal assumptions used in the derivation of the Navier–Stokes equations. These are as follows: 1. The gas evolves on two time scales, a mean free time scale, t , and a hydrodynamic time scale, tL , with corresponding length scales,  and L, respectively.

3.4 The Distribution Function to First Order in μ: The Navier–Stokes Equations 87

2. The time and length scales are such that t tL and  L. 3. For t tl , the system is close to a local equilibrium state and the distribution function f¯(r,c,t) may be expanded as a power series in the parameter μ ∼ /L as ¯ 1 + μ2  ¯ 2 + · · · ), f¯(r,c,t) = f¯loc (r,c,t)(1 + μ

(3.4.35)

where f¯loc (r,c,t) is the local equilibrium distribution function. 4. To order μ, the distribution function is given by  

  c2 f¯(r,c,t) = f¯loc 1 + A(c2 )c · ∇ ln T + f¯loc B(c2 ) cc − 1 :D . d (3.4.36) ¯ j is of order j + 1 and higher in the gradients. 5. The spatial gradient of μj  With the aid of these assumptions we were able to decompose both sides of Eq. (3.1.2) into a formal power series in μ, and then we equated coefficients of equal powers of μ on each side of the equation. Since these coefficients depend upon n,u, and T , which also may depend upon μ, the equations obtained in this way contain, in general, more than one power of μ. Therefore, the equating of equal powers of μ in Eq. (3.3.7) is not done in order to collect the coefficients of a single power of μ consistently in one equation, but instead to generate at each ¯ j that is soluble without having to assume that all quantities stage an equation for  are analytic functions of μ. This is accomplished by requiring that the solubility ¯ j are identical to the conservation laws, conditions for the equation determining  j written to order μ . Consequently we have no a priori guarantee that the resulting hydrodynamic equations give a good description of the behavior of the fluid, unless it can be proved that the μ expansion of the hydrodynamic equations is convergent. Such a proof was given by A. J. McLennan for the linearized hydrodynamic equations that result from applying the Chapman–Enskog procedure to the linearized Boltzmann equation, in the special case that the system consists of molecules with a finite total collision cross section, such as hard spheres, placed in an unbounded region [476, 477]. The linearized Boltzmann equation applies to a gas that is close enough to a total equilibrium state that quadratic deviations from equilibrium can be neglected.28 However, in general, the μ expansion is asymptotic and not convergent [265, 266, 267, 82, 83, 85, 667]. In such cases, it is not at all clear that going further in the Chapman–Enskog procedure would lead to useful results, especially in view of the fact that the Navier–Stokes equations provide an excellent description of fluid flows. Nevertheless, the equations for the next two orders in the μ expansion of the hydrodynamic equations are known, although their nonlinear forms are of limited utility, especially when boundaries are involved. The ¯ 2 lead to the Burnett equations containing among other equations for the function 

88

The Boltzmann Equation 2: Fluid Dynamics

¯ 3 lead to the terms third-order gradients of the hydrodynamic fields, and those for  super-Burnett equations with fourth-order gradients [221, 222, 239]. The linearized versions of these equations are used for a treatment of sound propagation in dilute gases, to be discussed later in this chapter. Finally, we mention again the particular form of the Chapman–Enskog solution, which depends on space and time only through the spatial and temporal dependence of the local thermodynamic variables. That is to say, f¯(r,c,t) is a functional of n(r,t),u(r,t) and T (r,t). This result is a direct consequence of the assumptions listed previously. The functional dependence has a clear physical origin, which is connected to the time scales, t and tL , mentioned before. Consider the evolution of the gas from some initial state. Then, for t > t , the particles will typically have collided several times. Each collision changes the velocities of the colliding particles, and therefore, almost all functions and moments of the velocities of the particles in the vicinity of a point r will change rapidly with time as collisions take place. However, the particular moments n(r,t),u(r,t), and T (r,t) will not change in time as collisions take place, since they are each determined by quantities that remain constant when collisions take place in the region close to the point r. In fact, the hydrodynamic fields change only on much longer time scales of order tL . Consequently, we might expect that the local distribution function will adjust to the prevailing values of n,u, and T and thus will become functionally dependent on just these slowly varying quantities after times greater than the mean free time between collisions, t . 3.5 The Rate of Entropy Production Throughout the preceding discussion, we have assumed that the distribution function is close to the local equilibrium form, with well-defined local quantities, namely the local density, energy, and velocity. Moreover, we have assumed the validity of local thermodynamic relations connecting the various densities that have forms identical with those given by equilibrium thermodynamics, albeit with local densities and other parameters [121]. All of these quantities are supposed to vary on time and space on scales large compared to microscopic scales. Recalling the requirement that any irreversible process in a macroscopic system, from one equilibrium state to another should be accompanied by an increase of entropy, we may ask if there is a version of this statement that applies to the local, timedependent entropy expressed in terms of ρ(r,t),u(r,t),e(r,t). A positive answer to this question is provided by the Navier–Stokes equations. One starts with the equation for the entropy density, Eq. (3.1.28), applies the linear laws for the heat flow vector and stress tensor, Eqs. (3.1.22) and (3.1.24), and finds, upon integrating

3.5 The Rate of Entropy Production

89

the entropy density over all space, that the total entropy of the system is given by the spatial integral of positive definite quadratic form in gradients given by29

2     (∇T )2 d η 2 ρ(r,t)s(r,t)dr = λ ∇u + (∇u)T − (∇ · u)1 dr, + dt T2 2T d (3.5.1) where the square of the tensor in Eq. (3.5.1) is the defined by the scalar product of the two tensors. The question immediately presents itself. Is there a microscopic version of this expression for the irreversible production of entropy in the flow of a dilute gas? The answer, of course, is closely connected with Boltzmann’s H -theorem, and is affirmative. Boltzmann’s H -theorem reduces to Eq. (3.5.1) for the normal solution expanded to first order in the gradients [541, 121]. We will obtain this result for monatomic dilute gases, as a simple example of the procedure that we will use to obtain the rate of entropy production and the Onsager symmetry relations [508, 509] for dilute gas mixtures. We recall that for an equilibrium system, Boltzmann’s H -function is directly proportional to the thermodynamic entropy for an ideal gas, apart from a constant term that gives no contribution to entropy differences. The precise relation is Seq = −kB Heq + c.

(3.5.2)

We may define a nonequilibrium entropy in terms of the H -function, using the time-dependent single-particle distribution function as   drdvf˜(r,v,t)[ln f˜(r,v,t) − 1]. (3.5.3) S(t) = −kB H (t) = −kB We know from the H -theorem that S(t) given before will increase with time unless the system is in a total equilibrium state. We tentatively take this as a useful expression for the nonequilibrium entropy and check to see if, for the normal solution, the  entropy grows in time according to Eq. (3.5.1). We write S(t) = drρ(r,t)s(r,t), and we use the Boltzmann equation to show that s(r,t) satisfies ∂(ρ(r,t)s(r,t)) + ∇ · Js (r,t) = σ (r,t), ∂t

(3.5.4)

where the entropy current consists of two parts, an entropy density carried along by the fluid plus a kinetic current with respect to the local fluid flow, as  (3.5.5) Js = uρs − kB cf¯[ln f¯ − 1]dc,

90

The Boltzmann Equation 2: Fluid Dynamics

where the local entropy density is defined in Eq. (2.2.10) as  ρ(r,t)s(r,t) = −kB h(r,t) = −kB dvf˜[ln f˜ − 1].

(3.5.6)

The irreversible entropy production rate, is given by30    ˆ k,g) ˆ σ (r,t) = −kB dv1 dv d kB( × [f˜(v1,r,t)f˜(v2,r,t) − f˜(v1,r,t)f˜(v2,r,t)]]   f˜(v1,r,t)f˜(v2,r,t) , × ln f˜(v1,r,t)f˜(v2,r,t)

(3.5.7)

which is always nonnegative, and positive in nonequilibrium situations. To make the connection with the Chapman–Enskog method, we use the gradient expansion of the distribution functions about local equilibrium, given by Eq. (3.3.1). This expansion, to lowest order in the gradients, is of order μ2 and is given by    ˆ k,g) ˆ f¯loc (r,c,t)f¯loc (r,c1,t) σ (r,t) = kB dc1 dc d kB(   ¯ 1 (r,c,t) +  ¯ 1 (r,c1,t) −  ¯ 1 (r,c,t) −  ¯ 1 (r,c1,t) 2 + O(μ3 ). ×  (3.5.8) We have not explicitly written the powers of the ordering parameter, μ, in the preceding expression, since its meaning is understood. The correction terms are at least of cubic order in the gradients, or combinations of powers of gradients and higher derivatives of second and higher orders. Expression (3.5.8) can be evaluated ¯ 1 given by Eq. (3.4.12). by using the result of the Chapman–Enskog procedure for  This makes it possible to obtain an explicit expression for the local rate of entropy production. After some algebra, which we will not reproduce here, we reproduce the integrand of Eq. (3.5.1), i.e.

2 (∇T )2 η 2 T σ =λ ∇u + (∇u) − 1∇ · u , + (3.5.9) T2 2T d as one would expect. The entropy current term vanishes in the spatial integral of Eq. (3.5.4) provided there is no flow of entropy at the boundary of the container. We can conclude that the Boltzmann equation together with the Chapman–Enskog expansion produce an equation for the local rate of entropy production that takes the form given by the theory of irreversible thermodynamics, at least to lowest order in the gradients. This result is entirely expected, since the Chapman–Enskog expansion itself produces the equations of fluid dynamics for a dilute gas. It is also worth pointing out that one can expand the entropy density, itself, in powers of the gradients, using Eq. (3.5.6) and

3.6 Boundary Conditions on the Hydrodynamic Densities

91

¯ 1 (r,c,t) + · · · ], f¯(r,c,t) = f¯loc (r,c,t)[1 +  to obtain ρ(r,t)s(r,t) = ρ(r,t)s0 (r,t) −

kB 2



¯ 1 (r,c,t)]2 + · · · , dcf¯loc (r,c,t)[ (3.5.10)

where

 ρ(r,t)s0 (r,t) = −kB

dvf¯loc (r,c,t)[ln f¯loc (r,c,t) − 1]

(3.5.11)

is the local equilibrium entropy density. The deviation from the local entropy density can be obtained from Eq. (3.4.12) and is given by  kB ¯ 1 (r,c,t)]2 dcf¯loc (r,c,t)[ − 2  2    −kB c2 2 2 ¯ dcfloc A(c )c · ∇ ln T + B(c ) cc − 1 :D = (3.5.12) 2 d The deviation from the local equilibrium entropy density in Eq. (3.5.10) is negative and proportional to the second power of the gradients. 3.6 Boundary Conditions on the Hydrodynamic Densities Now we turn our attention to the effect of the boundary conditions on the normal solution. In particular, we must determine the circumstances, if any, under which the solution, f˜(r,c,t), given by Eq. (3.4.36), satisfies the boundary equation, Eq. (3.1.3). To do this, we must give a specific form for the particle–wall collision kernel KW (v,v )31 appearing in the expression, Eq. (2.1.38), for the wall collision operator T W . Although there are many possible forms for the collision kernel that can model the particle–wall collisions and that describe these collisions with varying degrees of attention to the specific details of this interaction, we consider here only the two typical examples of simple model kernels that were introduced already in subsection 2.2.1, plus linear combinations of these.32 As a reminder, these models were as follows: 1. The specular collision model, where particles are reflected elastically and specularly33 from the wall with which it collides. In this case, we had KW = Ksp with ˆ n), ˆ Ksp (v,v ) = δ(v − v + 2(v · n)

(3.6.1)

where nˆ is a unit vector normal to the surface and pointing into the container.

92

The Boltzmann Equation 2: Fluid Dynamics

2. The diffuse collision model, where particles striking the surface are absorbed and then instantly re-emitted with a velocity determined by a Maxwell– Boltzmann distribution with wall temperature TW (ρ S ) = (kB βW (ρ S ))−1 . Here KW (v,v ) = KD (v,v ), where 1/2 ˆ  · n|(2πmβ ˆ φW (v,ρ S ), KD (v,v ) = |v · n||v W (ρ S ))

(3.6.2)

where φW is the Maxwell–Boltzmann distribution function for the reflected molecules given by Eq. (2.2.5), and ρ S is the point on the wall where the collision takes place. One can also consider kernels that are linear combinations of these two forms, namely, Kα = (1 − α)Ksp + αKD .

(3.6.3)

The coefficient, α, is called the accommodation coefficient, and this boundary condition is usually called the Maxwell boundary condition34 since it was first introduced by Maxwell [465, 79, 396]. If we now insert the expression for f˜ given by Eq. (3.4.36) into Eq. (3.1.3), use one of the three possible forms of the kernel K(v,v ) given earlier, and compare equal powers of μ on the right and left sides of Eq. (3.1.3), we find that 1. For specular reflections, the order μ0 equation leads to u(ρ S ,t) · nˆ = 0,

(3.6.4)

ˆ : ∇u(ρ S ,t) = 0, (nˆ nˆ ⊥ + nˆ ⊥ n)

(3.6.5)

nˆ · ∇ ln T (ρ S ,t) = 0,

(3.6.6)

and the order μ equation gives

and

ˆ Equation (3.6.4) expresses the where nˆ ⊥ is any unit vector perpendicular to n. fact that there is no net flow of the fluid into or out of the container, Eq. (3.6.5) says that there is no tangential stress exerted by the fluid on the wall at ρ S , and Eq. (3.6.6) indicates that there is no temperature gradient normal to the wall at ρ S . Equation (3.6.4) should hold for any boundary condition where particles cannot leave or enter the container, but Eqs. (3.6.5) and (3.6.6) are valid only when particles make elastic, specular collisions with the wall and there is no transfer of momentum from the particles to the wall in a plane tangent to the wall. These boundary conditions, Eqs. (3.6.4)–(3.6.6), are referred to as slip boundary conditions since there is no momentum exchange tangent to the walls. They are sufficient for finding solutions to the Navier–Stokes equations for a given geometrical configuration of the boundaries.

3.6 Boundary Conditions on the Hydrodynamic Densities

93

One might try to extend this method to higher orders in μ in order to obtain the boundary conditions on the hydrodynamic densities appropriate for solutions of the Burnett and higher-order equations for specular reflection of particles by the boundaries. However, to order μ2 and beyond, the normal solution no longer satisfies the boundary conditions. Consequently, boundary conditions for Burnett and higher-order equations cannot be obtained in this way. Instead, one must conclude that there is a region close to the walls where the normal solution is not a solution to the Boltzmann equation, including the specular interactions with the walls, beyond order μ. This region is called the kinetic boundary layer, and in it, the hydrodynamic fields, n,u, and T vary over distances of the order of the mean free path length rather than on macroscopic scales, as was assumed in the derivation of the normal solution. 2. For diffuse reflections, the order μ0 equations for the boundary conditions lead to u(ρ S ,t) = 0

(3.6.7)

T (ρ S ,t) = TW (ρ S ).

(3.6.8)

and

These boundary conditions are called stick boundary conditions. For fixed walls, the stick boundary conditions imply that the fluid is at rest at the boundary. If the boundary is moving, the stick boundary conditions would apply in the rest frame of the boundary.35 These boundary conditions also lead to solutions of the Navier–Stokes equations, when particles make diffuse collisions with the walls. Unlike the case discussed before for specular reflections, the normal solution breaks down near the walls already at order μ. Thus, there is a kinetic boundary layer of order μ in the stick case, but of order μ2 in the slip case. The existence of this boundary layer leads to corrections of order μ in the boundary conditions, such that the boundary conditions on the tangential velocity and temperature at the wall take the form, for three-dimensional systems,

  2 1/2 (nˆ ⊥ · ∇) ln T u · nˆ ⊥ = ζS nˆ nˆ ⊥ + nˆ ⊥ nˆ : ∇u + ω mβ

 mβ 1/2  T − TW = τ nˆ · ∇ ln T + χ 3nˆ nˆ − 1 : ∇u. (3.6.9) TW 2 Here ζS ,ω,τ, and χ are constants having the dimension of a length that are in magnitude, of the order of a mean free path. Here the unit vector, nˆ ⊥ , is ˆ normal to the surface. The constant, ζS perpendicular to the unit vector, n, is called the slip coefficient, ω the thermal creep coefficient, and τ is the the temperature jump distance [465, 164, 637, 638, 83].

94

The Boltzmann Equation 2: Fluid Dynamics

From these equations, we see that the corrections to Eqs. (3.6.7) and (3.6.9) are of order μ and vanish in the continuum limit where the ratios of the mean free path to the macroscopic lengths approach zero. The determination of the exact values of the constants appearing in the preceding boundary conditions is quite difficult in most cases, since the solution of the Boltzmann equation including the distribution function in the kinetic boundary layer must be known. For some simple model equations, such as the BGK form of the Boltzmann equation [40], to be discussed in Section 3.11, and for simple geometries, such as the stationary flow over a flat plate in a semi-infinite space, solutions and explicit expressions for the slip lengths can be obtained. Some approximate results can be obtained using variational or moment expansion methods. For a more detailed treatment, we refer the reader to the books of Cercignani and the references contained therein [79, 598, 81, 83, 85, 599, 600]. Maxwell boundary conditions, Eq. (3.6.3), also lead to boundary equations of the form of Eq. (3.6.9), with constants ζS ,ω,τ,χ as in the stick boundary case. As long as α /L, the scattering from the surface is essentially diffusive, and the corrections to stick boundary conditions are small. However, if α is very small, on the order of /L, boundary conditions are obtained that are significantly different from the “stick” conditions. This is an indication why stick boundary conditions are very useful in hydrodynamic calculations for realistic systems: a reflection mechanism that is mostly specular is not very likely to occur in nature because of surface irregularities at the boundary and of thermal motions of the surface molecules. The boundary conditions to be used with Burnett equations have been determined for the special BGK model Boltzmann equation by Y. Sone and co-workers [597, 24, 598, 599, 600], and for more general Boltzmann equation models by M. de Wit, using variational methods [136]. These boundary conditions allow for nonphysical solutions exhibiting spatial variations on the scale of the mean free path. Since boundary conditions are not available that reject these unphysical solutions, one must eliminate them in an ad hoc manner and use the available boundary conditions to determine the solutions of the hydrodynamic equations. In most cases, the addition of Burnett and higher-order terms in the hydrodynamic equations leads to only small improvements, if any, over the Navier–Stokes results. In cases where boundary conditions can be avoided, such as in the description of sound propagation in a gas, the Burnett and higher-order terms do improve the comparisons of theoretical results with experiments done at very high sound frequency, where the wavelength is on the order of a mean free path. We discuss this in the next section. We end this section with an observation that we will amplify further in Chapters 4 and 7 when we discuss the kinetic theory for gas mixtures as obtained from

3.7 Comparison of the Results of the Normal Solution Method with Experiment

95

the Boltzmann, Enskog, and revised Enskog equations. These equations, and for our present discussion, we focus on the Boltzmann equation, together with the H -theorem can be used to derive the equations of nonequilibrium thermodynamics. For mixtures, one can show that the set of diffusion and cross-coefficients36 satisfy the Onsager reciprocal relations of irreversible thermodynamics. 3.7 Comparison of the Results of the Normal Solution Method with Experiment The normal solution method just outlined leads to two principal results, both of which can be tested experimentally. These are (1) explicit expressions for the coefficients of shear viscosity, η, and of thermal conductivity, λ, for dilute, monatomic gases in terms of the intermolecular potential energy, φ(r), and (2) explicit forms of the Burnett and higher-order corrections to the Navier–Stokes equations, together with expressions for the associated transport coefficients in terms of the intermolecular potential. The latter equations and results can be tested in experiments on sound propagation in the gas.37 3.7.1 The Transport Coefficients η and λ In the previous section, we have noted that one can make two predictions about λ and η without having to obtain their explicit forms.38 These predictions are that both coefficients are independent of the gas density, at least over the region of densities where the Boltzmann equation is appropriate, and that they are related, to a good approximation by the Eucken factor, Eq. (3.4.30), namely λ 5 = . ηcv 2

(3.7.1)

Both of these predictions have been borne out by experiments. In fact, the independence of λ and η on density was one of the first important results in the history of the kinetic theory of gases. To begin our discussion of experimental results, we present in Table 3.1 some typical results for λ/ηcv for several noble gases at various temperatures. The agreement with the predicted value of about 2.5 is quite good and within a few percent. Some more accurate determinations of this ratio, which improve the agreement between experiment and theory, will be presented momentarily. Of course, if we wish to make detailed comparisons of theoretical results for the coefficients of thermal conductivity and shear viscosity with experimental values, we must know the intermolecular potential, φ(r). In principle, the intermolecular potential can be obtained from quantum mechanical calculations or, failing that, by

96

The Boltzmann Equation 2: Fluid Dynamics

Table 3.1. The Eucken factor for three noble gases. Data taken from W. G. Kannuluik and E. H. Carman [339]. Gas

T, K

λ/ηcv

Gas

T, K

λ/ηcv

Gas

T, K

λ/ηcv

Ar

90.18 194.65 273.15 373.15

2.50 2.52 2.51 2.51

Xe

273.15 373.15 551.15 579.5

2.58 2.61 2.58 2.54

Ne

90.18 273.15 373.15 491.15

2.49 2.48 2.49 2.44

means of scattering experiments. However, due to the existence of serious difficulties and uncertainties when obtaining intermolecular potentials in either of these ways, it is more useful to determine the potential by indirect methods, of which we discuss the two most important ones here. The model Potential Method Here one assumes a specific analytical form for φ(r), such as a Lennard–Jones potential, with as yet unspecified numerical coefficients. Then, using this potential, one calculates an equilibrium property of the gas, such as the second virial coefficient, B(T ) given by  1 B(T ) = − (3.7.2) dr(e−βφ(r) − 1), 2 for this potential function in terms of the undetermined coefficients. By varying the coefficients to get the best possible fit to experimental data, one obtains an explicit form of a potential energy, which can be used to calculate λ and η [306]. The Corresponding States Method This method assumes that the potential energy, φ(r), has the same analytic form for a number of gases, but with parameters that vary from one gas to another. By dimensional analysis, one can see that any such potential must be of the form

 r i fi , (3.7.3) φ(r) = σ i i where the i have the dimension of energy, the σi have the dimension of length, and all the fi may be different functions of their argument. Consider the simplest case, where φ(r) = f (r/σ ) with only one energy constant and one length constant. We can easily show then that the shear viscosity and thermal conductivity satisfy simple scaling relations for all gases with this functional form of the potential energy. If we define a reduced temperature T ∗ = kB T /, then the quantities η∗ (T ∗ ) =

σ 2η σ 2λ ∗ ∗ and λ (T ) = (mkB T )1/2 (m−1 kB3 T )1/2

(3.7.4)

3.7 Comparison of the Results of the Normal Solution Method with Experiment

97

Figure 3.7.1 The reduced second virial coefficient, B ∗ (T ∗ ), as a function of the reduced temperature [357, 356]. This figure is taken from the paper of J. Kestin, S. T. Ro, and W. A Wakeham [357]

should be functions only of the reduced temperature T ∗ and should have the same values for all gases whose interaction potential has the same functional form. Consequently, one tries to find values for the parameters  and σ for each gas such that the experimental data for η and λ can be reduced to universal functions for η∗ and λ∗ [357]. In addition, B ∗ (T ∗ ) = B(T )/σ 3 should be a function only of T ∗ . In practice, one determines  and σ by finding universal forms for B ∗,η∗ , and λ∗ . This method does not test predictions for the magnitudes of these quantities, but only the form of their temperature dependence. The Comparisons The Lennard–Jones potential, given by [306]   σ 12  σ 6 , φLJ (r) = 4 − r r

(3.7.5)

is strongly repulsive at short distances, attractive at longer distances, and a good model potential, at least for monatomic gases. One fits  and σ using the second virial coefficient and then determines values for T ∗ , η∗ (T ∗ ), and λ∗ (T ∗ ) using experimental values for η(T ) and λ(T ). One expects universal curves for η∗ (T ∗ ) and λ∗ (T ∗ ) vs. T ∗ for all gases for which the Lennard–Jones potential gives a good fit for B(T ). In Figure 3.7.1, results are shown for the reduced second virial coefficient B ∗ (T ∗ ) = B(T )/(2πσ 3 /3) as a function of T ∗ for a variety of gases [356]. The quantities  and σ are adjusted for each gas so that the data all lie on one universal curve. Then these quantities are used to obtain values for η∗ and λ∗ . In

98

The Boltzmann Equation 2: Fluid Dynamics

Figure 3.7.2 The reduced inverse coefficient of viscosity for the noble gases and some binary mixtures of noble gases in Figure 3.7.1 [357, 356]. This figure is taken from the paper of J. Kestin, S. T. Ro, and W. A. Wakeham [357]

Figure 3.7.2, we show experimental values, obtained by J. Kestin, S. T. Ro, and W. Wakeham [358, 356], for the inverse of the reduced viscosity (η∗ )−1 as a function of T ∗ for the same gases as in Figure 3.7.1. In both figures, we see a striking confirmation of the prediction that B ∗ and η∗ should be universal functions of T ∗ . In Figure 3.7.3, we show a plot of 2λ/(5ηcv ) versus the reduced temperature T ∗ . If the first Sonine polynomial approximation were correct, the value of this quantity would be unity, and in any case the data from all measurements should fall on the same curve again. In the figure, one can see that the data do not really collapse onto a single line, and fluctuate about unity by as much as 7%. However, the thermal conductivity data obtained by J. W. Haarman leads to a curve that is much closer to unity [288]. Apparently, Haarman was able to avoid the disturbing effects of thermal convection that tend to plague measurements of thermal conductivity in fluids. Notwithstanding the fundamental issues raised after its derivation, the Boltzmann equation leads to results that compare so well with experimental data that

3.8 Projection Operator Methods for the Linearized Boltzmann Equation

99

Figure 3.7.3 The Eucken factor for the noble gases. The dashed line is the theoretical result obtained using the 11–6–8 potential, Eq. (3.7.6), of Klein and Hanley [394, 295]. This figure is taken from the paper of B. Najafi, E. A. Mason, and J. Kestin [500]

the predictions are regarded as verified. Instead of using experiments on transport coefficients to test the Boltzmann equation, the Boltzmann equation predictions are now used to test model potential energy functions. For example, a potential energy function proposed by M. Klein and H. J. M. Hanley, called the 11–6–8 potential, has the form   σ 11  σ 6  σ 8   φ(r) = (6 + 2γ ) , (3.7.6) − (11 − 3γ ) − 5γ 5 r r r with three adjustable parameters, γ ,, and σ . This potential provides excellent fits for the experimental data the the second virial coefficient and transport coefficients. In Figure 3.7.3, we also show results for 2λ/(5ηcv ) as obtained by using this potential and including the contributions of higher Sonine polynomials in the expansion of the appropriate distribution functions [394, 295]. 3.8 Projection Operator Methods for the Linearized Boltzmann Equation There are some elegant methods for studying the solutions of the linearized Boltzmann equation. A method that we shall apply quite often in the course of this

100

The Boltzmann Equation 2: Fluid Dynamics

book is the projection operator method, introduced by R. Zwanzig and by H. Mori [708, 489, 705]. This method will enable us to derive the linearized Navier–Stokes equations from the Boltzmann equation. Since the projection operator method in its most simple from requires a linear equation as a starting point, we will use the linearized Boltzmann equation (LBE), obtained by expanding the distribution function about its total equilibrium form, as f (r,v,t) = feq (v) (1 + χ (r,v.t)) , where



βm feq (v) = nφ0 (v) = n 2π

(3.8.1)

d/2 exp(−βmv 2 /2),

(3.8.2)

where we drop the tilde notation on the distribution functions, and feq (v) is the Maxwell–Boltzmann velocity distribution function for a gas at equilibrium. We insert this form in the Boltzmann equation, and keep only terms linear in χ (r,v,t). In the absence of wall terms – that is, for infinite or periodic systems – this leads to the linear equation for χ given, in the external field free case, by ∂χ (r,v,t) + v · ∇r χ (r,v,t) = (v)χ (r,v,t), ∂t

(3.8.3)

where (v) is the Boltzmann collision operator, linearized about a total equilibrium solution, given by   ˆ ˆ eq (v1 )B(k,g) (v)χ (r,v,t) = dv1 d kf   × χ (r,v,t) + χ (r,v1,t) − χ (r,v,t) − χ (r,v1,t) ,

(3.8.4)

and, as usual, the primes denote restituting velocities. The linear operator, (v), is a version of the operator, loc (v), defined by Eq. (3.8.4), where the local equilibrium distribution function is replaced by the total equilibrium distribution. The hydrodynamic equations obtained from the LBE will also be linear and will describe small deviations of the local mass density, the local velocity, and the local temperature from their total equilibrium values. These small deviations, respectively, are given by  (3.8.5) δρ(r,t) = mn dvφ0 (v)χ (r,v,t),  (3.8.6) δu(r,t) = n dvvφ0 (v)χ (r,v,t), d nkB δT (r,t) = n 2

 dv

mv 2 φ0 (v)χ (r,v,t). 2

(3.8.7)

3.8 Projection Operator Methods for the Linearized Boltzmann Equation

101

Here it is important to note that these quantities are among the first few velocity moments of the function χ (r,v,t). The projection operator method allows us to derive equations for these moments in an especially simple way by projecting the full distribution function at a given point r onto the space spanned by the conserved quantities, 1,v,mv 2 /2 . We begin by constructing an orthonormal set of d + 2 functions ψi (v), using the d + 2 conserved quantities. This set will be orthonormal under the inner product39  f (v)|g(v) = dvφ0 (v)f (v)g(v). (3.8.8) This set of functions may be chosen to be ψ1 (v) = 1

,

ψi (v) = (βm) vi−1, i = 2, . . . ,d + 1,  12  mv 2 d 2 . β ψd+2 (v) = − d 2 2 1/2

(3.8.9) (3.8.10) (3.8.11)

We wish to project the distribution function on to the space spanned by these functions by means of a projection operator denoted by P. An operator P is said to be a projection operator if P 2 = P. For our discussion, the projection operator needed is P=

d+2 

|ψi  ψi | ,

(3.8.12)

i=1

with an obvious use of bra-ket notation. All of the local hydrodynamic variables can be obtained from the function Pχ (r,v,t). To obtain an equation for it, we first define the complement operator P⊥ : P⊥ = 1 − P.

(3.8.13)

We now express the function χ (r,v,t) as the sum of two functions χ (r,v,t) = Pχ (r,v,t) + P⊥ χ (r,v,t), rewrite the kinetic equation Eq. (3.8.3) as two coupled ˆ equations, one for P χ (r,v,t) and one for P⊥ χ (r,v,t), as ∂Pχ (r,v,t) + PLPχ (r,v,t) + PLP⊥ χ (r,v,t) = 0, ∂t ∂P⊥ χ (r,v,t) + P⊥ LPχ (r,v,t) − +P⊥ LP⊥ χ (r,v,t) = 0, ∂t

(3.8.14) (3.8.15)

where L = v · ∇r − . The projection operator method proceeds by solving Eq. (3.8.15) for the compleˆ ˆ mentary function, P⊥ χ (r,v,t), in terms of the function, P χ (r,v,t), and its initial

102

The Boltzmann Equation 2: Fluid Dynamics

value, and using this in Eq. (3.8.14) to obtain, apart from initial value terms, a closed equation for P χˆ (r,v,t). This, in turn, will enable us to obtain equations for the local hydrodynamic densities. This procedure is most simply carried out by taking advantage of the linearity and translation invariance of the equations and using the Fourier–Laplace transforms of the distribution function and the equations for Pχ and P⊥ χ . We define χk (v,z) by   ∞ dt exp [−zt − ik · r] χ (r,v,t). (3.8.16) χk (v,z) = dr 0

Then the equations for Pχk (v,z) and P⊥ χ k (v,z) are (z + PLk P) Pχk (v,z) = Pχk (v,t = 0) − PLk P ⊥ χk (v,z), (z + P⊥ Lk P⊥ ) P⊥ χk (v,z) = P⊥ χk (v,t = 0) − P⊥ Lk Pχk (v,z),

(3.8.17) (3.8.18)

where the operator Lk is Lk = ik · v − (v).

(3.8.19)

We will also use a projected form of this operator onto the perpendicular subspace, Lˆ k , given by Lˆ k = P⊥ Lk P⊥ .

(3.8.20)

We proceed to obtain a closed expression for Pχk (v,z), apart from an initial condition term for P⊥ χk (v,t = 0) by solving Eq. (3.8.18) for P⊥ χk (v,z) in terms of Pχk (v,z) and then inserting this expression in Eq. (3.8.17). This leads directly to zPχk (v,z) + PLk Pχk (v,z) = Pχk (v,t = 0)

−1 P⊥ Lk Pχk (v,z) + PLk P⊥ z + Lˆ k

−1 P⊥ χk (v,t = 0). (3.8.21) − PLk P⊥ z + Lˆ k One may obtain the linearized hydrodynamic equations by taking inner products of Eq. (3.8.21) with the functions ψj (v) defined in Eqs. (3.8.9)–(3.8.11). To obtain these equations, it is important to note that the conserved quantities are right and left eigenfunctions of the linearized Boltzmann collision operator, (v), with eigenvalue zero. Thus, PLk P = ik · PvP, PLk P⊥ =ik · PvP⊥, P⊥ Lk P =ik · P⊥ vP.

(3.8.22)

Consequently, Pik · vPχk (v,z) =

d+2  i,j =1

   |ψi  ψi | ik · v ψj ψj  χk .

(3.8.23)

3.8 Projection Operator Methods for the Linearized Boltzmann Equation

103

The second term on the right-hand side of Eq. (3.8.21) is at least of order k 2 , as follows from the identities, Eq. (3.8.22), and thus corresponds to terms that in position space would be second order in the gradients of the hydrodynamic densities. These are the dissipative terms, and the coefficients of the order k 2 terms are proportional to the transport coefficients. We can drop the initial condition term for the perpendicular part of the distribution, either by setting it equal to zero by assuming a special initial condition or by showing that for times large in comparison to the mean free time it gives only small corrections to the initial conditions for the hydrodynamic equations.40 Then in a very condensed form, the linearized Navier–Stokes equations can be expressed as z ψi | χk  +

d+2 

d+2      2  ij ψj χk  = ψi | χk (t = 0) − k Uij (k,z) ψj  χk ,

j =1

j =1

(3.8.24) where   ij = ψi | ik · v ψj

(3.8.25)

Uij (k,z) = − ψi | kˆ · vP⊥ [z + Lˆ k ]−1 P⊥ kˆ · v|ψj .

(3.8.26)

and

Here kˆ is a unit vector in the direction of k. The matrix ij is first order in the wave number and contributes non-dissipative terms to the linearized equations, and the matrix Uij (k,z) contains the dissipative terms in the equation, which take the form of linear combinations of transport coefficients in the limit k, z → 0. This latter limit corresponds to the long-time limit of these equations and is an excellent approximation as soon as z becomes small compared to the first nonzero eigenvalue of the operator − – in other words, for times long compared to the inverse of this eigenvalue. We can express the matrix elements Uij in terms of projected currents, Ji , given by Ji = P⊥ (kˆ · v)ψi

(3.8.27)

Uij = −Ji [z + Lˆ k ]−1 Jj .

(3.8.28)

and

In this way, we obtain the following linearized hydrodynamic equations, expressed in terms of Fourier–Laplace transforms of the hydrodynamic variables: zδρ(k,z) + iρk · δu(k,z) = δρ(k,t = 0),

(3.8.29)

104

The Boltzmann Equation 2: Fluid Dynamics

zρδu(k,z) + ik [kB T δn(k,z) + nkB δT (k,z)]   (d − 2) ˆ ˆ 2 k(k · δu(k,z)) , = ρδu(k,t = 0) − ηk δu(k,z) + d

(3.8.30)

d d z nkB δT (k,z) + inkB T (k · δu(k,z)) = nkB δT (k,t = 0) − λk 2 δT (k,z). 2 2 (3.8.31) Then the k,z-dependent coefficients of shear viscosity, η(k,z), and thermal conductivity, λ(k,z), become41   (i) (i) η(k,z) = βm (kˆ · v)(kˆ ⊥ · v)|[z + Lˆ k ]−1 |(kˆ · v)(kˆ ⊥ · v) (3.8.32)



 2 βmv βmv 2 (d + 2) ˆ (d + 2) −1 − (k · v)|[z + Lˆ k ] |(kˆ · v) − . λ(k,z) = kB 2 2 2 2 (3.8.33) (i) ˆ For distribution functions changing on Here kˆ ⊥ is a unit vector perpendicular to k. macroscopic time and length scales only, the transport coefficients may be replaced by their limits for k → 0 and z → 0. These are42   η = −βm vx vy  −1 |vx vy , (3.8.34) 



2 2 kB βmv (d + 2) (d + 2) βmv λ=− − v · |−1 |v − . (3.8.35) d 2 2 2 2

Then an inverse Fourier–Laplace transform of Eqs. (3.8.29)–(3.8.31) yields the standard linearized, Navier–Stokes equations for a dilute gas. ∂δρ(r,t) + ρ∇ · δu(r,t) = 0, ∂t ρ

∂ δu(r,t) + ∇ [kB T δn(r,t) + nkB δT (r.t)] ∂t   d −2 2 ∇ (∇ · δu(r,t)) , = η ∇ δu(r,t) + d

(3.8.36)

(3.8.37)

∂ d nkB δT (r,t) + nkB T ∇ · δu(r,t) = λ∇ 2 δT (r,t). (3.8.38) 2 ∂t The expressions, Eq. (3.8.34) and (3.8.35), agree with those obtained from the Chapman–Enskog method and given earlier by Eqs. (3.4.21) and (3.4.18) with the exception of the replacement here of the local equilibrium distribution function by the total equilibrium one. projection operator,  Note the action  of the perpendicular   P⊥, on the function v βmv 2 /2 − d/2 replaces it by v βmv 2 /2 − (d + 2)/2 , and it is this function that appears in the expression for the thermal conductivity, in agreement with our earlier results based on the Chapman–Enskog method.

3.8 Projection Operator Methods for the Linearized Boltzmann Equation

105

3.8.1 Hydrodynamic Modes of the Linearized Boltzmann Equation As we have noted, the equations of fluid dynamics describe long-wavelength, collective behavior of the fluid when the characteristic lengths of the gradients are large compared to microscopic lengths. The LBE provides us with a very useful way to directly describe the microscopic properties of these long-wavelength collective modes of a dilute gas. Applications of this analysis will be found throughout this book, particularly in descriptions of the decay to equilibrium of microscopic fluctuations in the gas in Chapter 8 and in the development of mode-coupling theory in Chapter 13 [156, 187, 150]. The importance of the hydrodynamic modes of the LBE for a description of long-wavelength collective effects in gases can be seen by considering a solution of the equation ∂fk (v,t) = [−ik · v + ] fk (v,t), ∂t

(3.8.39)

fk (v,t) = et (−ik · v + ) fk (v,0).

(3.8.40)

which is

Now if we assume that the operator, −ik · v + , has eigenfunctions, k(i) (v), with corresponding eigenvalues, ω(i) (k), the solution of this time-dependent equation can be expressed as   (i)  (i)    (v) eω (k)t  (i) (v)|fk (v,0) . (3.8.41) fk (v,t) = k k i

It is clear that in this solution the eigenvalues determine the decay of an initial disturbance of the system away from equilibrium. The slowest time decays, for small wave numbers, are governed by the eigenvalues with the smallest real part. The corresponding eigenfunctions are the hydrodynamic modes. They are constructed from the eigenstates of the operator  with zero eigenvalues by considering −ik · v as a perturbation. This operator is symmetric with respect to the inner product defined by Eq. (3.8.8). In Chapter 7 on the Enskog theory we will encounter an operator that is not symmetric and, as a consequence, has both right and left eigenfunctions corresponding to the same eigenvalue. To zeroth order in the wave number, these are the conserved quantities, 1,v,v 2 . The perturbation expansion in powers of the wave number has the general form (−ik · v + )[0(i) + 1(i) + · · · ] = (ω0(i) + ω1(i) + · · · )[0(i) + 1(i) + · · · ], (3.8.42) where the subscripts on n(i) and ωn(i) indicate the power of the wave number associated with these terms. By equating equal powers of the wave number on each

106

The Boltzmann Equation 2: Fluid Dynamics

side of this equation, we find that the first few equations to determine the terms in the wave number expansions of the eigenvalues and eigenfunctions are 0(i) = ω0(i) 0(i),

(3.8.43)

−ik · v0(i) + 1(i) = ω0(i) 1(i) + ω1(i) 0(i),

(3.8.44)

−ik · v1(i) + 2(i) = ω0(i) 2(i) + ω1(i) 1(i) + ω2(i) 0(i),

(3.8.45)

and so on. The hydrodynamic modes are obtained by perturbing about the conserved quantities given by Eqs. (3.8.9)–(3.8.11), so that to obtain these modes, we set ω0 = 0, and 0(i)

=

d+2 

aj(i) ψj (v),

(3.8.46)

j =1

with coefficients aj(i) to be determined presently. The first-order equation becomes ⎛ ⎛ ⎞ ⎞ d+2 d+2   −ik · v ⎝ aj(i) ψj (v)⎠ + 1(i) = ω1(i) ⎝ aj(i) ψj (v)⎠ . (3.8.47) j =1

j =1

Using the fact that the ψi form an orthonormal set, we can obtain the following equation for the coefficients al(i) by multiplying this equation by ψl and integrating over v. This yields −

d+2 

  aj(i) ψl | ik · v ψj = ω1(i) al(i) .

(3.8.48)

i=1

This is a simple matrix equation for the coefficients, and the d + 2 values for ω1(i) are obtained by diagonalizing the matrix ψl | ik · v |ψi  . One finds d − 1 shear mode eigenfunctions are given by (η ) 0 i (v) = (βm)1/2 (kˆ ⊥,i · v), i = 1, . . . ,d − 1,

(3.8.49)

(η )

with ω1 i = 0, and having components of the velocity perpendicular to the wave ˆ kˆ ⊥,i form a d-dimensional, orthonormal set of unit vectors. The vector k. Here k, three other hydrodynamic modes are easily found to be: two sound modes

βmv 2 βm 1/2 ˆ + σ 0(σ ) (v) = k · v, σ = ±1, (3.8.50) 2 [2d(d + 2)]1/2 with corresponding first-order eigenvalues ω1(σ ) = −iσ kc, where c is the lowdensity value for the sound velocity c = [(d + 2)/(dβm)]1/2 , and one heat mode

3.8 Projection Operator Methods for the Linearized Boltzmann Equation

0(H ) =

1/2

mv 2 d + 2 2 , β − d +2 2 2

107

(3.8.51)

with ω1(H ) = 0. These d + 2 functions 0(i) with i = ηi ,σ,H are the zeroth-order terms in the wave number expansion of the hydrodynamic modes. One can easily continue this process to higher orders in the wave number. To second order in the wave number, the hydrodynamic mode eigenvalues are [156, 187] ω(ηi ) (k) = −νk 2 + O(k 4 ), s 2 k + O(k 3 ), 2 ω(H ) (k) = −DT k 2 + O(k 4 ). ω(σ ) (k) = −ikcσ −

Here ν = η/nm is the kinematic viscosity, DT = λ/nCp , is the thermal diffusivity, with Cp being the specific heat at constant pressure per particle, and s = d −1 DT + (d − 1)ν/d is the sound damping constant. For use in later chapters, particularly Chapter 8, we will need the formal expression for the first-order terms, 1 , in the expansion of the hydrodynamic eigenfunctions, which can easily be obtained for Eq. (3.8.47) as 1(i) = −1 (ik · v + ω1(i) )0(i) .

(3.8.52)

The hydrodynamic modes represent microscopic descriptions of the long time, long wavelength decay of density, velocity and temperature fluctuations in the gas. They lead to the hydrodynamic description of the gas obtained by carrying out a diagonalization of the linearized hydrodynamic equations. Thus the O(k) terms correspond to the Euler equations, the order O(k 2 ) terms correspond to the Navier– Stokes equations, and so on. These modes will play a very important role in the later chapters in this book and they have been crucial for our understanding of some remarkable properties of transport processes, called long-time tails, which we describe in detail in Chapter 13. The sound and heat modes in a gas, resulting from fluctuations in density, temperature, and velocity taking place in the gas, even in equilibrium, can be probed by light scattering. In such scattering experiments, microscopic sound modes in a gas are responsible for two Brillouin peaks on either side of a central peak, and the central peak, called the Rayleigh peak, is produced by the decay of heat modes resulting from fluctuations [59, 38]. We will discuss fluctuations, light scattering, and these peaks in Chapter 8. Now we turn our attention to the case of sound modes in a dilute gas where higher terms, beyond order k 2 , in the wave number expansion of the sound mode eigenvalues are needed for an explanation of experimental results on sound propagation in the gas.

108

The Boltzmann Equation 2: Fluid Dynamics

3.8.2 Sound Propagation and the Higher Order Transport Equations An experimental check on the higher order corrections to the Navier–Stokes equations is faced with some serious difficulties. As we have mentioned earlier, these corrections involve third and higher order spatial derivatives of the hydrodynamic fields. The solutions of such equations require additional boundary conditions which are generally unknown. In addition the length scale on which the effects of higher order terms in the hydrodynamic equations is comparable to the width of the kinetic boundary layer near the walls. This makes it hard to avoid the effects of the latter. Therefore one might look one might look for experimental situations in which the boundaries play a minor or unimportant role in the description of the gas flow. Such a situation occurs in experiments on the propagation of sound designed to measure the dispersion relation for this propagation, namely the relation between the sound frequency and the wavelength. To determine this dispersion relation using kinetic theory we can make use of the previous discussion of hydrodynamic modes. In particular, the expansion of the sound mode eigenvalue, ω(σ ) (k), as a function of the wave number, k, is precisely the dispersion relation for sound in the gas we wish to determine. We consider an experimental situation where the amplitude of the sound waves is sufficiently small that the disturbance of the gas from an equilibrium state is small and nonlinear effects may be ignored. The relation between ω(σ ) and k, for small wave number, is determined by continuing the perturbation expansion described earlier to higher order in the wave number. To arbitrary order in the wave number, this dispersion relation has the form s 2 (3.8.53) k − iσ Bk 3 + Ck 4 + · · · , 2 where c is the sound velocity in an ideal gas. The coefficients B,C, . . . require expanding the perturbation expansion beyond second order in the wave number. As mentioned earlier, McLennan has proved that the series on the right-hand side of Eq. (3.8.53) is convergent for hard sphere molecules, at least [476].43 In experiments one fixes the frequency ω and then measures the real and complex parts of k(ω) ω(σ ) = −iσ ck −

k(ω) = kR (ω) + ikI (ω),

(3.8.54)

where kR is the wave number of the pressure wave and kI the damping. For a wave traveling in the z-direction, say, the deviation of the pressure from its equilibrium value for a propagating sound wave has the form δp(z,t) = ae[i(kR (ω)z−ωt)−kI (ω)z] .

(3.8.55)

The phase velocity, U , of the sound wave is then U = ω/kR . In Fig. 3.8.1 we compare experimental measurements by M. Greenspan [276] of U and kI as functions

3.8 Projection Operator Methods for the Linearized Boltzmann Equation

109

Figure 3.8.1 The dispersion of and absorption of sound in neon. In these figures, U0 = c, the velocity of sound, and α = kI , the sound damping coefficient. The solid curves are obtained from kinetic theory, J. D. Foch, G. W. Ford, and G. E. Uhlenbeck assuming that the particles are Maxwell molecules [224, 222]. The data points are due to to the experiments of M. Greenspan [276]

of frequency for sound waves in neon with theoretical values of these quantities obtained using Maxwell molecules44 [224, 222]. The dimensionless sound frequency in these figures, ξ , is defined by ξ=

ωη c2 n

eq m

,

(3.8.56)

where neq is the equilibrium density, c = (5kB T /3m)1/2 , and η the shear viscosity of the gas. The quantity ξ is on the order of the product of the sound frequency and the mean free time between collisions in the gas, that is the ratio of the mean free path to the wavelength of the sound wave. The theory should be good up to ξ ∼ 1, which is confirmed reasonably well. Note that the Burnett and super-Burnett terms noticeably improve the agreement over that obtained using the Navier–Stokes equations alone. One may not conclude from this that the higher order equations always improve Navier–Stokes results, since, among other reasons, we have not needed to use boundary conditions to obtain these results. These results strongly suggest that under such circumstances – no boundaries and near equilibrium flows – the

110

The Boltzmann Equation 2: Fluid Dynamics 1.0 0.9 UQ / U

0.8 0.7 HELIUM NEON ARGON KRYPTON XENON

0.6 0.5 0.4 0.3

a UQ / w

0.2 0.1 0 0.01

0.1

x

1

10

Figure 3.8.2 Comparison of theoretical values for sound dispersion with the experimental results of M. Greenspan for a range of wave numbers and frequencies and for the noble gases [276]. The upper curve is a plot of U0 /U, as a function of the dimensionless sound frequency, ξ, where U0 = c, the ideal gas velocity of sound, and U is the phase velocity of the sound wave. The lower curve is a plot of the absorption coefficient, αU0 /ω, where α ≡ kI , as a function of ξ . The solid curves are results for Maxwell molecules. This figure is taken from the paper of J. D. Foch and M. F. Losa [223]

higher order hydrodynamic equations yield important corrections to the Navier– Stokes equations.45 In Fig. 3.8.2 we present the results of J. Foch and J. M. Losa who calculated the sound dispersion relations for Maxwell molecules [223]. Since the eigenvalues and eigenfunctions of the linearized Boltzmann equation are known exactly for these molecules, these authors were able to look at the dispersion relations for a large range of values of the wave number and frequency of the sound. They used summation techniques designed to obtain an approximate expression for the sum of a large number of terms in the power series to express the dispersion relations in terms of functions of the wave number or frequency. Even given the approximate form of the intermolecular potential, their results are quite striking. 3.9 Other Solutions of the Boltzmann Equation and Models for the Collision Operator In addition to the normal solutions of the Boltzmann equation just discussed in detail, other solutions, both exact and approximate are known. We will discuss some of these later. In addition, model Boltzmann collision operators have been constructed that preserve some, but not all features of the original equation and allow one to obtain exact solutions or to explore features that are not otherwise accessible to analytic treatment. An example of the latter situation are simplified

3.9 Other Solutions, Other Models

111

nonlinear Boltzmann equation models, invented by A. V. Bobylev, that have exact solutions [44, 45]. Further on in this chapter we give a brief review of these models and their solutions. 3.9.1 Remarks on Rigorous Derivations of, and Existence Theorems for Solutions to, the Boltzmann Equation The Boltzmann equation has been derived here on the basis of intuitive arguments involving a number of assumptions about the frequency of binary collisions as well as the assumptions that collisions involving more than two particles interacting simultaneously can be ignored, as well as two-body collisions that do not obey the Stosszahlansatz due to correlations resulting from previous collisions and from differences in position of the colliding particles. It is not yet clear that there is a truly satisfactory derivation of the Boltzmann equation from first principles in which the intuitive assumptions are given a rigorous underpinning. Progress in this direction was made by O. E. Lanford III [415, 413, 414] who derived the Boltzmann equation in the Boltzmann–Grad limit [264, 265], valid for a short time after some initial state, proportional to the mean free time. The Boltzmann–Grad limit is one in which the the number density, n of the gas approaches infinity, the size of the molecules, a approaches zero such that the mean free path is kept constant, implying that the combination na d−1 remains finite and nonzero. This has the effect of forcing the gas to be very dilute, since na d approaches zero. In this limit, there are no complications due to many-body or correlated binary collisions, since their contributions vanish as the density of the gas approached zero. Lanford’s results have been improved in some respects by other workers, but remain essentially the strongest rigorous derivation available of the Boltzmann equation. For a careful discussion of such derivations we refer to the papers of C. Villani [667]. Lanford’s results have also been used to obtain other properties of dilute gases such as time correlation functions for a gas of hard spheres by H. van Beijeren, O. E. Lanford III, J. L. Lebowitz, and H. Spohn [647]. For a much simpler model, the dilute random Lorentz gas, to be discussed in Chapter 5, much more work has been done. The derivation of the Boltzmann equation for the Lorentz gas, in the Boltzmann–Grad limit, has been developed by a number of authors, including G. Gallavotti [236, 237], H. Spohn [604, 606], and by C. Boldrighini, L.A. Bunimovich, and Ya. G. Sinai. [56] By approaching the Boltzmann–Grad limit ever closer, either for a Lorentz model or for a general gas, one may try to extend the range of validity of the Boltzmann equation to longer times, expressed in mean free times. However, at each density there is a maximum time, beyond which density corrections to the Boltzmann equation become more important.46 As a consequence of this no proof of the existence of positive and finite transport coefficients (or for the Lorentz gas,

112

The Boltzmann Equation 2: Fluid Dynamics

the diffusion coefficient) has been obtained up till now. We refer the reader to the original papers and to the book of Spohn [606] for more detail. In the discussions so far we have operated under the assumptions that the Boltzmann equation actually has smooth47 solutions, one of which is the normal solution. The validity of this assumption is not at all obvious. In fact, only very few results are known about the existence of solutions to the nonlinear, spatially inhomogeneous Boltzmann equation [76, 79, 81, 606, 85, 667]. From the mathematical viewpoint proving the existence of solutions to this equation is an extremely hard problem and many questions remain open. For example, among the many questions one might pose are: 1. Can one first prove the existence of solutions of the nonlinear Boltzmann equation in the simpler, spatially homogeneous case where the interplay between the effects of spatial inhomogeneities and those of collisions is absent? 2. Can one prove the existence of solutions to simplified, model Boltzmann equations, where the effects of collisions are represented in some mathematically more tractable way than might be possible for realistic systems? 3. Can one describe the conditions on the initial value of the distribution function, and the boundary conditions that will lead to a solution of the Boltzmann equation that is close to the Chapman–Enskog solution48 ? 4. Can one provide rigorous estimates for the rate at which the distribution function for a dilute, isolated gas in a container with simple boundary conditions, approaches equilibrium? Under what conditions and to what approximation is the approach to equilibrium exponential or power law or uniform in velocity? 5. Under what conditions is the late stage of the approach of a system to thermal equilibrium described by the linearized Boltzmann equation? One should not be surprised that only a few mild results about the existence of solutions to the full, nonlinear, spatially inhomogeneous Boltzmann equation are known. A detailed review of the mathematical literature has been prepared by C. Villani [667] and we refer the reader to his papers as well as to the books by Cercignani for more complete discussions [81, 83, 85]. In the subsections beyond the next one we will make some observations about solutions to some model Boltzmann equations. 3.10 Moment Expansions and Variational Methods for the Boltzmann Equation The earliest work on moment methods for constructing solutions to the Boltzmann equation was carried out by Maxwell [465]. In fact, his development of moment equations to describe transport processes predated Boltzmann’s derivation of the

3.10 Moment and Variational Methods

113

equation for the distribution function by a few years.49 Maxwell’s equations took an especially simple form for molecules interacting with forces that vary as the inverse fifth power of their separation (the Maxwell molecules we encountered already). Moment methods are, or can be, more general than the normal solution method of Chapman and Enskog. The normal solution leads to closed equations for the first d + 2 moments of the distribution function. However, it is but unable to describe boundary layers, processes that decay on shorter timescales than the hydrodynamic time, the dynamics of shock fronts and other situations that can be treated, to some approximation by moment expansions. In the moment expansion method one expands the distribution function  in a complete set of polynomial functions of velocity, with space- and time-dependent coefficients. The Boltzmann equation leads to an infinite set of coupled equations for these coefficients, which has to be solved to some degree of approximation. The most common approximation consists of setting all but the first n-coefficients equal to zero, and then solving the resulting coupled, but closed set of equations for these n-coefficients. The best known and most used form of the moment equations is due to H. Grad [264, 265] who, for three-dimensional systems, suggested that the 13 moments of the polynomials of lowest degree in a Hermite polynomial expansion are the most important ones. These moments, correspond to linear combinations of the quantities n(r,t),n(r,t)u(r,t),n(r,t)kB T (r,t),q(r,t), and P(r,t), involving the density, local velocity, temperature, heat flow vector, and pressure tensor, respectively. These add up to 14 independent quantities (where the symmetry of the pressure tensor has been taken into account), but one of them, the trace of the pressure tensor, is determined by the hydrostatic pressure which is the product of n and kB T . This method is called the Grad thirteen moment method. Equations for these moments are obtained from the Boltzmann equation. The set of equations for the moments are not closed, since further moments are needed in order to solve the equations. Thus various closure schemes have been proposed, each of which can be problematic and subject to various methods of improvement. Moment expansion methods, particularly the Grad moment system, have been used to describe the decay of an initial nonequilibrium disturbance to a state where the normal solution is appropriate, to discuss flow phenomena where the various time and length scales are not well separated, to describe boundary layer phenomena in Couette, Poiseuille, and Stokes flows, and to describe shock waves at low Mach number [83]. The failure of the thirteen-moment method to describe shock waves above a certain Mach number is one indication that this method has limited application and very few, if any, rigorous results are known about the quality of this approximation to the distribution function. Another important technique for constructing solutions to the Boltzmann equation is based on variational principles that can be derived for the linearized

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The Boltzmann Equation 2: Fluid Dynamics

Boltzmann equation. The fact that the linearized collision operator is self-adjoint with a simple definition of an inner product implies that variational methods, familiar from quantum mechanics, can be applied to the Boltzmann equation also. One particularly useful feature of variational methods is that quantities of physical interest, such as the drag on an object or the flow rate through some bounded system can be related directly to the stationary point in a variational equation. One application of this method has been to obtain the drag on a sphere in a gas stream at low Mach numbers for all values of the ratio of the mean free path length of the gas to the radius of the sphere. The results are in very good agreement with experimental results from Millikan oil-drop experiments [83]. 3.11 Model Boltzmann Collision Operators In view of the difficulties encountered when trying to solve the Boltzmann equation, one often tries to simplify the mathematical structure while retaining some features that reflect the physical behavior of the system. Such simplifications might include a linearization of the equation for systems close to equilibrium, or a consideration of spatially homogeneous situations to focus on the relaxation of some velocity disturbance to equilibrium, or a simplification of the collision term, replacing it by a simple model form. This way the collision integral may be made more tractable mathematically, even at the cost of losing contact with a realistic intermolecular potential. All of these methods have been pursued by several researchers, and here we will discuss just a few of the model systems that are useful for gaining insights into the general properties of solutions of the Boltzmann equation. 3.11.1 The BGK Equation There are three features of the Boltzmann collision integral that one would like to preserve when constructing simple models for it. These are: (a) The H -theorem, leading to a decrease of the H -function with time, unless the distribution function is of the Maxwell–Boltzmann type. Closely connected to the H -theorem are two properties that one would also like to preserve in models, namely (b) The existence of conserved functions of velocity which give rise to the value of zero whenever the model collision operator acts upon a distribution function that is a form of the Maxwell–Boltzmann distribution function or, for linearized models, the first order distribution function  is a linear combination of conserved functions; and (c) The negative definiteness of the collision operator whenever the distribution function is not a Maxwell–Boltzmann distribution (or  is not a linear combination of conserved functions); (d) the eigenvalue spectrum has a gap between zero and the real parts of the other eigenvalues. A simple model with all of these features

3.11 Model Boltzmann Collision Operators

115

was introduced by P. L. Bhatnagar, E. P. Gross, and M. Krook [40, 280] and by P. Welander [678] in 1954. It is generally called the BGK model. The model is constructed by replacing the full collision integral J (f ,f ) by Jbgk (f ) where Jbgk (f ) = ν[floc (f ) − f (r,v,t)],

(3.11.1)

where floc (f ) is a local equilibrium distribution function constructed to have the same average values of the conserved quantities as the distribution function f . That is   (3.11.2) dvfloc (f )ψi (v) = dvf (r,v,t)ψi (v), where, as before, ψi (v) is any one of the d + 2 conserved quantities, 1,v, and v 2 . The quantity ν plays the role of a collision frequency, and is usually treated as an adjustable parameter. In principle it may depend on the local values of the moments of the distribution function. The BGK equation is self-consistent since the distribution function f determines the local hydrodynamical variables, n(r,t),u(r,t), and T (r,t), in the local equilibrium distribution function. For a spatially inhomogeneous system, the BGK equation is, ignoring boundary effects, ∂f (3.11.3) + v · ∇f = ν[floc − f ]. ∂t The H -theorem for the BGK equation follows immediately from the observation that ln floc is a linear combination of conserved quantities, so that, with the help of Eq. (3.11.2) one sees that  dv (floc − f ) ln floc = 0, (3.11.4)   hence − ∂t∂ dvf (ln f − 1) = ν dv(ln floc − ln f )(floc − f ) ≥ 0. One can also apply the Chapman–Enskog procedure to the BGK equations and obtain Euler, Navier–Stokes, and higher order equations, albeit with transport coefficients that differ in some details from those for more realistic systems.50 A particularly useful form of the BGK equation is obtained by linearizing both f and floc about their total equilibrium forms. One writes f = feq (v)[1 + φ(r,v,t)] + · · · 

 δn δβ d mv 2 floc = feq (v) 1 + + mβ0 v · u + − β0 + ··· . n0 β0 2 2

(3.11.5) (3.11.6)

Here, the total equilibrium distribution function is given by the usual Maxwell– Boltzmann form

d β0 m 2 − β0 m v 2 e 2 , (3.11.7) feq = n0 2π

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The Boltzmann Equation 2: Fluid Dynamics

where the subscript zero denotes the uniform equilibrium value of the density or the temperature. If we ignore all terms of quadratic and higher order in the deviations from total equilibrium, we obtain the linearized BGK equation

  ∂φ δβ d δn β0 mv 2 . − β0 mv · u − + v · ∇φ = −ν φ − − ∂t n0 β0 2 2

(3.11.8)

The important feature of the right-hand side of this equation is that the collision operator projects the distribution function φ onto the space orthogonal to the d + 2 dimensional space spanned by the conserved quantities. Therefore any function that is orthogonal to the space of conserved quantities is an eigenfunction of the linearized BGK collision operator, with eigenvalue −ν. One might say that the linearized BGK operator is a simplification of any linearized Boltzmann collision operator with a gap between zero and the negative eigenvalues, in which all of the nonzero eigenvalues have been replaced by a common eigenvalue for all of the corresponding eigenfunctions.51 In a more formal notation, the linearized BGK operator takes the form −ν(1 − P0 ) where P0 is the projection operator onto the space spanned by the conserved quantities. This allows for the mathematical treatment of many situations of physical interest, such as a determination of the structure of kinetic boundary layers. Although the numerical results obtained this way for quantities such as velocity profiles and the values of slip lengths are likely to be different from those for more realistic models, one expects that the overall physical features will be essentially the same. 3.11.2 Maxwell Models, the Kac Model Boltzmann Equation and BKW Modes Often one-dimensional forms of complicated equations may be simpler to solve and yield valuable insights into physical phenomena. For example, a one dimensional form of the Navier–Stokes equation for the velocity of a fluid, invented by J. Burgers and known as the Burgers equation [75] was designed to yield insights into the description of turbulent behavior in a fluid. Although that has not turned out to be the case, the Burgers equation does describe the formation and behavior of solitons in fluids. Thinking along these lines, M. Kac proposed a one-dimensional, nonlinear model Boltzmann collision operator which has received a considerable amount of attention in the literature [328, 329]. The Kac–Boltzmann equation is ∂f (v,t) ∂f (v,t) +v = JK (f ,f ), ∂t ∂x

(3.11.9)

3.11 Model Boltzmann Collision Operators

where the Kac–Boltzmann collision operator JK (f ,f ) is given by  ∞  π dw dθσ (θ)[f (v  )f (w ) − f (v)f (w)], JK (f ,f ) = −∞

117

(3.11.10)

−π

where v,w are taken to be one-dimensional velocity variables, θ is an angle that determines the effect of a binary collision, σ (θ) = σ (−θ) plays the role of a one-dimensional “scattering cross section,” and the restituting velocities, v ,w are given by v  = v cos θ − w sin θ; w = v sin θ + w cos θ.

(3.11.11)

This Kac operator conserves particles and energy but not momentum. The Kac model is in the class of Maxwell models discussed earlier. These models have the ˆ is independent of the relative velocproperty that the collision rate, |v − v1 |B(g, k) ity, or equivalently, the relative energy of the two particles, and depends only on the scattering angle. The Kac model is somewhat simpler than most Maxwell models since the velocities are all one-dimensional. Nevertheless its spatially homogeneous version possesses a solution which is of considerable general interest, since related solutions exist for all of the nonlinear, spatially homogeneous Maxwell models. In 1976 Bobylev, and independently, Krook and Wu [44, 402, 403, 177, 178, 45] published exact solutions, called BKW modes, for the spatially homogeneous, nonlinear Boltzmann equation for Maxwell models. Their work stimulated a great deal of activity with the result that other, related solutions to these models equations have been discovered. Moreover, the spectrum for the linearized version has been studied in some detail with the result that, among other things, there are eigenfunctions of the linearized collision operator outside the Hilbert space of square integrable functions, but within the space of integrable functions, with eigenvalues forming a continuous spectrum including the value zero. There is no spectral gap between zero and the first non-zero eigenvalue, meaning that there is no slowest decay mode to equilibrium for these models if one expands the space of allowable functions beyond L2 . We refer to the literature for details [177]. In order to illustrate the method we present a simple case, the BKW mode solution for the Kac–Boltzmann equation, Eq. (3.11.9).52 We consider the spatially homogeneous Kac equation  ∞  π ∂f (v,t) = JK (f ,f ) = dw dθσ (θ)[f (v  )f (w ) − f (v)f (w)], ∂t −∞ −π (3.11.12) with restituting velocities given by Eq. (3.11.11). We assume that the total energy of the gas is finite and since it is constant, we can normalize the velocity according to

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The Boltzmann Equation 2: Fluid Dynamics

 

∞

dvf (v,t) = 1.

(3.11.13)

dvv 2 f (v,t) = 1.

(3.11.14)

−∞ ∞

−∞

A simple version of the BKW method looks for solutions of Eq. (3.11.12) of the form f (v,t) =

exp[−v 2 /2s(t)] (A(t) + v 2 B(t)), (2πs(t))1/2

(3.11.15)

where A(t),B(t) and s(t) are to be determined by fitting this trial solution to Kac model equation, Eq. (3.11.12). The normalization conditions alone lead to A + sB = 1,

(3.11.16)

sA + 3s B = 1,

(3.11.17)

2

which can be solved for A,B in terms of s, leading to   exp[−v 2 /2s(t)] 3s(t) − 1 2 1 − s(t) . +v f (v,t) = (2πs(t))1/2 2s(t) 2s(t)

(3.11.18)

The function s(t) is determined by the Kac equation, by comparing powers of v on the right- and left-hand side of Eq. (3.11.12) using Eq. (3.11.18) as a trial solution. This procedure yields a very simple differential equation for s(t) given by ds(t)/dt = λ(1 − s(t)), where λ is  π dθσ (θ)(sin θ cos θ)2 . (3.11.19) λ= −π

This yields an expression for s as s(t) = 1 + ηe−λt ,

(3.11.20)

where η is a constant of integration. If one takes t = 0 as the initial time, positivity of the distribution function requires that − 23 < η < 0. This solution is the BKW mode for the Kac equation. It is called a similarity solution because the solution retains the same functional form in the velocity variable for all times t, the only change with time comes through the variation of the function s(t) with time as it changes exponentially from the initial value 1 + η to its asymptotic value, s(t = ∞) = 1, and the solution of the Kac equation approaches the Maxwell–Boltzmann form. The BKW solution has the feature

(b)

1.00

1.20

R

119

1.40

3.12 Other Models

.80

t = 75

0.00 .20

.40

.60

t = 60

t = 45 t = 30 t=15 0 100

200

300

400

500

800 v

700

Figure 3.11.1 BKW modes. The plot represents the ratio of the BKW solution to the equilibrium solution as a function of velocity for various times. The approach to equilibrium is not uniform in velocity and approaches the equilibrium from below. This figure is taken from the paper of M. H. Ernst [177]

that the approach to the Maxwell–Boltzmann distribution becomes slower as the velocity v increases. The ratio, R(v,t), of the exact solution to the equilibrium distribution function is plotted in Fig. 3.11.1, and one can see that the approach to equilibrium is not uniform in the velocity and the equilibrium value R = 1 is approached from below. There are a number of sophisticated methods that can be used to find the BKW modes for nonlinear Boltzmann equations for a variety of Maxwell models. Moreover, additional solutions are known for the spatially homogeneous case, for gas mixtures, for systems with external forces, and other cases. At one time it was conjectured that any typical solution to model Boltzmann equations of the Maxwell type would reach the final equilibrium state via BKW similarity modes. That is, an arbitrary solution would rapidly approach a BKW mode and then decay to equilibrium via this mode. This conjecture turns out to be false [303]. Nevertheless, in view of the rarity and importance of finding solutions to the nonlinear Boltzmann equation, the work stimulated by the discoveries of Bobylev, Krook, and Wu cannot be underestimated. 3.12 Other Models The work described earlier does not nearly cover the range of different models for the nonlinear and the linear Boltzmann equation. Discrete velocity models,53 Lorentz models,54 lattice gas models, stochastic models of various sorts, and many others have been investigated [606, 559].

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The Boltzmann Equation 2: Fluid Dynamics Notes

1 The mean free path length for a particle in a dilute gas is on the order of (na d−1 )−1 in d dimensions. 2 Mathematical discussions and proofs related to the asymptotic properties of the Chapman–Enskog expansion are given by H. Grad, C. Cercignani, and C. Villani [264, 265, 266, 83, 85, 667]. Villani’s paper, in particular, has a very extensive and useful bibliography. 3 So we represent the walls by a “hard” potential, which jumps to ∞ abruptly. One could also choose to represent them by smooth potentials that increase from zero to ∞ over an atomic size. This potential would enter the Boltzmann equation through the v˙ · ∇v -term. The problem with this is that the length scale on which the external potential varies in this is case is short instead of long compared to the mean free path. This makes the term hard to handle within the standard framework for solving the Boltzmann equation. And as representation of an actual wall, it is hardly more realistic than a hard potential. 4 Van Kampen has written: “The miracle that statistical mechanics is called upon to explain is the empirical fact that systems having 1023 degrees of freedom can be described, on a coarse, macroscopic scale by a much smaller number of variables, in such a way that these macroscopic variable again obey deterministic equations of motion.” [654]. 5 For example, this requirement rules out any term in the expression for the heat current that is linear in the gradient of the local gas pressure. For a discussion of this point see Landau and Lifshitz [412], p. 186. 6 This assumes that the fluid consists of mechanically identical particles. We consider multicomponent fluids in Chapter 4. 7 The relative velocity c is often called the peculiar velocity. 8 In cases of penetrable or adsorbing walls, there may also be nonvanishing contributions to the mass current, but for simplicity, we will ignore this possibility in the sequel 9 Some care is needed when defining the currents for an arbitrarily dense fluid, and there is some dependence upon the type of ensemble used when taking averages [488, 272]. 10 This is correct apart from a term in the diagonal elements of the pressure tensor that is proportional to the hydrostatic pressure in the fluid. See Eq. (3.1.20). 11 As mentioned in Chapter 1, there are circumstances, such as those occurring in the phenomenon of Landau damping associated with the reversible Vlasov equation, in which reversible equations can exhibit decays to equilibrium. This can occur under circumstances where the solution of the reversible equation has a fractal structure. For a detailed and careful discussion of this subject, we refer the reader to the book by N. van Kampen and B. U. Felderhof [656]. It is useful to know that what are now referred to as fractal properties of distribution function were called “corrugated functions” by these authors. The word “fractal” had not yet been coined when their book was written. A more modern discussion of the mathematics of Landau damping can be found in the paper by C. Mouhot and C. Villani [494]. 12 This is done in order to ignore wall contributions at the moment. 13 On tensorial grounds an additional term proportional to the gradient of ρ would be entirely admissible. However, such a term immediately would lead to conflicts with the second law of thermodynamics, as it would lead to terms proportional to ∇p · ∇T in the expression for the rate of entropy production, which are not necessarily positive, and reflect the fact that a ∇n term in the expression for the heat flow vector would allow for heat currents in the direction opposite to the temperature gradient. For discussions of possible gradient terms in the heat current, for example, see the book of Landau and Lifshitz [412]. 14 The entropy density is defined in terms of the local densities of mass and internal energy, with coefficients determined by the local equilibrium distribution. Consequently, there is no ambiguity in the definitions of these quantities. 15 For sound propagation, this corresponds to the inverse sound frequency. For diffusive phenomena, such as heat conduction or shear flow, this is actually not the most relevant macroscopic time scale. In such a case, a good characteristic time would be the average time it

Notes

16

17 18 19

20 21

22 23 24 25 26

27 28

121

takes for a particle performing a random walk, with step length  and jump frequency t−1 to move over a distance L. This latter time is proportional to L2 rather than to L. Exceptional cases would include descriptions of an initial transient state of the gas before it settles into a state described by the Chapman–Enskog solution, the description of gas flows near a strong shock front, or near a wall. That is to say that even in the hydrodynamic regime indicated by the inequalities, Eq. (3.2.1), the Chapman–Enskog solution must be extended to a more microscopic description of the gas under any circumstances where the distribution function changes appreciably over short times and/or over small distances [79, 81, 83, 85, 84]. This argument is due to E. G. D. Cohen [103]. Our presentation of the normal solution method is based upon, and is a some what shortened version, of the standard presentations found in the basic references on the Boltzmann equation, a few of which are listed here [306, 631, 103, 89, 83]. Here we use the symbol μ in two ways: as the small parameter defined before, and as an ordering parameter in the Chapman–Enskog expansion that indicates the order of the gradients of the hydrodynamic fields. At the end, we set μ = 1 since the order of the gradients for each term will then be clear. A very useful discussion of the Hilbert expansion can be found in the book of Cercignani [83]. See also [266, 297]. This situation is addressed mathematically by Hilbert’s Uniqueness Theorem [83]. The normal solution method that we develop in this chapter leads to a solution of the Boltzmann equation that depends only upon the initial values of the hydrodynamic fields, n(r,0),u(r,0),T (r,0). Hilbert’s Uniqueness Theorem states that if the distribution function f˜(r,v,t) can be expanded in a convergent series in powers of some small parameter, then this function is uniquely determined by the initial values of the hydrodynamic fields. Under such circumstances, we would obtain the same distribution function whether we were to expand the distribution function about a total equilibrium distribution in powers of some small parameter characterizing the deviations from total equilibrium, or about local equilibrium in powers of μ. This all assumes that the series expansions converge. Otherwise, we must look upon the requirement Eq. (3.3.4) as a solubility condition for the μ-expansion about local equilibrium. In view of the fact that in many cases the expansions are likely to be asymptotic rather than convergent, Hilbert’s theorem is then of little use, the mathematical foundations of the method are not secure, and one must rely on physical intuition and experiment to validate the procedure. However, there are examples, notably in the case of hard-sphere gases or other potential models with finite total cross sections, and some periodic Lorentz models, where one can argue for the convergence of some of the expansions. Important results for these cases are given J. A. McLennan [476] and C. Dettmann [143, 144]. It is worth noting that the local equilibrium distribution function is not a solution of the Boltzmann equation. It has the important property that it does not change in binary collisions, i.e. it satisfies Eq. (3.3.2). In what follows and whenever there is no ambiguity, we will use ∇ for ∇r . This idea was taken up and generalized by N. N. Bogoliubov in his seminal work on extending the Boltzmann equation to higher order in the density [55]. Here, and in the following steps in the Chapman–Enskog method, it is more convenient to use the peculiar velocity, c, as the velocity variable, instead of the laboratory velocity, v. J. Piasecki and Y. Pomeau have discussed the circumstances under which solutions to Eqs. (3.4.13) and (3.4.14) exist [526] for hard-sphere particles. They show that one must tame the large-velocity behavior of the solutions. The solutions, more precisely, approximate solutions, to be constructed in what follows are well behaved at large velocities and not problematic. In the second line of Eq. (3.4.17), we have inserted a vanishing term so as to write the expression for λ in a symmetric form. Similar convergence results are known for systems that are intrinsically linear such as in the case of diffusion of tracer particles in an equilibrium fluid [363] and that of diffusion of moving particles in Lorentz gases with fixed scatterers [144].

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The Boltzmann Equation 2: Fluid Dynamics

29 This form applies only to dilute gases. For denser gases, there is an additional term proportional to the bulk viscosity, a quantity that vanishes to lowest order in the density. 30 We suppose that r is in the interior of the gas, and we can drop the wall term. 31 In this notation, v is the incident velocity, and v is the velocity of the particle immediately after collision with the wall. The collisions are taken to be instantaneous. 32 Excellent and detailed discussions of boundary conditions, with many references, can be found in the books by C. Cercignani [83, 85]. 33 That is, the angle of reflection of the particle’s velocity is equal to the angle of incidence. 34 Recently, it has been shown by Fan and Manson [213] that this reflection model is actually realized to a very good approximation in the scattering of argon from a monolayer of decanethiol adsorbed on a gold surface, as observed by Gibson et al. [250]. 35 As would occur if a macroscopic, solid body, were moving in the fluid. Such a situation is encountered in the derivation of Stokes’ law, for example. 36 Such as the coefficient of thermal diffusion, the thermal-diffusion ratio, etc. 37 There are a number of books and articles that contain data on the transport properties of monatomic gases. Here we list a representative sample of them. The literature is sufficiently large that rather than listing all of the relevant papers, we recommend that the interested reader use Google Scholar or the Science Citation Index to find recent papers that cite some of those listed here [306, 292, 293, 61, 358, 215, 288, 356, 531]. 38 Since we are now discussing the results of laboratory experiments, we will drop the superscript 3 that denotes the number of spatial dimensions of the systems under consideration. 39 Under this inner product,  is symmetric. 40 Some discussions of this point can be found in the papers [298, 188]. 41 General expressions for the transport coefficients for fluids can be found in [188]. 42 On extending the Boltzmann equation by adding so-called ring terms, one obtains transport coefficients that, for gases in two dimensions, diverge in this limit. This will be discussed in Chapter 14. 43 L. Sirovich and J. K. Thurber have discussed other models where the wave number expansion is divergent [596]. 44 The results depend only weakly on the intermolecular potential. Similar results for Lennard-Jones potentials differ from those for Maxwell molecules by at most 7%. 45 There is a large literature on obtaining the dispersion relations for sound using the Boltzmann equation. A number of different methods have been used and some approximate methods lead to very good agreement with experimental data. Here, we list a few of the papers [521, 595, 594, 74, 224, 87, 222, 223, 225, 83]. 46 These events include multiparticle collisions taking place over short distances and short times, as determined by the range of the intermolecular forces, as well as correlated sequences of collisions that take place over much longer times and distances, determined by the mean free path length between collisions. 47 That is, solutions for the distribution functions that are continuous and have continuous derivatives with respect to the space, time, and velocity variables. This assumption plays an important role in the derivation of the H -theorem. 48 In Chapter 5 we will present an exact solution for Boltzmann equation for particles in the three-dimensional Lorentz gas with random, hard sphere scatterers given by E. H. Hauge [298]. For this model, at least, one can compare the exact solution with the normal one. 49 Maxwell’s work dates from 1866 and Boltzmann’s from 1872. Boltzmann’s achievement was to derive an equation for the full distribution function, not just some of its moments, and to derive the H -theorem. 50 The ratio of the viscosity to the thermal conductivity differs numerically from that for a more realistic model, for example. 51 There is at least one physical model with a collision operator of the BGK form, namely, the dilute, hard-sphere Lorentz gas, in three dimensions, discussed here in Chapter 5 [298]. 52 Here we follow the presentation of Ernst [177, 178]. 53 At this point it is not inappropriate to quote Marc Kac, “Be wise, discretize.” 54 We discuss Lorentz gas models in Chapter 5.

4 Transport in Dilute Gas Mixtures

4.1 Introduction In this chapter, we will consider the general theory of dilute gas mixtures and pay particular attention to their hydrodynamic properties. After developing the normal solution of the Boltzmann equation for mixtures, we discuss the phenomena associated with the diffusion of the different species in the gas. There is a close coupling of the flow of particles and the flow of energy, leading to a number of interesting cross effects such as thermal diffusion, whereby a temperature gradient can produce a differential flow of particles of different species. Similarly, a concentration gradient can produce a heat flow in the fluid. The relations between the transport coefficients characterizing these cross effects are determined by the Onsager reciprocal relations, to be discussed later in this chapter. Cross effects are of considerable practical importance. For example, kinetic theory, particularly the first explanation1 of the phenomenon of thermal diffusion, has been used to devise methods to separate nuclear isotopes. These methods have been, and continue to be, of primary importance for nuclear technology. In later chapters, we will consider other, not necessarily dilute mixtures that are of particular interest for a variety of reasons. Such mixtures will include the following2 : • Lorentz gas models, in which light particles collide with much heavier ones. In the limit that the ratio of the light particle mass to the mass of the heavy particles becomes zero, we may imagine that on time scales characteristic for the motion of the light particles, the heavy particles are fixed in space. In this case, one can require that the fixed particles either form a regular lattice or are placed at random in space, with or without allowing them to overlap each other. Typically, the light particles are treated as ideal gas particles with no mutual interactions, but this is not necessary. These models will be discussed in the next chapter.

123

124

Transport in Dilute Gas Mixtures

• Tracer particle models. Models of large particles in a gas of smaller ones may be used to describe the motion of a Brownian particle in a gas. One can vary the ratio of the mean free path of the gas particles to some characteristic size of the Brownian particle, ranging from hydrodynamic behavior when this ratio is very small to rarefied gas behavior when it is very large. Rarefied and hydrodynamic flows around large objects, as well as Brownian motion will be discussed, briefly, in Chapter 15, Sec. 15.2. 4.2 The Boltzmann Equation for Dilute Gas Mixtures The Boltzmann transport equation for mixtures of dilute gases is obtained by a simple generalization of the arguments given by Boltzmann for the pure gas case.3 We assume that the Stosszahlansatz applies to all of the varieties of binary collisions that can take place in such a gas – that is, collisions between two particles of like species and collisions of two particles of unlike species. We assume that the colliding particles interact with central forces.4 In the description of the mixtures, we will denote the different species by the subscript α and the distribution function for particles of species α by fα (r,v,t). The Boltzmann equation for the distribution functions fα (r 1,v1,t) can be simply obtained. To do this, we need to give expressions for the restituting velocities or momenta. The equations for the restituting momenta in terms of the apse vector are found in the same way as for like-particle collisions, by combining the laws for conservation of momentum and energy with the time reversibility of the equations of motion. In this way, one easily finds that a collision between a particle of mass, m1, and another of mass, m2, produces a change in velocity given by ˆ k, ˆ v1 = v1 − 2(μ12 /m1 )(v12 · k)

(4.2.1)

ˆ k. ˆ v2 = v2 + 2(μ12 /m2 )(v12 · k)

(4.2.2)

ˆ the apse vector, is a unit vector in the relative coordinate system, from the Here k, center of the action sphere to the symmetry point of the trajectory along the line of symmetry in the plane of the binary collision. Also, μ12 is the reduced mass of the two colliding particles, and g = v12 = v1 − v2 is the relative velocity of the colliding particles, the magnitude of which is readily shown to be conserved in a binary collision, as follows from ˆ k. ˆ g  = v1 − v2 = g − 2(g · k)

(4.2.3)

Since the two-body problem is solvable for central forces, no matter what the masses of the two particles are, the direction of the vector kˆ is well defined for any central potential. One can easily show, using these collision equations, that the ˆ Jacobian is unity for changing variables from v1,v2 to v1,v2 at fixed k.

4.2 The Boltzmann Equation

125

The Boltzmann equation for fα (r 1,v1,t) is given by ∂fα (r 1,v1,t) + v1 · ∇fα (r 1,v1,t) + Fα · ∇v1 fα (r 1,p 1,t) ∂t  Jαβ (fα,fβ ) + T¯ W,α fα (r 1,v1,t). =

(4.2.4)

β

We suppose that the external forces per unit mass, Fα, do not depend upon velocity but do depend upon the particle species. The term Jαβ describes the effects of binary collisions between a particle of species α with a particle of species β on the distribution function fα (r 1,v1,t). It is given explicitly by   ˆ αβ (g, k) ˆ Jαβ (fα,fβ ) = dv2 d kB   × fα (r 1,v1,t)fβ (r 1,v2,t) − fα (r 1,v1,t)fβ (r 1,v2,t) .

(4.2.5)

ˆ kˆ = gbαβ dbαβ d Here we have defined the binary scattering function Bαβ (g, k)d in terms of the differential scattering cross section, as was done in Chapter 2 for ¯ W,α fα (r 1,v1,t) represents the the case of collisions of like particles. The quantity T change in fα (r,v1,t) due to collisions with the walls. We include the particle species subscript on the wall collision operator, since the outcome of such a collision might depend upon the particle species. 4.2.1 The H -Theorem for Mixtures An important ingredient for understanding the physical consequences of the Boltzmann equation is the H -theorem, describing the rate of increase in the nonequilibrium entropy, S(t) = −kB H (t), due to irreversible processes taking place in the gas. Here, the H (t) function is defined by   H (t) = dr 1 h(r 1,t) = dr 1 dv1 fα (r,v1,t)[ln fα (r,v1,t) − 1]. (4.2.6) α

The quantity h(r,t) is the local H -function. Using Eq. (4.2.4), we obtain an equation for h(r,t) as  ∂h(r 1,t) dv1 ln fα (r 1,v1,t)Jαβ (fα,fβ ) + ∇ · Jh (r 1,t) = ∂t α,β  + dv1 ln fα (r 1,v1,t)T¯ W,α fα (r 1,v1,t), (4.2.7) α

where we have assumed that the external forces do not depend on the velocities of the moving particles. The current of the h-function, Jh , is simply the net flow of h per unit volume in a small region around the point r 1 and is given by

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Transport in Dilute Gas Mixtures

Jh (r 1,t) =



dv1 v1 fα (r 1,v1,t)(ln fα (r 1,v1,t) − 1).

(4.2.8)

α

The binary collision contribution to ∂h/∂t can be transformed to the, nonpositive quantity  1 dv1 ln fα (r 1,v1,t)Jαβ (fα,fβ ) 4 α,β  1 ˆ αβ (g, k) ˆ dv1 dv2 d kB (4.2.9) = 4 α,β × [fα (r 1,v1,t)fβ (r 1,v2,t) − fα (r 1,v1,t)fβ (r 1,v2,t)]   fα (r 1,v1,t)fβ (r 1,v2,t) ≤ 0. × ln fα (r 1,v1,t)fβ (r 1,v2,t)

(4.2.10)

As is the case for one-component gases, the binary collisions decrease the value of h(r,t). The analysis of the contribution to the change in h from the wall collisions is essentially the same as in the pure gas case and will not be considered any further here. The basic conclusions from the H -theorem are the same as in the pure gas case, namely that if the wall collision operator for each species of particles describes collisions that result in the re-emission of particles with a distribution characterized by a wall temperature, TW (ρ S ), at the point ρ S on the wall and the same temperature for each species, then the change in the total value of H satisfies   1  dH (t) mα v2 1 ˆ dv(v · n) dS (4.2.11) ≤− fα (ρ S ,v,t). dt kB ∂V TW (ρ S ) 2 α Here the notation is the same as that used in Chapter 2, Eqs. (2.2.8–2.2.9). The property of most importance to us for the development of the fluid dynamics of gas mixtures is that the binary collision contribution to the change in the h-function vanishes for local equilibrium distribution functions whenever the local temperature and the local mean velocity are the same for all the species. 4.3 The Chapman–Enskog Solution Now we turn our attention to the Chapman–Enskog derivation of the equations of fluid dynamics for dilute gas mixtures. This derivation provides a set of equations that describe a rich variety of phenomena in addition to viscous and thermal flows due to the addition of variables describing the concentrations and flows of different species in the gas. Among these are cross effects mentioned earlier, in which gradients in one hydrodynamic density can produce a current of another hydrodynamic density.5 A general description of such cross effects was provided by Onsager who

4.3 The Chapman–Enskog Solution

127

showed that the transport coefficients satisfy a set of symmetry properties, called Onsager reciprocal relations, which are based on the time-reversal invariance of the microscopic equations of motion of the constituent particles [508, 509]. We discuss the Onsager relations in the next subsection. 4.3.1 The Onsager Reciprocal Relations The calculations in the previous chapter for the rate of entropy production can be put in a form that is part of a very general description of irreversible process in systems that are close to thermodynamic equilibrium, with small deviations about it. This description of irreversible processes is due to Onsager, and his main result is a set of symmetry relations called Onsager reciprocal relations. These reciprocal relations are very important for describing irreversible processes in fluid mixtures, and we shall have occasion to refer to and use them. They are of crucial importance for finding the correct formulation of the Enskog theory for dense gas mixtures, and this aspect will be discussed in Chapter 7, which is devoted to the Enskog theory. Here we will discuss the kinetic theory of dilute gas mixtures as a mesoscopic theory underlying the Onsager relations. Much of our discussion of the kinetic theory foundations of nonequilibrium thermodynamics for dilute gases was first presented by I. Prigogine in 1949 [541]. In order to set the stage for this discussion, we make some general remarks.6 We assume that the system of interest is free of external forces and close to a total equilibrium state. The nonequilibrium state may be a consequence of fluctuations in the system or produced in some other way. Motivated by the gradient expansion of the local entropy density for a one-component fluid given by Eqs. (3.5.10)– (3.5.12) of the previous chapter, we describe the deviation of the entropy density from its local equilibrium value in terms of the gradients of a set of extensive variables or, more precisely, the gradients of local hydrodynamic densities that is the number densities, the momentum density, and the energy density. We will denote the gradients of the local densities by ai (r). We express the deviation of the entropy from its local equilibrium value, due to the presence of gradients, as sne (r) − sloc (r) = −

1 bi,j ai (r)aj (r) + · · · i,j 2

(4.3.1)

where the coefficients, bij , are elements of a positive definite matrix. Onsager makes the hypothesis, called the Onsager regression hypothesis, that the regression of the small deviations from equilibrium follows the linear laws of hydrodynamics, namely the linear relations between the currents of the local densities and the local gradients of the densities. These linear relations for a one-component gas include Fourier’s law of heat conduction and Newton’s law of friction, but they can be

128

Transport in Dilute Gas Mixtures

generalized to more complex systems and phenomena, as we will see later for mixtures. The linear relations of the currents of conserved quantities, Ji , to the gradients of the hydrodynamic fields denoted here have the form  Lij Xj . (4.3.2) Ji = j

The quantities Xi are called thermodynamic forces, defined by  ∂s =− bij aj , Xi = ∂ai j

(4.3.3)

where we used (4.3.1). The rate of entropy production, σ, will be shown in subsection 4.5 to be of the form   dS Ji Xi = Lij Xi Xj . (4.3.4) = σ = dt i ij The symmetries of the matrix of coefficients, Lij , known as the Onsager reciprocal relations, will be presented in Eqs. (4.3.10) and (4.3.11) [508, 509]. As an example of an expression for the rate of entropy production in a dilute, onecomponent gas, we quote the result obtained in the previous chapter’s Eq. (3.5.9), using Boltzmann’s H -theorem and the linear laws q = −λ∇T ,

2 η (∇u) + (∇u)T − 1(∇ · u) . P=− 2 d The entropy production, σ (r,t) is σ (r,t) = λ

η (∇T )2 + T2 2T



(4.3.5) (4.3.6)

2

(∇u) + (∇u)T −

2 1(∇ · u) d

,

(4.3.7)

which is a positive definite quadratic form depending on the square of the gradient of temperature, T , and the gradient of local fluid velocity, u, with transport coefficients λ,η, the coefficients of thermal conductivity and shear viscosity, respectively. Here one can identify the thermodynamic forces since the entropy production and the linear laws are given. For example, the thermodynamic force connected with the heat flow is Xq = −∇T /T 2 , and the corresponding Onsager coefficient is Lqq = λT 2 . In general, we say that the current, Ji , is conjugate to the force Xi . Given the linear relations, Eq. (4.3.2), and the positivity of the entropy production rate, we may easily conclude from Eq. (4.3.4) that the matrix of transport coefficients, Lij , must be positive definite. However, one can say much more about these coefficients and about the equations that describe irreversible processes, at least for processes that take place when the system is close to a local equilibrium state. The first statement that can be made is called the Curie symmetry principle. This principle

4.3 The Chapman–Enskog Solution

129

is based on spatial symmetries of the system and the fact that there are currents and gradients of different irreducible tensorial character. For example, the mass and energy currents, as well as the gradients of the mass and energy densities, are all vectors, while the momentum currents and the gradients of the local fluid velocity form second-order tensors. The Curie principle states that, for a spatially isotropic system, the relations between the currents and gradients, of the form given by Eq. (4.3.2), are such that different irreducible tensorial forms do not couple to each other. That is to say, vector currents are linearly related only to vector gradients, and irreducible second-order tensor currents are linearly related only to irreducible second-order tensor gradients. An example of this for a one-component gas can be seen in the expression, Eq. (4.3.7) where there is no coupling of the heat current to the gradient of the local velocity, to first order in the gradients because any such terms vanish due to to the different tensorial character of the terms in the gradient expansion of the single-particle distribution function. The Onsager reciprocal relations, in general, follow from symmetries of the system – in particular, the time-reversal invariance of the underlying microscopic equations of motion of the constituent particles. Recall that under a time-reversal transformation, the velocities of all the constituent particles change sign. This change does not affect some of the densities of conserved quantities, such as mass and energy densities, and their spatial gradients. However, mean velocities, for example, change sign under a time-reversal transformation. Now suppose that the current of a density that does not chance sign under time reversal is denoted by Ji(e) and its conjugate force by Xi(e) , and, similarly, the currents and conjugate forces of a density that does change sign under time reversal are denoted by Ji(o) and Xi(o) , respectively. One can then express the linear laws in the form Ji(e)

=

ne 

(e) L(ee) ik Xk

+

k

Ji(o) =

no 

no 

(o) L(eo) il Xl ,

(4.3.8)

(e) L(oe) ik Xk ,

(4.3.9)

l (o) L(oo) il Xl +

l

ne  k

where ne,no are the number of densities that are even or odd under time reversal, respectively.7 Onsager proved that, in the absence of an external magnetic field, the following identities are satisfied by the transport coefficients, Lij , (ee) L(ee) ij = Lj i , (oe) L(eo) ij = −Lj i .

L(oo) = L(oo) ij ji ,

(4.3.10) (4.3.11)

These are the well-known Onsager reciprocal relations. They take a somewhat different form when external magnetic fields are present, but we will not consider such cases at the moment. From a consideration of the rate of entropy production,

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Transport in Dilute Gas Mixtures

Eq. (4.3.4), we see that the elements L(ee) ij must form a positive definite matrix, as (oo) (oe) must the elements, Lij , while, Eq. (4.3.11) shows that the cross-terms L(eo) ij ,Lij make no contribution to the rate of entropy production in the system. We will show in what follows that the Boltzmann equation for mixtures leads to linear relations of the form Eq. (4.3.2) with coefficients that satisfy the Onsager relations. 4.3.2 Conservation Equations We begin our construction of the equations of fluid dynamics for mixtures by deriving the equations for conservation of particle numbers, energy, and momentum using the Boltzmann equation, Eq. (4.2.4). First, we remove the boundary terms so as to concentrate on the equations for the behavior of the fluid in the interior of the container. We do this by writing fα (r 1,v1,t) = W (r 1 )f˜α (r1,v1,t),

(4.3.12)

where W (r 1 ) is the Heaviside function for the container. That is, it is unity within the container and vanishes outside it. We then use the delta functions appearing in Eq. (4.2.4) to separate the Boltzmann equation into two equations, one containing the effects of collisions and external forces upon f˜α (r1,v1,t) and the other containing the boundary conditions on f˜α (r1,v1,t) at the walls. We assume the external forces vanish, for simplicity, and obtain  ∂ f˜α (r1,v1,t) Jαβ (f˜α, f˜β ) + v1 · ∇ f˜α (r1,v1,t) = ∂t β

(4.3.13)

and f˜α (r1,v1,t)v1 · ∇W (r 1 ) = TW f˜α (r1,v1,t).

(4.3.14)

For the time being, we focus our attention on Eq. (4.3.13). We next derive equations for ρα (r 1,t), the mass density of species α at point r 1 , at time t, for the local mean velocity, u(r 1,t), of the fluid, and for eK (r 1,t), the local mean kinetic energy of the fluid. These equations are obtained from the Boltzmann equation by multiplying the equation by mα,mα v1 and mα v12 /2, integrating over all velocities and then using the facts that masses, momentum, and kinetic energy are all conserved in binary collisions. We easily obtain a group of equations that follow from the conservation of masses. First, we multiply Eq. (4.3.13) by mα and integrate over the velocity, v1 to obtain ∂ρα (r 1,t) + ∇r 1 · [Jα (r 1,t) + ρα u(r 1,t)] = 0, ∂t

(4.3.15)

4.3 The Chapman–Enskog Solution

131

where ρα , the mass density of particles of species α, and Jα , the specific mass current of particles of species α, are given by  (4.3.16) ρα (r 1,t) = dv1 mα f˜α (r1,v1,t),  Jα (r 1,t) = ρα V α = dv1 mα (v1 − u(r 1,t)) f˜α (r1,v1,t). (4.3.17) In order to simplify the equations and notation in this chapter, we will often, but not always, omit the space and time variables, r1 and t, in local quantities, particularly in the local mass densities, ρ and ρα and the local mean velocity, u. We have defined V α , the specific velocity of particles of species α, by Eq. (4.3.17), where the mean velocity of the fluid, u, with respect to which the currents are to be measured is defined by  (4.3.18) dv1 mα v1 f˜α (r1,v1,t), ρ(r 1,t)u(r 1,t) = α

 where the total mass density ρ is simply ρ = α ρα . It follows from Eqs. (4.3.17) and (4.3.18) that the sum of the specific currents vanishes,  Jα (r 1,t) = 0. (4.3.19) α

By summing Eq. (4.3.17) over all the species, we obtain a conservation law for the total mass density ∂ρ(r 1,t) + ∇r 1 · [ρu(r 1,t)] = 0. ∂t

(4.3.20)

The equation for the time evolution of the mean velocity, u(r 1,t), is obtained in a similar manner, by using the function mα v1 and summing over all species. We obtain ∂(ρu(r 1,t)) + ∇r 1 · [ρuu + P (r 1,t)] = 0, ∂t

(4.3.21)

where the number density, nα , is given by nα = ρα /mα . The pressure tensor, P , appearing in Eq. (4.3.21) is defined by   mα dc1 c1 c1 f¯α (r 1,c1,t), (4.3.22) P (r 1,t) = α

where the peculiar velocity, as in previous chapters, is c1 = v1 − u, and we have expressed the distribution function in terms of the peculiar velocities, c. Finally,

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Transport in Dilute Gas Mixtures

we derive the conservation law for the kinetic energy density, eK (r 1,t), defined through the relation  ρu2 + eK (r 1,t) = 2 α

 dv1

mα v12 ˜ fα (r1,v1,t). 2

(4.3.23)

A simple integration of the Boltzmann equation then leads to ∂ ∂t



  2

 ρu2 ρu + eK (r 1,t) + ∇ · u + eK (r 1,t) 2 2

+ ∇ · Jq + ∇u : P = 0.

(4.3.24)

Here the heat current, Jq is defined by Jq (r 1,t) =

 α

dc1 c1

mα c12 ¯ fα (r 1,c1,t). 2

(4.3.25)

We define the local temperature, T (r 1,t) in analogy with the one-component gas case, through the relation d eK (r 1,t) = n(r1,t) kB T (r1,t), 2 with n =

 α

(4.3.26)

nα . Then Eq. (4.3.24) describes the rate of change of temperature. 4.3.3 The Chapman–Enskog Expansion

We wish to obtain expressions for the particle currents, Jα ; the pressure tensor, P ; and the heat current, Jq as expansions in the gradients of the hydrodynamic variables, ρα,u,T . Such expansions will allow us to obtain the Euler, Navier– Stokes, and higher-order, hydrodynaic equations for the flows in dilute gas mixtures, under the circumstances that the mean free paths of the particles are typically much smaller than macroscopic lengths that characterize the distances over which there are changes in the hydrodynamic densities. In this case, we expand the distribution functions in powers of the gradients, taking the leading terms to be local equilibrium distribution functions. We also expand the time derivatives of the hydrodynamic densities in terms of the powers of their gradients, as we did in the case of pure gases. Using only the local equilibrium distribution functions in the expressions for the currents and pressure tensor in the conservation equations, we obtain the Euler equations, which provide the leading terms in the gradient expansions of the time derivatives of the hydrodynamic densities as expansions in powers of their gradients. We then follow the Chapman–Enskog procedure to obtain the

4.3 The Chapman–Enskog Solution

133

first correction to the local equilibrium distribution in the expansion of f¯α (r1,v1,t), which – when inserted in the expressions for the currents and pressure tensor – gives us the linear laws referred to earlier, the Navier–Stokes equations, Fourier’s law, diffusion equations, and explicit formulae for the transport coefficients. The procedure follows that used for the case of pure gases; namely, we assume that a solution of the Boltzmann equation exists with the form ˜ (1) f˜α (r1,v1,t) = f˜α,0 (r 1,v1,t)[1 +  α (r 1,v1,t) + · · · ],

(4.3.27)

where the local equilibrium distribution function is given by β(r 1,t)mα (v1 − u(r 1,t))2 2 f˜loc.α (r 1,v1,t) = nα (r 1,t)e ≡ f¯loc.α (r 1,c1,t). (4.3.28) −

For simplicity, we assume that the local temperature is the same for all of the ¯ (1), . . ., are taken to be proportional species of particles. The correction terms,  to the first and higher orders in the gradients as well as powers of gradients. The zeroth-order term in this expansion, namely f¯loc,α (r 1,c1,t), is the solution to Eq. (4.3.13) when the left-hand side is set equal to zero – that is, when all terms of first and higher orders in the gradients are dropped. Then the equation becomes  Jαβ (f¯loc,α, f¯loc,β ) = 0, (4.3.29) β

with local equilibrium solution given by Eq. (4.3.28). As in the case with the Boltzmann equation for one-component gases, in order to get soluble equations for the higher-order equations, we impose the condition that the local quantities appearing in the local equilibrium function – nα (r 1,t),u(r 1,t), and T (r 1,t) – are the total values of these quantities, so that the higher-order terms, (j ) , for j = 1,2, . . ., give no contributions to the average densities, the average velocity, and average temperature of the fluid. 4.3.4 The Euler Equations In order to obtain an equation for the first-order gradient correction to the local ¯ (1) (r 1,c1,t), we must evaluate the left-hand side equilibrium distribution function,  of Eq. (4.3.13) to first order in the gradients. This, in turn, requires us to obtain expressions for the time derivatives of the hydrodynamic fields appearing in the local equilibrium distribution to first order in the gradients. These derivatives are, of course, given by the Euler equations, obtained by using only the local equilibrium distribution to evaluate the particle currents, Jα (r 1,t), Eq. (4.3.17), the pressure tensor, P (r 1,t), Eq. (4.3.22), and the heat current, Jq (r 1,t), Eq. (4.3.25). We easily find that

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Transport in Dilute Gas Mixtures

J(0) α (r 1,t) = 0, P (0) (r 1,t) = p(r 1,t)I =

(4.3.30) n(r 1,t) 1 = n(r 1,t)kB T (r 1,t)1, β(r 1,t)

J(0) q (r 1,t) = 0.

(4.3.31) (4.3.32)

The superscript merely denotes the fact that these are the currents and pressure tensor to zeroth order in the gradients. Here, of course, p(r 1,t) is the local pressure. The Euler equations easily follow after substituting the preceding expressions in the general conservation laws for mass, momentum, and energy, given by Eqs. (4.3.15), (4.3.21), and (4.3.24), respectively. With this substitution, the Euler equations are obtained as ∂ρα (r 1,t) + ∇r 1 · [ρα u(r 1,t)] = 0, ∂t

(4.3.33)

for the time rate of change of the local mass density of particles of species α, to first order in the gradients ∂(ρu(r 1,t)) + ∇r 1 · [ρuu + n(r 1,t)kB T (r 1,t)1] = 0, ∂t

(4.3.34)

for the time rate of change of the local fluid velocity to first order, and    2

 ρu dn ∂ ρu2 d + n(r 1,t)kB T (r 1,t) + ∇ · u + n(r 1,t)kB T (r 1,t) ∂t 2 2 2 2 1 + ∇ · [un(r 1,t)kB T (r 1,t)] = 0.

(4.3.35)

After some manipulations, these equations can be written in a simplified form, where we define the total time derivative of some function g(r 1,t), Dg/Dt by ∂g(r 1,t) Dg(r 1,t) ≡ + u(r 1,t) · ∇g(r 1,t). Dt ∂t We then obtain the Euler equations in the form   Dρα + ρα ∇ · u = 0, Dt 1   Dρ + ρ∇ · u = 0, Dt 1   Du ρ + ∇(nkB T ) = 0, Dt 1   D(ρT −d/2 ) = 0. Dt 1

(4.3.36)

(4.3.37) (4.3.38) (4.3.39) (4.3.40)

4.3 The Chapman–Enskog Solution

135

We have used a notation for these equations that calls attention to the fact that, in the Euler equations, the time derivatives of the hydrodynamic variables are given only to first order in their gradients. These equations are reversible. That is, the equations are invariant under the transformation t → −t and u → −u. 4.3.5 The First Correction to the Distribution Function ¯ (1) , the We are now in a position to obtain, after some algebra, the equation for  first-order correction in the gradients to the local equilibrium distribution function.8 We insert the expansion of the distribution function given by Eq. (4.3.27) into the collision operator appearing on the right-hand side of Eq. (4.3.13), and evaluate the left-hand side of the equation using only the local distribution function and the Euler equations for the time derivatives. To this order in the gradients, the lefthand side of Eq. (4.3.13) can be written as

  D ln nα (r 1,t) ¯ + c1 · ∇ ln nα floc,α (r 1,c1,t) Dt 1



 D ln T (r 1,t) βmα c12 d − + (c1 · ∇ ln T ) + 2 2 Dt 1  

Du + βmα c1 · + (c1 · ∇)u . (4.3.41) Dt 1 Here the subscripts on the time derivatives mean that the Euler equations are to be used to express these time derivatives to first order in the gradients of the hydrodynamic fields. When this is done, we obtain, after some algebra, the Boltzmann equation for the first order in gradients correction to the local equilibrium distribution function:

 1 2 n βmα c1 c1 − c1 1 : ∇u(r 1,t) + c1 · dα d nα 

 βmα c12 d + 2 ¯ + αγ (), (4.3.42) − c1 · ∇ ln T = 2 2 γ where the vector quantity dα is defined as

n n ρα α α ∇ ln p dα = ∇ + − n n ρ where we have used the simple identity 

n α + ln p − ln T . ∇ ln nα = ∇ ln n

(4.3.43)

(4.3.44)

136

Transport in Dilute Gas Mixtures

The Boltzmann collision operator, linearized about a local equilibrium distribution function αγ , appearing on the right-hand side of Eq. (4.3.42), is given by   ˆ αγ (g, k) ˆ f¯loc,α (r 2,c2,t) ¯ αγ (α ) = dc2 d kB ¯ γ (r 1,c2,t) −  ¯ α (r 1,c1,t) −  ¯ γ (r 1,c2,t)]. ¯ α (r 1,c1,t) +  × [ (4.3.45) The vectors dα are not linearly independent, since they satisfy  dα = 0.

(4.3.46)

α

In order to construct the solution to the linear equation, Eq. (4.3.42), we note that the terms on the left-hand side of this equation consist of inner products of two dyadic tensors or inner products of two vectors. Since the Boltzmann collision operator is linear and rotationally invariant, we could immediately write the solution for ¯ α , as we did in the one-component case, as a uniquely defined sum of terms,  were it not for the fact the inhomogeneous terms are not linearly independent. J. H. Ferziger and H. G. Kaper [215] devised a simple method to deal with this problem. They replace the vectors, dα by a linearly independent set, d∗α , cleverly designed so that the original dα reappear after a few steps.9 The set d∗α satisfies the relation ρα  ∗ dα = d∗α − d . (4.3.47) ρ β β Notice that the vectors d∗α are not completely defined; that is, one cannot invert relation Eq. (4.3.47) to express them in terms of the dα . Nevertheless, by a series of small tricks, one can overcome this problem and reintroduce the original vectors. If we insert the expressions, Eq. (4.3.47), into the left-hand side of Eq. (4.3.42), we obtain a set of equations with inhomogeneous terms that are linearly independent ¯ α takes and orthogonal to the conserved quantities. In this case, the solution for  the form  ∗ ¯ α (r 1,c1,t) =  D(λ) (4.3.48) α · dλ + Aα · ∇ ln T + Bα : ∇u, λ

where the functions D(λ) α ,Aα ,Bα , respectively, satisfy the equations

 n ρα αγ (D(λ) ) = c − , δ 1 αλ α n ρ α γ

 βmα c12 d + 1 c1, αγ (Aα ) = − 2 2 γ

(4.3.49)

(4.3.50)

4.3 The Chapman–Enskog Solution



 1 2 αγ (Bα ) = βmα c1 c1 − c1 1 . d γ

137

(4.3.51)

The trick of Ferziger and Kaper is based on the observation that the functions satisfy the identity    ρλ  (λ) = 0, (4.3.52) αγ Dα ρ γ λ

D(λ) α

which one obtains from Eq. (4.3.49) by multiplying both sides by ρλ /ρ and then  summing over the index λ. Thus, the quantity, λ (ρλ /ρ)D(λ) α is a linear combination of conserved quantities. Since solutions of linear equations are only defined up to the addition of an arbitrary solution of the corresponding homogeneous equations, we can, without loss of generality, require this sum to be equal to zero:  ρλ

D(λ) (4.3.53) α = 0. ρ λ This condition allows us to replace the vectors d∗λ in Eq. (4.3.48) with d∗λ −  (ρλ /ρ) κ d∗κ , which in turn are just the known quantities, dλ . Thus, the solution for the first-order distribution functions can be written as  ¯ (1) D(λ) (4.3.54)  α (r 1,c1,t) = α · dλ + Aα · ∇ ln T + Bα : ∇u, λ

and the quantities D,A,B are solutions of the Boltzmann equations, Eqs. (4.3.49)– (4.3.51), respectively, which are uniquely determined by the conditions that they do not contribute to the local hydrodynamic fields – that is, they are each orthogonal to the conserved quantities – and that the D satisfy Eq. (4.3.53). Using the tensor forms of the right-hand sides of the equations, Eq. (4.3.49)– Eq. (4.3.51), and using the fact that the Boltzmann collision operator is rotationally invariant, we can write ) (γ ) 2 D(γ α (c1,t) = Dα (c1,t)c 1,

(4.3.55)

Aα (c1,t) = Aα (c12,t)c1,

1 2 2 Bα (c1,t) = B(c1,t) c1 c1 − c1 1 , d

(4.3.56) (4.3.57)

where D,A,B are scalar functions of c12 . The condition that A,D give no contribution to the local momentum of the fluid, mentioned earlier, requires that these functions satisfy

138

Transport in Dilute Gas Mixtures





mα

α



dc1 f¯α,0 (r 1,c1,t)c12 Dα(γ ) (c1 ) = 0, 

dc1 f¯α,0 (r 1,c1,t)c12 Aα (c1 ) = 0.

mα

(4.3.58) (4.3.59)

α

As we did in the case of one-component gases, we construct the function α by expanding each of the functions D,A,B in a series of Sonine polynomials. 4.3.6 Thermal Conduction, Diffusion, and Thermal Diffusion We note that the first-order distribution function for mixtures, α , has a property that is not shared by its counterpart for pure gases. That is, for a mixture, the average value of the velocity of particles of each species, as measured with respect to the mean fluid velocity, u, does not vanish! The differences in these average velocities for the various species lead to the phenomenon of particle diffusion. As we will see next, the average velocity for each species depends on density gradients, external forces, and the temperature gradient. This last dependence indicates the presence of thermal diffusion in the system, whereby a temperature gradient can produce differential species velocities leading to the spatial separation of particles of different species. According to the Onsager relations discussed earlier, there should also exist a conjugate effect; namely, gradients in the concentrations of the various species should be able to produce a heat flow. This phenomenon is referred to as the diffusion thermo-effect. To describe these phenomena, we consider the specific velocity of particles of species α, defined by Eq. (4.3.17) and denoted by V α (r 1,t),  1 ¯ (1) dc1 c1 f¯α,0 (r 1,c1,t)[1 +  V α (r 1,t) = α + ···] nα  1  1 ) dc1 f¯α,0 (r 1,c1,t)c1 · D (γ dγ = α nα γ d  1 dc1 f¯α,0 (r 1,c1,t)c1 · Aα ∇ ln T + dnα  Dαγ dγ − DT α ∇ ln T . (4.3.60) =− γ

The final line in Eq. (4.3.60) is the macroscopic expression for the specific velocity that follows from the laws of irreversible thermodynamics [121]. The quantity, Dαγ , defined in what follows, is called a coefficient of mutual diffusion. Here we

4.3 The Chapman–Enskog Solution

139

(γ )

used the vector forms for the functions Dα and Aα given by Eqs. (4.3.55) and (4.3.56), respectively. Since the current is the sum of vectors proportional to dγ and ∇ ln T , with dα defined by Eq. (4.3.43), we see that the current can be produced by density and temperature gradients, and if present, external forces. The coefficient DT α appearing in Eq. (4.3.60) is called the coefficient of thermal diffusion. The coefficients of mutual diffusion, Dαγ are given by   1 1 (γ ) Dαγ = − dc1 f¯α,0 (r 1,c1,t)c1 · Dα = − dc1 f¯α,0 (r 1,c1,t)c12 Dα(γ ), dnα dnα (4.3.61) and the coefficient of thermal diffusion, DT α , is given by10   1 1 ¯ dc1 fα,0 (r 1,c1,t)c1 · Aα = dc1 f¯α,0 (r 1,c1,t)c12 Aα . DT α = dnα dnα (4.3.62) The Symmetry of Diffusion Coefficients As mentioned earlier, the Onsager relations, which are certainly of more general validity than the Boltzmann equations, require that the coefficients of mutual diffusion obey the symmetry relation: Dαγ = Dγ α . Here we show that this relation is indeed satisfied by the diffusion coefficients given earlier. This will require some further manipulations with the Boltzmann collision operators, which we now present. We will do this in a somewhat indirect way, but it leads to the desired conclusion. Consider the quantity [D (α),D (γ ) ], called a bracket integral, defined by   ) μν (D (γ [D (α),D (γ ) ] ≡ dc1 D (α) μ (r 1,c 1,t) · μ ) μ

=−

 1 4

ν

ˆ dc1 dc2 d kˆ f¯μ,0 (r 1,c1,t)f¯ν,0 (r 1,c2,t)B(g μν , k)

μ,ν

 (α)  (α) (α) × [D (α) μ (r 1,c 1,t) + D ν (r 2,c2,t) − D μ (r 1,c1,t) − D ν (r 2,c 2,t)] )  (γ )  (γ ) (γ ) × [D (γ μ (r 1,c1,t) + D ν (r 1,c2,t) − D μ (r 1,c1,t) − D ν (r 1,c2,t)]. (4.3.63)

The result, Eq. (4.3.63), follows, as in the proof of the H -theorem, from interchanging particle labels and direct and restituting collisions. It is clear that the bracket integral is symmetric in the indices α and γ . That is, [D (α),D (γ ) ] = [D (γ ),D (α) ].

(4.3.64)

140

Transport in Dilute Gas Mixtures

Next we apply Eqs. (4.3.49) and (4.3.55) to obtain    (α) (γ )   ) dc1 D (α) μν (D (γ D ,D = μ (r 1,c1,t) · μ ) μ

=

 μ

n = nγ −



ν

n ρμ (α) ¯ dc1 fμ,0 (r 1,c1,t)D μ (r 1,c1,t) · c1 δμγ − nμ ρ dc1 f¯γ ,0 (r 1,c1,t)D (α) γ · c1

 n ) dc1 mμ f¯μ,0 (r 1,c1,t)c1 · D (γ μ . ρ μ

(4.3.65)

The final term on the right-hand side of Eq. (4.3.65) vanishes, as follows from the condition expressed in Eq. (4.3.53). The remaining integral is equal to −(n/d)Dγ α , with the diffusion coefficient, Dγ α , defined by Eq. (4.3.61). However, the symmetry in the indices α and γ of the bracket integrals leads to the conclusion that Dαγ = Dγ α . In effect, we exploited the symmetry property of integrals of the Boltzmann collision operator, whereby restituting and direct collision velocities can be interchanged, to prove the symmetry of the diffusion coefficients. This is, of course, very close to the arguments based on microscopic reversibility that Onsager used to derive the reciprocal relations. Thermal Diffusion Ratios The bracket integrals themselves are quite useful. We will need almost immediately the easily derived result that the coefficient of thermal diffusion is proportional to sums of the bracket integrals, [Aα,D α ]. From this observation, the fact that  the diffusion coefficients, Dαβ , are proportional to the bracket integrals, D α,D β , and the condition given by Eq. (4.3.52), it follows that  ρα (4.3.66) Dαβ = 0 and ρ α  ρα (4.3.67) DT α = 0. ρ α We now define a set of quantities, kT α , called thermal diffusion ratios, as solutions of the inhomogeneous equations  Dαγ kT γ = DT α . (4.3.68) γ

Of course, the solutions for thermal diffusion ratios are only unique up to the solutions of the homogeneous equations corresponding to Eq. (4.3.68). This latter

4.3 The Chapman–Enskog Solution

141

set has, according to Eq. (4.3.52), the solution ργ /ρ. Equation (4.3.67) shows that the vector ρα /ρ is also orthogonal to the DT α . In order to obtain unique values for the kT α , one can impose the condition that [215]  kT α = 0. (4.3.69) α

The Heat Current Finally, we consider the expression for the energy current as defined by Eq. (4.3.25). From the preceding calculations, we obtain an expression for the heat current to first order in the gradients given by J(1) q (r 1,t)  mα c12 ¯ ¯ (1) = floc,α (r 1,c1,t) dc1 c1 α (r 1,c 1,t) 2 α    mα c12 d + 2 ¯ ¯ (1) dc1 c1 − floc,α (r 1,c1,t) = α (r 1,c1,t) 2 2β α  d +2 ¯ (1) dc1 c1 f¯α,0 (r 1,c1,t) + α (r 1,c1,t) 2β α    mα c12 d + 2 ¯ 1 dc1 = − floc,α (r 1,c1,t) d α 2 2β    (λ) (c1 · D α )dλ + (c1 · Aα )∇ ln β + d +2 + 2dβ



λ



dc1 f¯loc.α (r 1,c1,t) (c1 · Aα )∇ ln β +

α

= −λ ∇T − p

 α

d +2 D T α dα − nα Vα . 2dβ α



 (c1 ·

D (λ) α )dλ

λ

(4.3.70)

Here the quantity λ is called the partial coefficient of thermal conductivity, since, as we show next, it is only one of a number of contributions to what is properly called the coefficient of thermal conductivity. The issue that must be resolved before one can meaningfully define the coefficient of thermal conductivity for mixtures is how to avoid the fact that a temperature gradient will lead to the diffusion of particles, which in turn will lead to heat flows due to concentration gradients. The resolution of this issue is found by expressing the heat current in terms of temperature gradients and the specific velocities, V α . Then one defines the coefficient of thermal

142

Transport in Dilute Gas Mixtures

conductivity in terms of the heat flow that takes place when all of the specific velocities vanish. To obtain the appropriate expression for the heat current in terms of ∇T , and the V α,i , we use Eqs. (4.3.60) and (4.3.67) to write    D T α dα = Dαγ kT γ dα = Dγ α kT γ dα α

α

γ

γ

=



α

  kT γ V γ + DT γ ∇ ln T .

(4.3.71)

γ

If we insert this expression in Eq. (4.3.70), we obtain a more useful expression for the heat current   d + 2 nα (1) (4.3.72) V α, kT α + Jq = λ∇T + p 2 n α where the coefficient of thermal conductivity, λ, of a multicomponent mixture is defined as  kT α DT α . (4.3.73) λ = λ − nkB α

[460, 459, 464, 572, 463, 358, 357, 356, 353, 531]. This quantity can be measured in an experiment where the gas is at rest so that all of the V α = 0. 4.3.7 The Pressure Tensor ¯ (1) By substituting the first-order distribution function,  α in the expression for the pressure tensor, Eq. (3.1.15), we obtain an expression for the dissipative part of the pressure for a dilute gas mixture. That is,   ¯ (1) mα dc1 c1 c1 f¯loc,α (r 1,c1,t) P (1) (r 1,t) = α (c1,t) α

  c12 ¯ mα dc1 c1 c1 floc,α (r 1,c1,t)Bα c1 c1 − 1 : ∇u = d α   kB T dc1 f¯loc,α (r 1,c1,t)Bα αγ (B) : ∇u. (4.3.74) = 



α

γ

One can easily show that the pressure tensor takes the standard form for a fluid with a non-zero coefficient of shear viscosity and a zero coefficient of bulk viscosity, namely

∂uj ∂ui 2 (1) + − δij ∇ · u . (4.3.75) Pij = −η ∂xi ∂xj d

4.3 The Chapman–Enskog Solution

143

The coefficient of shear viscosity, η, depends upon the composition of the gas but not upon its total density. Transport Coefficients Returning to Eq. (4.3.57), we have already seen in Chapter 3 that useful Sonine (l) (βmc12 /2) polynomials for the expansion of Bα (c12 ) are the polynomials S(d+2)/2 since these satisfy the orthogonality relation



 ∞ βmc2 βmc12 βmc12 (l  ) d−1 − 2 1 4 (l) S(d+2)/2 dc1 c1 e c1 S(d+2)/2 2 2 0

−( d2 +2) 1 βm = δll  (1 + l + (d + 2)/2). (4.3.76) 2l! 2 It is important to note the weight factor c14 appearing in the orthogonality condition. This factor comes from the velocity dependence of the dyadic tensors, [c1 c1 − (1/d)c12 1], and c1 c1 , which appear in the velocity integrals when the pressure tensor is evaluated by means of the expansion of the functions Bα in Sonine polynomials. We then write

∞  βmc12 (l) (l) , (4.3.77) bα S(d+2)/2 Bα (c1 ) = 2 l=0 where the coefficients, bα(l) , are to be determined by expressing Bα as an expansion in Sonine polynomials through the expansion of Bα given by Eq. (4.3.77) and then substituting the resulting expansion in the inhomogeneous Boltzmann equation, Eq. (4.3.51). This substitution leads to the equation



∞  d 2 1 2 (l) l ,l (d) . (4.3.78) bγ Bαγ = nα δl ,0 1 −  (1) 2 + d (mα γ )2 π d/2 2 γ l=0 Here (d) (1) is the total solid angle in a d-dimensional space – or, equivalently, the l ,l is the surface area of a unit sphere embedded in d-space – and the quantity Bαγ collision integral

 ) 1 (l l ,l 2 2 Bαγ = dc1 f¯α,0 (r 1,c1,t)(c1 )S(d+2)/2 (βmα c1 /2) c1 c1 − c1 1 d

1 (l) : αγ S(d+2)/2 (βmγ c12 /2) c1 c1 − c12 1 . (4.3.79) d 

l ,l The quantities Bαγ are proportional to the product nα nγ through the presence of the local equilibrium distribution functions in the collision integrals. This observation,

144

Transport in Dilute Gas Mixtures

together with Eq. (4.3.78), leads to the conclusion that the constants, bγ(l) in the l ,l Sonine polynomial expansion, must be of order n−1 γ . The quantities, Bαγ , for certain values of l and l  have been tabulated by Chapman and Cowling [89], Ferziger and Kaper [215], Mason [460, 459], and others, all for three-dimensional systems. As in the case of pure gases, these coefficients are needed to obtain good approximations to the coefficients of shear viscosity for dilute gas mixtures. Sonine polynomial expansions are also useful for determining good approximations (γ ) to the functions Aα (c2 ) and Dα defined in Eqs. (4.3.56) and (4.3.55) needed for the various diffusion coefficients and the thermal conductivity of dilute gas mixtures. 4.4 Transport Coefficients for Binary Mixtures As one might surmise from the preceding discussion, the full development of the set of equations leading to explicit expressions for the various transport coefficients for general multicomponent mixtures becomes algebraically complex. The results of this analysis can easily be found in the literature. For these results, we refer to the books by Hirschfelder, Curtiss, and Bird [306]; Ferziger and Kaper [215]; as well as to a number of papers in the literature [356]. Here we will consider the simple case of a binary mixture and present only the explicit expressions for the coefficient of mutual diffusion in the first Sonine polynomial approximation. As one might expect, this approximation is a good one when the mass ratio of the two species is close to unity [613], as would be appropriate for dilute gaseous mixtures of different isotopes of one atomic system. Even for binary mixtures, and with the simplest approximations for the appropriate collision integrals, the explicit expressions for the transport coefficients can be complicated, so here we consider the case of binary mixtures of hard-sphere particles. We will need to define some quantities needed for these expressions. We will use the notation of Chapman and Cowling [89]. Let m1,m2 be the atomic masses of each of the two species, m0 = m1 + m2, Mi = mi /m0, and σ12 = (σ1 +σ2 )/2 , where σ1,σ2 are the diameters of the particles of the two species. We will also need the total density of the gas, n = n1 + n2 . Then (1) , in the first Sonine approximation is the coefficient of mutual diffusion, D12 (1) = D12

where

E=

3E , 2nm0

2m0 βπM1 M2

1/2

(4.4.1)

1 . 2 8σ12

(4.4.2)

4.4 Transport Coefficients for Binary Mixtures

145

The expressions for the coefficients of viscosity, thermal conductivity, thermal diffusion and thermal diffusion ratio are all rather complicated. If one wishes to use the method of corresponding states to develop universal curves as functions of reduced units, one must take into account the three different interaction potentials in a binary mixture. The important new ingredient is the contribution of the interactions of unlike particles to the transport coefficients. One can use the theoretical expressions to express this contribution in terms of a collision integral. For the case of shear viscosity, the relevant collision integral is denoted as 22 , and it is related to the unlike binary collision contribution to the viscosity, μ12 , in the first Sonine polynomial approximation by 5 22 (T ) = 16 ∗



kB MT ∗ π

1/2

1 2 σ12 μ12

,

(4.4.3)

where M = 2m1 m2 /(m1 +m2 ) is the reduced mass. We use the reduced temperature T ∗ = kB T /12 , where 12 is the energy scaling parameter in the Lennard–Jones potential, for example, as obtained from experimental values of the second virial coefficient of the binary mixture. If the experimental values for the viscosities of the pure gases and for the binary mixture are known, one can obtain experimental values for μ12 by using the theoretical expressions for the viscosity of the binary mixture. In Fig. (4.4.1), we show the values of the comparison of the experimental values of the reduced collision integral, 22 (T ∗ ), for mixtures of Xe with the other noble gases, as reported by J. Kestin, H. E. Khalifa, and W. A. Wakeham, with the theoretical curve obtained using Lennard–Jones potentials [354]. The agreement is excellent. Due to the complicated nature of the explicit expressions for the transport coefficients of binary and multicomponent mixtures, we refer the interested reader to the literature for the explicit expressions for various molecular models and for further comparisons with experimental results [460, 459, 464, 572, 463, 89, 462, 215, 358, 357, 356, 355, 531]. Here we will only mention that a very thorough and painstaking theoretical analysis of the theoretical expressions for the transport coefficients of mixtures of dilute gases was carried out over many years by E. A. Mason and his co-workers [461]. The most careful experiments on the transport properties of dilute gases and their mixtures was carried out by J. Kestin and coworkers, also over a period of many years. Here we list a few of their publications [358, 357, 356, 353, 354, 355]. Figure 4.4.1 is an example of the results obtained by Kestin’s laboratory. We would be remiss if we did not call attention to the significant contributions to this field of research made by E. A. Mason, J. Kestin, and their co-workers.

146

Transport in Dilute Gas Mixtures

Figure 4.4.1 The inverse of the unlike-interaction contribution to the viscosities of binary mixtures of Xe with other noble gases. Experimental data are given by open circles for He–Xe, bottom-filled circles for Ne–Xe, side-filled circles for Ar–Xe, and filled circles for Kr–Xe. The vertical axis is defined by Eq. (4.4.3). This figure is taken from the paper of J. Kestin, H. E. Khalifa, and W. A. Wakeham [354]

4.5 The Rate of Entropy Production As we did in the case of a pure gas, we can use the first-order solution of the Boltzmann equation to obtain an expression for the local rate of entropy production in the interior of the gas. The results of this calculation will allow us to compare them with the expression for the rate of entropy production as obtained from the general arguments of irreversible thermodynamics and to check that these results are consistent with the Onsager reciprocal relations. We return to Eq. (4.2.7) and note that the expression for the irreversible rate of ` local entropy production, σ (r,t), is given by  dv1 ln f˜α (r 1,v1,t)Jαβ (f˜α, f˜β ). (4.5.1) σ (r 1,t) = −kB α,β

To proceed, we expand the distribution functions appearing in Eq. (4.5.1) about their local equilibrium values and keep only the first-order gradient corrections to the local equilibrium functions. We then obtain  ¯ (1) ¯ (1) σ (r 1,t) = −kB (4.5.2) dc1 f¯loc,α (r 1,c1,t) α (r 1,c 1,t)αβ (α ), α,β

¯ (1) where αβ ( α ) is given by Eq. (4.3.45). We have defined the rate of entropy production to be the negative of the irreversible rate of production of the local

Notes

147

function, h(r 1,t), supplemented by the inclusion of Boltzmann’s constant, kB . It is clear from many previous calculations that the rate of entropy production must be positive unless all gradients are zero, and the entropy production rate vanishes. If we now use Eqs. (4.3.45), (4.3.49), (4.3.50), (4.3.51), and (4.3.54) to express the action of the linearized collision operators on the distribution functions for each species, we find that the rate of entropy production can be expressed as  (1) σ (r 1,t)/kB = −P(1) : ∇u − J(1) (4.5.3) α · dα − Jq · ∇ ln T , α (1) where the quantities, P(1),J(1) α ,Jq , are expressed in terms of the transport coefficients and gradients of the hydrodynamic fields by Eqs. (4.3.75), (4.3.60), and (4.3.72), respectively, with J(1) α = ρα Vα . The rate of entropy production for dilute gas mixtures has the form required for Onsager’s analysis, and as we have seen, the symmetry relations are indeed satisfied by the relevant transport coefficients. We emphasize that for dilute monatomic gases, the validity of the Onsager reciprocal relations follows directly from the symmetries of the collision processes, namely the symmetry between direct and restituting collisions. For denser gases, the situation is more interesting and more complicated. The Enskog equation for dense hard spheres, as originally formulated by Enskog, when applied to mixtures in naive ways, leads to transport coefficients that do not satisfy the Onsager relations, as was first noted by L. Garcia–Colin and co-workers [22]. This situation was remedied by the development of the revised Enskog equation by H. van Beijeren and M. H. Ernst [644, 645]. This will be discussed in Chapter 7.

Notes 1 Although the phenomenon of thermal diffusion had been discovered experimentally in the nineteenth century, the first theoretical explanation of it was provided by David Enskog in 1911 for a simple system. S. Chapman generalized this work to arbitrary dilute gas mixtures and collaborated on an experimental study to verify the predictions of the theory. Perhaps Maxwell would have noticed this phenomenon but for the fact that the effect vanishes for Maxwell molecules. For a discussion of this history, see the books of S. Brush [72, 71] 2 An important example of mixtures is provided by the Widom–Rowlinson mixtures of hard spheres, in which two species of hypothetical particles are mixed. These have the properties that particles of like species do not interact with each other but particles of different species repel each other with hard-sphere interactions. This leads to a separation into two phases, as one increases the particle density from low to high values. We refer to the literature for a discussion of this model [679, 342]. 3 Excellent treatments of the kinetic theory of dilute gas mixtures with more details than presented here can be found in the books [306, 89, 215]. As the explicit results for the transport coefficients can be quite complicated, we refer to these books for the expressions, and here we outline the main features of the theory leading up to these expressions. 4 As usual, this assumption simplifies the description of the restituting processes. For non-central potentials and forces, the restituting processes require a more elaborate treatment. These occur when one studies transport processes in gases of polyatomic molecules [215, 297].

148

Transport in Dilute Gas Mixtures

5 Recall that the differentials of thermodynamic potentials, such as free energy can be written as sums of terms of the form AdB, with A and B pairs of conjugate variables. Familiar examples include pressure–volume, temperature–entropy, etc. Very useful discussions of the thermodynamics of irreversible processes, as well as the related kinetic theory, can be found in books of S. de Groot and P. Mazur [121] and J. Keizer [352] and in articles by E. G. D. Cohen, G. H. Weiss, and K. E. Grew in the book edited by H. J. M. Hanley [294]. 6 There are several excellent presentations of the foundations of nonequilibrium thermodynamics and the Onsager reciprocal relations. These reciprocal relations, in general, follow from the microscopic reversibility of the system. The standard reference is the book by S. R. de Groot and P. Mazur, but more recent presentations are given by J. Keizer and in a book edited by D. Bedeaux, S. Kjelstrup, and J. V. Sengers [28]. Here we only make a few remarks, without derivation, based on a discussion by Coleman and Truesdell [109]. 7 We note that the linear laws are irreversible and the currents do not satisfy the requirements that would be imposed on their actual counterparts by the reversibility of the equations of motion for the particles. For example, the microscopic expression of particle currents would change sign under time reversal, but the macroscopic current as given by Fick’s law, −D∇n, does not. 8 Much of the discussion to follow makes use of the excellent treatment of the kinetic theory of mixtures to be found in the book by J. H. Ferziger and H. G. Kaper [215]. 9 Chapman and Cowling avoid this problem in their book by considering only binary mixtures. In that case, the two dα differ only in sign. One just picks one of them, and it appears in both of the two equations for the two different . The method of Ferziger and Kaper applies to a general mixture of monatomic, dilute gases. 10 Note that a positive sign appears in the definition of DT α that follows, since Fourier’s Law is formulated with respect to the gradient of temperature, T , and not to the gradient of the inverse temperature, β.

5 The Dilute Lorentz Gas

5.1 Introduction Much of our understanding of the basic assumptions made when deriving kinetic equations, such as the Boltzmann equation, is due to the construction of simple models. These models are designed to illustrate the assumptions made in such derivations in a simplified context, where basic steps can be viewed without the complications generated by more complex and realistic models. We have already encountered such a model when we discussed the Kac ring model. The ring model bears very little resemblance to a dilute gas but provides a way to understand the role of the Stosszahlansatz in the derivation of the Boltzmann equation, as well as the role of the large system limit, tacitly made in this derivation. In this chapter, we will introduce and discuss another model that has been very useful for kinetic theory, both as an aid to understanding more complicated systems and as a model for certain types of physical systems, particularly for studying electrical conduction in amorphous solid systems or diffusion in binary mixtures where the diffusing particles are present in low concentrations and have a mass much smaller than the other species in the mixture. This is the Lorentz gas introduced by H. A. Lorentz in 1905 [440]. In its simplest form, the model consists of a collection of heavy particles considered to be fixed in space together with a collection of moving particles that interact with the heavy particles but have such a low density that the moving particles do not interact with each other. As a model of electrical conduction, the light particles represent electrons, while the heavy particles represent atoms in an amorphous solid or, for quantum systems, random impurities in a crystalline solid.1 As we shall see later in this chapter, the Lorentz gas is a useful model for the conduction of electrons in such a solid and allows for a derivation of Ohm’s law, where the electrical current, j, depends linearly on the electric field strength, E, as j = σ E. However, the application of the Lorentz model for a derivation of Ohm’s law and for the determination of the properties of electrical conduction in a laboratory situation

149

150

The Dilute Lorentz Gas

requires some modifications of the model. First of all, electrons in solids behave strongly quantum mechanically, so that classical scattering cross sections must be replaced by quantum mechanical ones, and the fact that electrons are fermions must be taken into account. The quantum nature of the electrons in an amorphous solid is also responsible for the phenomenon of localization, which can prevent electrical conduction from taking place. At low temperatures, electrons in a threedimensional solid with randomly placed scatterers can undergo a metal–insulator phase transition whereby the solid acts as an insulator for low Fermi energies, but as the Fermi energy of the electrons increases, the solid becomes a conductor above a certain energy [422, 699, 410]. Returning to the Lorentz model as a model for conduction in a classical system, or for quantum systems above the metal–insulator phase transition, we must take into account the fact that real systems are finite and have boundaries. This is especially important for a description of electrical conduction since the finite size of the system allows for the dissipation of Joule heat at the boundaries. Without some mechanism for this dissipation, an external electric field will continually heat the solid with a concomitant breakdown of Ohm’s law. In this chapter, we will discuss the properties of the classical Lorentz gas, but we will postpone the treatment of boundary effects in a Lorentz gas until Chapter 14. There we will argue that a steady state can be produced in a Lorentz gas of charged, moving particles, acted upon by a constant electric field, with heat-absorbing boundaries that remove the Joule heat generated by the field acting on the particles. In this chapter, we will discuss the Lorentz gas, as described before, with fixed scatterers distributed randomly in space. We will first present the Lorentz– Boltzmann equation for the distribution function of the light particles and discuss its solution for the case where the scatterers are hard spheres. In the context of our discussions in previous chapters, we point out that the Lorentz gas, with fixed scatterers, is a very useful model for understanding the validity as well as the limitations of the Chapman–Enskog procedure, and its transport properties are very interesting and have some special features that are worth knowing about and understanding.2 We will investigate the transport properties, such as diffusion, electrical conduction for a system in an electric field, and electrical conduction for a system in an external magnetic field. We will assume that all of the moving particles carry identical electric charges. Finally, we will take advantage of the simplicity of the Lorentz gas to investigate some important properties of chaotic systems, particularly Lyapunov exponents [243, 152]. These exponents characterize the exponential separation rate of two infinitesimally close trajectories in the phase space of a moving particle in a fixed environment of scatterers. We will show how to calculate the spectrum of Lyapunov exponents for two-dimensional hard-sphere Lorentz gases using a procedure that can be generalized to higher dimensions [649].

5.2 The Lorentz–Boltzmann Equation

151

5.2 The Lorentz–Boltzmann Equation We begin our treatment of the kinetic properties by assuming that the scatterers are fixed in space and interact with the moving particles with a central potential energy function. To avoid the complications arising from interactions with walls, we assume that the system either is infinite (but homogeneous) or satisfies periodic boundary conditions. The interaction potential between a scatterer and a moving particle is required to vanish whenever the moving particle is at a distance greater than a from the center of the scatterer (and is required not to vanish at smaller distances). For our purposes, the moving particles can always be treated as point particles, thanks to the assumption that they do not interact with each other but only with the fixed scatterers. The scatterers are taken to be placed completely at random in space, with number density ns subject to the condition that the scatterers are dilute, in the sense that ns a d 1, where the radius of the interaction sphere for particle– scatterer collisions is a. We take the radius of the interaction sphere between two scatterers to be σ , and we impose the condition ns σ d 1, so the scatterers may be treated as mutually non-interacting. The mean free path  between collisions of a moving particle with a scatterer is of the order  ∼ (ns a d−1 )−1 . It satisfies the inequality,  > a but will also be assumed to be small compared to some macroscopic length,3 L. The Lorentz gas is illustrated schematically in Fig. 5.2.1. Since the scatterers are fixed and we require the collisions to be elastic, the kinetic energies of a moving particle before and after a collision are equal. Since the scattering potential is central, depending only on the separation between the moving ˆ along particle and a scatterer, their collision may be described by a unit vector, k, the apse line. The relation between the particle’s velocity before collision, v, and its velocity after collision, v, is given by ˆ k. ˆ v = v − 2(v · k)

(5.2.1)

In fact, all of the collision dynamics discussed in Chapter 2 can be immediately applied to the Lorentz gas, with the simple adjustment that the scatterers remain fixed, with zero velocity, before and after the collision with the moving particle. Using all of the arguments familiar from the derivation of the Boltzmann equation, including the Stosszahlansatz, we can write an equation for the distribution function f (r,v,t) for the moving particles, called the Lorentz–Boltzmann equation, as F ∂f (r,v,t) + v · ∇r f (r,v,t) + · ∇v f (r,v,t) ∂t m  ˆ ˆ ˆ H (−v · k)B( k,v)[f (r,v,t) − f (r,v,t)]. = ns d k

(5.2.2)

152

The Dilute Lorentz Gas

Figure 5.2.1 Fixed scatterers are placed at random in space. Moving particles interact with the scatterers but not with each other.

ˆ Here F is the external force acting on the particles, and B(k,v) is the scattering function describing the collision with the scatterer, given for three-dimensional systems by    db  ˆ (5.2.3) B(k,v) = 2bv   [cos(θ/2)]−1, dθ where θ is the scattering angle, v · v = v 2 cos θ, illustrated for the case of hard spheres in Fig. 5.2.2, b is the impact parameter for the collision, and v is the constant speed of the moving particles. Since the scatterers are distributed randomly, as illustrated in Fig. 5.2.1, the probability of finding a scatterer in a small volume dV is ns dV . This accounts for the prefactor of the density on the right-hand side of the Lorentz–Boltzmann equation. It is important to note that the Lorentz– Boltzmann equation is a linear equation for the distribution function of the moving particles. This is a feature that greatly simplifies the construction of solutions and, together with translation invariance of the distribution of scatterers, allows the use of familiar methods such as Fourier–Laplace transforms for obtaining the position and time dependence of the solutions. There are some special cases where a complete solution is possible. The most interesting of these is the situation where the scatterers are hard spheres (d = 3) and there are no external fields. These complete solutions can be used to gain some general insights, which are also valid for other particle–scatterer interactions, into the validity and the restrictions of approximate solutions, such as the Chapman–Enskog solution. We discuss this case later in this chapter.

5.2 The Lorentz–Boltzmann Equation

153

Figure 5.2.2 The direct collision with apse line vector, σˆ , and the restituting collision with apse line vector, −σˆ , for moving particles colliding with hardsphere scatterers in three dimensions. Here b is the scattering impact parameter, and  denotes the azimuthal angle for the collision plane.

5.2.1 The Hard-Sphere Lorentz–Boltzmann Equation Much of our discussion of the kinetic theory for the random Lorentz gas will be devoted to the case of a particle moving among a random distribution of scatterers with which it makes hard-sphere collisions – that is, instantaneous, elastic, specular collisions. For this case, the Lorentz–Boltzmann equation has a simple form given by F ∂f (r,v,t) + v · ∇r f (r,v,t) + · ∇v f (r,v,t) ∂t m  d−1 = ns a d σˆ H (−v · σˆ )|v · σˆ |[f (r,v,t) − f (r,v,t)].

(5.2.4)

Here σˆ is a unit vector in the direction from the center of the scatterer to the point of collision with the moving particle,4 a is the radius of a scatterer, and the restituting velocity is given, as usual, by v = v − 2(v · σˆ )σˆ . The collision dynamics for hard-sphere scatterers is illustrated in Fig. 5.2.2 . 5.2.2 Conserved Quantities and the Equilibrium Distribution Function In an elastic collision between a moving particle and a fixed scatterer, the momentum of the moving particle is not conserved, but the mass and the kinetic energy are5 . Consequently, the collision term on the right-hand side of Eq. (5.2.2) will vanish whenever the distribution function, denoted by f0 (r,v,t), depends only on the mass m, the position r, and the magnitude of the velocity, v, of the moving

154

The Dilute Lorentz Gas

particles – that is, whenever f (r,v,t) = f0 (r,v,t). The precise form of the distribution function for a system in equilibrium under a conservative external force, F, will depend upon the total energy – kinetic plus potential – of the particles. In the next section, we will consider the simple case where there is no external potential, all of the particles have the same kinetic energy, and the scatterers are distributed randomly but uniformly in space. 5.2.3 The H -theorem for the Lorentz–Boltzmann Equation It is quite easy to show that an H -theorem obtains for the Lorentz–Boltzmann equation. As usual, we define the H function by    H (t) = drh(r,t) = dr dvf (r,v,t)[ln f (r,v,t) − 1], (5.2.5) where the definition of h(r,t) is clear from the definition of the H -function. The time rate of change of h(r,t) follows from Eq. (5.2.2) as   ∂h(r,t) ˆ ˆ H (−kˆ · v)B(k,v) + ∇.jh (r,t) = ns dv d k ln f (r,v,t) ∂t × [f (r,v,t) − f (r,v,t)], where the current jh (r,t) of the function h(r,t) is given by  jh (r,t) = dvvf (r,v,t)[ln f (r,v,t) − 1].

(5.2.6)

(5.2.7)

There is no contribution to this equation from the external field if the external force is independent of velocity or if it has the form of the Lorentz force that is proportional to the vector product of the velocity with an external magnetic field. Using the fact that the Jacobian of the transformation from v to v is unity, we can transform the collision term on the right-hand side of Eq. (5.2.6) to   ns ˆ H (−kˆ · v)B(k,v) ˆ dv d k 2 × [ln f (r,v,t) − ln f (r,v,t)][f (r,v,t) − f (r,v,t)] ≤ 0,

(5.2.8)

where we have used the identity (ln x − ln y)(y − x) ≤ 0. From this, it follows that the right-hand side of Eq. (5.2.6) is never positive, and after an integration over all space, dH (t) ≤ 0. dt

(5.2.9)

5.3 Diffusion in the Lorentz Gas

155

Since the Lorentz–Boltzmann collision operator is linear and satisfies an H -theorem, its spectrum is nonpositive definite, just like the spectrum of the linearized Boltzmann collision operator. As only mass and energy are conserved for particles with fixed speed, it has only one zero eigenfunction, the unit function. 5.3 Diffusion in the Lorentz Gas As the first application of the Lorentz–Boltzmann equation, we consider the evolution of a spatially inhomogeneous initial distribution toward an equilibrium distribution function. We again consider the case that all moving particles have the same mass and energy and the scatterers are distributed uniformly. When the density varies on scales large compared to the mean free path, one may expect that after a few mean free times, this evolution is dominated by hydrodynamics. The hydrodynamic flow would be the diffusion of the moving particles through the system, leading – for long times and far from any boundaries – to a spatially homogeneous final state. That this is indeed the case will be shown in two ways. First, we will apply the Chapman–Enskog method to the Lorentz–Boltzmann equation, and then, for the case of hard-sphere scatterers in three dimensions, we will discuss the complete solution for all times and for all density gradients, following the method of E. H. Hauge [298]. We recall the general strategy for the Chapman–Enskog procedure. We suppose that the distribution function for the moving particles starts from some not too singular initial form. After a time of the order of several mean free times, the particles will have had, on the average, several collisions, and the distribution function will be close to a local equilibrium distribution function. When this stage is reached, any further time dependence of the distribution function is governed by the time dependence of the local conserved densities, which in turn are governed by the hydrodynamic equations. We say that the distribution function is a functional of the local densities, which in equation form may be expressed as f (r,v,t) = f (r,v, n(r,t),T (r,t) ).

(5.3.1)

For the Lorentz gas, where all of the moving particles have the same kinetic energy when they are not in a collision with a scatterer, there is only one independent hydrodynamic density, the local number density of the moving particles, n(r,t). We imagine an initial state where the local density of the moving particles varies with position and the distribution of velocity directions may be nonuniform. The mean free time is denoted by t = /v0 , where v0 is the speed of each of the moving particles with , the mean free path, proportional to 1/(ns a d−1 ), and the

156

The Dilute Lorentz Gas

hydrodynamic time scale is denoted by th = L/v0 , where L is some macroscopic length that characterizes the density gradient. A conservation law for the local density of particles may be obtained by integrating the Lorentz–Boltzmann equation over all velocities. Since the collision term, i.e. the right side of this equation, conserves the particle number, we obtain ∂n(r,t) + ∇r · Jn (r,t) = 0, ∂t where the particle current, Jn (r,t), is Jn (r,t) =

(5.3.2)

 dvvf (r,v,t).

(5.3.3)

We then look for solutions of Eq. (5.2.2) in the form of an expansion in the gradient and higher spatial derivatives of the density. Setting the external force, F, equal to zero, we can write this expansion as f (r,v,t) = Ad n (r,t) (1 + )δ(v − v0 )

(5.3.4)

where Ad is a normalization constant for the velocity integration, depending upon the spatial dimension of the system, given by 

−1 −1 δ(v − v0 )dv = (vd−1 Ad = 0 d ) , with d , as before, the surface area of the unit sphere in d dimensions6 . We require that the local number density, n(r,t), appearing in this form of the distribution function, is the exact number density of the particles, so there is no contribution to the density from the correction term (r,v,t). This correction term is assumed to have a gradient expansion of the form (r,v,t) = 1 + 2 + · · · ,

(5.3.5)

where each of the j is a product of a function of v and the j th spatial derivative, ∇ j n(r,t), of the local density. No products of gradients appear in this expansion since the Lorentz–Boltzmann equation is a linear equation for the distribution function. Inserting the expansion Eq. (5.3.5) in Eq. (5.2.2) and collecting terms of the same order in the derivatives of the local density, we find the first order equation  ˆ H (−v · k)B( ˆ ˆ δ(v − v0 )v · ∇n(r,t) = ns n(r,t) d kθ k,v) × [1 (r,v,t) − 1 (r,v,t)]δ(v − v0 ) ≡ n(r,t)λL 1 (r,v,t)δ(v − v0 ),

(5.3.6)

5.3 Diffusion in the Lorentz Gas

157

where the Lorentz–Boltzmann collision operator, λL , is defined by the right-hand side of Eq. (5.3.6), namely   ˆ H (−v · k)B( ˆ ˆ k,v)[φ(v ) − φ(v)], λL φ(v) = ns d k with the restituting velocity, v, given by Eq. (5.2.1). Notice that, in this order, the time derivative of the distribution function in Eq. (5.2.2) does not contribute to the left-hand side of Eq. (5.3.6), since both terms in it are orthogonal to the unit function.7 The equations for the functions n(r,t)j (r,v,t) for j = 2,3, . . . , follow immediately as  

  ∂n(r,t) + v · ∇ n(r,t)j −1 (r,v,t) δ(v − v0 ) ∂t j = n(r,t)λL j (r,v,t)δ(v − v0 ).

(5.3.7)

Each of these equations is soluble provided the left-hand side is orthogonal to the eigenfunctions of λL with zero eigenvalue. This solubility condition determines the values of the first terms on the left-hand sides of these equations; in this case, where the kinetic energy of each particle has the same value, the only zero eigenfunction of λL is a constant. It is apparent that with this solution of the Lorentz– Boltzmann equation, the conservation law, Eq. (5.3.2), becomes an equation for the time derivative of the local density expressed as a series in spatial derivatives of the density, starting with the second derivative and with all odd derivatives absent, leading to the macroscopic diffusion equation plus corrections. The first nonvanishing term, Jn,1 , in the expansion of the current and the corresponding term in the conservation equation are  Jn,1 = dvδ(v − v0 )vλL −1 (v · ∇n(r,t)),  ∂n(r,t) = −∇ · Ad dvδ(v − v0 )vλL −1 (v · ∇n(r,t)) + · · · , ∂t = DL ∇ 2 n(r,t) + · · · . Here the diffusion constant, DL , is given by  Ad dvδ(v − v0 )v · λL −1 v, DL = − d

(5.3.8)

(5.3.9)

where we used the fact that the system is isotropic, and the averages of the squares of all components of the velocity vector are equal. This expression can be evaluated in the same way as is done for the expressions obtained from the linearized Boltzmann equation for the coefficient of shear viscosity, the thermal diffusion

158

The Dilute Lorentz Gas

coefficient, for example. If the interaction potential is short ranged and isotropic, the diffusion coefficient can be evaluated exactly. We illustrate this remark for the case of hard-sphere scatterers. In order to evaluate the preceding expression for the diffusion coefficient, we need to solve the integral equation  λL ψ(v) = ns a d−1 d σˆ H (−σˆ · v)|σˆ · v|(ψ(v ) − ψ(v)) = v, (5.3.10) from which it follows that Ad DL = − d

 dvδ(v − v0 )v·ψ.

(5.3.11)

Since the operator λL is isotropic and the norm v0 is fixed, we may set ψ(v) = αv and determine the constant α by substituting this form in the integral equation. Multiplying the integral equation by v and carrying out an integration over the velocities v, we obtain  dvδ(v − v0 )v2 , (5.3.12) α= dvδ(v − v0 )v.λL v leading to Ad DL = −α d



  2 2 dvδ(v − v )v A 0 d  dvδ(v − v0 )v2 = − . d dvδ(v − v0 )v.λL v

(5.3.13)

For hard spheres, we can easily carry out all the integrations required for calculating α. For two dimensions, scattering by hard disks with radius a, we find DL(2) =

3v0 3 = v20 t(2), 16ns a 8

(5.3.14)

and for three-dimensional, hard-sphere scatterers with radius a, the diffusion coefficient is DL(3) =

v0 1 = v20 t(3) . 2 3πns a 3

(5.3.15)

Here t(d) is the mean free time between collisions in a d-dimensional dilute Lorentz gas. In particular, t(2) = (2ns av0 )−1, and t(3) = (πns a 2 v0 )−1 . These results can be generalized to hard hypersphere scatterers in any number of dimensions. One can go beyond the Navier–Stokes type of hydrodynamics for the Lorentz gas and obtain results for generalized diffusion coefficients that appear as coefficients, (j ) Dd , in the generalized hydrodynamic equation ∞

∂n(r,t)  (d) 2j Dj ∇ n(r,t). = ∂t j =1

(5.3.16)

5.4 Hard-Sphere Systems in Three Dimensions

159

The usual diffusion coefficient is D1(d) . Only even powers of the gradient operator appear in Eq. (5.3.16). For the special case of hard-sphere scatterers in three dimensions, an exact solution of the field free Lorentz–Boltzmann equation is available, which leads to a number of interesting results, including explicit values for all of (j ) the coefficients, D3 , appearing in Eq. (5.3.16). This will be the subject of the next section. 5.4 Hard-Sphere Systems in Three Dimensions The linearity of the Lorentz–Boltzmann equation, together with the restriction to a single speed, makes the construction of solutions much easier than for the nonlinear Boltzmann equation or even the linearized Boltzmann equation for almost all other inter-particle potentials. For the special case of hard-sphere scatterers in three dimensions, it has been shown that it is possible to transform the collision term into a form familiar from the BGK model for the linearized Boltzmann equation [657, 298]. Using this form, we can easily construct and analyze the exact solution for the distribution function of the moving particles. The transformation of the collision term is accomplished by changing the integration variables from the apse line unit vector, σˆ , to the solid angle into which the moving particle is scattered upon a collision. The collision term for particles moving at constant speed, v0 , expressed in terms of the apse line vector, is given by  2 a d σˆ H (−v · σˆ )|v · σˆ |[f (r,v,t) − f (r,v,t)] 



π/2

= a 2 v0

2π

dψ 0

d sin ψ cos ψ[f (r,v,t) − f (r,v,t)],

(5.4.1)

0

where π − ψ is the angle between v and σˆ in the plane of the collision, or, equivalently, ψ is the angle between v and σˆ in this plane. The angle  is the azimuthal angle that identifies the plane in which the scattering takes place. The scattering angle, θ – that is, the angle between v and v – is θ = π − 2ψ. We now express the collision integral in terms of the variables θ and  to find  2π  π/2 2 dψ d sin ψ cos ψ[f (r,v,t) − f (r,v,t)] a v0 0

a 2 v0 = 4



0 π



dθ 0

2π

d sin θ[f (r,v,t) − f (r,v,t)].

(5.4.2)

0

Now the integration can be understood as an integration over all possible directions of v in a spherical coordinate system with one axis in the direction of v and the plane indicated by the azimuthal angle. In other words, v = v0 [ˆv cos θ + vˆ ⊥,1 sin θ cos  + vˆ ⊥,2 sin θ sin ].

(5.4.3)

160

The Dilute Lorentz Gas

Here vˆ ⊥,1, vˆ ⊥,2 are orthonormal vectors in the plane perpendicular to the direction of v. Finally, we can express the field free Lorentz–Boltzmann equation, for this case, as ∂f (r,v,t) + v · ∇r f (r,v,t) = ns v0 a 2 π(PL − 1)f (r,v,t), (5.4.4) ∂t where the operator PL is a projection operator that averages the velocity variable of the function on which it acts over all possible directions,  dωv , (5.4.5) PL = 4π where the limits of integration over a solid angle of possible directions of the velocity vector, dωv = sin θdθd, are defined on the right-hand side of Eq. (5.4.2) and, obviously, PL2 = PL . To proceed further, we convert the partial differential equation, Eq. (5.4.4), to an algebraic equation by taking the Fourier–Laplace transform of this equation in the spatial and time variables. The function of interest is now fk (v,z), defined by  ∞  dt dr exp [−zt + ik · r] f (r,v,t). (5.4.6) fk (v,z) = 0

This function satisfies the equation (z − ik · v) fk (v,z) = fk (v,t = 0) + ν(PL − 1)fk (v,z),

(5.4.7)

where ν = ns πa 2 v0 is the collision frequency, and fk (v,t = 0) is the spatial Fourier transform of the initial value of the distribution function for the moving particles. We can rewrite this equation in a simple way: fk (v,z) = [z − ik · v + ν]−1 fk (v,t = 0) + [z − ik · v + ν]−1 νPL fk (v,z). (5.4.8) The projection operator produces a function only of the variables z,k, so the application of this operator to both sides of Eq. (5.4.8) leads to an expression for PL fk (v,z), which is −1    ν −1 kv0 tan × PL [z − ik · v + ν]−1 fk (v,t = 0) , PL fk (v,z) = 1 − kv0 z+ν (5.4.9) and the expression for the distribution function fk,z (v) is fk (v,z) = [z − ik · v + ν]−1 fk (v,t = 0) −1  ν −1 −1 kv0 + ν[z − ik · v + ν] tan 1− kv0 z+ν   −1 × PL [z − ik · v + ν] fk (v,t = 0) .

(5.4.10)

5.4 Hard-Sphere Systems in Three Dimensions

161

Here we have used the identity PL [A − ik · v]−1 =

1 kv0 tan−1 . kv0 A

(5.4.11)

Thus, we have found a solution of the Lorentz–Boltzmann equation for hardsphere scatterers, expressed in terms of the initial value of the distribution function. The space- and time-dependent distribution function, f (r,v,t), is obtained from fk (v,z) by inverting the Fourier–Laplace transform. We can now compare the properties of the exact solution to those of its Chapman–Enskog approximation. As we will see in what follows, the Chapman– Enskog expansion for the hard-sphere Lorentz gas converges for all k values for which it is defined. This situation is similar to that which obtains for the Chapman– Enskog solution of the linearized Boltzmann equation for hard spheres in three dimensions. We note that the inverse Laplace transform of the exact solution, as described by Eqs. (5.4.9) and (5.4.10), is determined by the analytic properties of this solution as a function of z in the complex z-plane. By inspecting the expression for fk (v,z), we see that, in the complex z-plane, there are the following: 1. A pole at z = −ν + ik · v. 2. A cut along a line parallel to the imaginary axis, but displaced to the left, from z = −ν − ikv0 to z = −ν + ikv0, due to the angular θ integration appearing in the last term on the right-hand side of Eq. (5.4.10), when the projection operator acts on the function to its right, namely the integral  2π  π d dθ sin θ [z + ν − ikv0 cos θ]−1 fk (v,t = 0). 0

0

3. Most importantly, provided that kv0 /ν < π/2, there is a pole on the negative real axis at z = H (k), where   kvo kv0 H (k) = ν −1 + cot . (5.4.12) ν ν This pole is near the origin for small values of the wave number. This is called the hydrodynamic pole, and it determines the long-time and large-wavelength behavior of the distribution function. The contributions to the inverse Laplace transform fk (v,t) from the pole at −ν + ik · v and from the cut in the z-plane decay exponentially, roughly as exp(−νt) and under typical conditions become very small after a few mean free times. For small values of k – that is, for processes taking place on a large-wavelength scale – the long-time behavior of the distribution is governed by the hydrodynamic pole at H (k), which to leading order is proportional to k 2 . Therefore, under typical

162

The Dilute Lorentz Gas

circumstances, the long-time behavior of the exact solution rapidly approaches to the normal solution form given by Eq. (5.3.7). Then, taking into account only the contributions from the hydrodynamic pole to the distribution, we find that the local density n(r,t) satisfies the generalized hydrodynamic equation ∞

 Bs ∂n(r,t) = −2v0 ∂t (2s)! s=1



2 πns a 2

2s−1



s −∇ 2 n(r,t),

(5.4.13)

where the Bs are the Bernoulli numbers, B1 = 1/6, B2 = 1/30, and so on. Equation (5.4.13) is the generalized hydrodynamic equation, the first term of which is the usual diffusion equation, Eq. (5.3.8). From this example, one can identify a number of important results. We see that the normal solution of the Lorentz–Boltzmann equation is not the complete, exact solution for a general initial state. Instead, there are contributions decaying exponentially with time that do not contribute to the asymptotic, long-time behavior of the local density. We have, in this case, solved a linear equation in which the operator ik · v + ns v0 a 2 π(P − 1) acts on the distribution function. The Lorentz–Boltzmann collision operator, nv0 a 2 π(PL − 1), has one zero eigenvalue corresponding to the conservation of the number of moving particles. It also has an infinitely degenerate non-zero eigenvalue, −ν, with eigenfunctions that are orthogonal to a constant. If we think of the additional term, ik · v, as a perturbation, then for small wave numbers, the perturbation moves the zero eigenvalue of the collision operator away from the origin and leads to the hydrodynamic pole. The infinitely degenerate eigenstates lead to the other pole and to the cut in the complex plane. As we will see in later chapters, hydrodynamic poles play an important role in the description of the hydrodynamic behavior of dilute as well as of moderately dense gases. Since the normal solution is only one term in the exact solution of the equation, we know that for short times, it may not give the leading contribution to the exact solution. This implies that the initial value of the density in the Chapman–Enskog solution must be modified in order that the exact solution approaches it for long times. The appropriate initial condition for the Chapman–Enskog solution may be obtained by adjusting the residue at the hydrodynamic pole of the normal solution so that it matches the residue at the hydrodynamic pole of the exact solution.8 Determining the exact residue at the pole z = H (k) takes some further calculation [298], and it is found to be

(kv0 /ν) (1 + Gk (v)) sin(kv0 /ν)

2 × PL [(1 + Gk (v))fk (v,0)],

(5.4.14)

5.4 Hard-Sphere Systems in Three Dimensions

163

where 

kv0 ik · v kv0 cot − Gk (v) = ν ν ν

−1

− 1,

(5.4.15)

which has the property that PL Gk (v) = 0. Thus, the effective initial value to be used for the Chapman–Enskog approximation to the exact solution is the spherical average of the exact residue, Eq. (5.4.14), ! "

2 /ν) (kv 0 P[(1 + Gk (v))fk (v,0)] . n∗k (0) = 4πPL (1 + Gk (v)) sin(kv0 /ν) (5.4.16) In the limit k → 0, the functions 1 + Gk (v) and kv0 /ν sin(kv0 /ν) approach unity, and the exact initial value, nk (0), differs from the adjusted initial value, n∗k (0), by a term of order k. This correction can be neglected in the Navier–Stokes approximation. Therefore, in most cases the adjusted initial value n∗k (0) will approach the actual initial value nk (0) in this limit. Since with increasing time an ever smaller range of k-values, centered at k = 0, will survive, the normal solution then becomes an increasingly better approximation with increasing time. However, some caution is needed here. In the case that PL fk (v,0) = 0, the effective initial value is given by approximately 4πPL G(v)fk (v,0) and for no k-value will it approach the actual initial density. An example of a physical situation for which this is relevant is a uniform equilibrium system that is exposed very briefly to an external field producing an non-isotropic and nonuniform initial velocity distribution while leaving the spatial density homogeneous.9 A possible solution to this difficulty in identifying the initial density field with the initial condition for the diffusion equation is to wait for a few mean free times and use the density field emerging then as initial condition. To conclude: we found that the main differences between the exact solution and the Chapman–Enskog solutions of the hard-sphere Lorentz–Boltzmann equation are found in the decay of short-wavelength excitations of the distribution function, with values of k that are comparable to or larger than the inverse mean free path, and that, typically after a few mean free times, all such excitations have decayed to zero. From the theory of the dilute-gas Boltzmann equation, it is known that further differences are to be expected near physical walls and shock fronts. Physical walls could be added, leading to additional terms in the Lorentz–Boltzmann equation. We have left them out here because their inclusion adds little to what we know already from the dilute-gas Boltzmann equation.

164

The Dilute Lorentz Gas

With this discussion, we have completed our treatment of the exact and Chapman– Enskog solutions of the field free Lorentz–Boltzmann equation for particles moving in a random, dilute array of three-dimensional, hard-sphere scatterers. Next, we return to the general case and consider solutions for the cases where a Lorentz gas is placed in an external field. 5.5 Lorentz Gas in External Fields In the preceding discussion of the Lorentz gas, we have used the simplifications provided by the Lorentz–Boltzmann equation, in particular its linearity, to gain some understanding of the functional assumption and of the time scales that characterize the Chapman–Enskog solution of the Boltzmann equation in general. Here we begin our consideration of a Lorentz gas with identically charged, moving particles, placed in an external gravitational, electric, or magnetic field. We suppose that the fields are constant in time and, unless otherwise specified, uniform in space. When placed in an unbounded space and under the influence of an electric or, equivalently, a gravitational field, the Lorentz gas has unexpected properties due to the acceleration of the particles by the external field. Several studies have shown that under these circumstances, there is no stationary solution of the Lorentz– Boltzmann equation, and the particle current does not obey Ohm’s law [527, 506, 400, 48, 401]. However, a stationary current in a Lorentz gas with fixed scatterers is achievable if there is some way to dissipate the Joule heat produced by the external electric field. There are at least two ways to accomplish this: (1) If the Joule heat can be transported to heat absorbing, boundary walls at a fixed temperature by thermal conduction, a spatially inhomogeneous, stationary state can be maintained with a constant current. This is, of course, the most physically interesting version of the model. In Chapter 14, we show how such a stationary state can be maintained in a system with boundaries, and for such a system, Ohm’s law can be realized. (2) One can also employ a “Gaussian thermostat” to artificially remove the Joule heat produced by the field. The Gaussian thermostat is actually a modification of the equations of motion for a Lorentz gas with elastic collisions, due to the application of a constraint that requires that the kinetic energy of the system remains constant in time. While such a constrained system is not realizable in a laboratory, it can be realized on a computer. A Lorentz gas with a Gaussian thermostat will be considered in some detail in Section 5.7. The motion of the moving particles in a magnetic field presents different challenges. For two-dimensional systems, with a magnetic field perpendicular to the plane in which the system resides, the Stosszahlansatz, a priori, cannot be used, except in the Grad limit, due to the circular motion of the particles in the field, which may easily give rise to systematic recollisions with the same scatterer.

5.5 External Fields

165

In such a case, one can no longer trust or apply this ansatz, and a more general approach must be developed. For three-dimensional systems and not too strong magnetic fields,10 motion in the direction of the magnetic field suppresses most trajectories giving rise to recollisions, so one can suppose that the Lorentz– Boltzmann equation will correctly describe the transport properties of such Lorentz gases. We will discuss each of these cases, starting with the motion of the particles in an electric (or, equivalently, gravitational) field. 5.5.1 Lorentz-Boltzmann Equation for Particles in Electric Fields Drude’s Law Before we begin a discussion of the solutions of the Lorentz–Boltzmann equation for charged particles in an external electric field, it is useful to consider one of the most useful applications of the Lorentz gas as a model for the conduction of electrons in a solid. The expression for the electrical conductivity of the electrons in such a solid is known as Drude’s law [167, 168]. This expression relates the electrical conductivity of electrons in a solid to their mean free path. The heuristic derivation of this formula is based on some simple arguments, which can be strengthened by a more formal [derivation based on the Lorentz–Boltzmann equation. To provide the heuristic derivation, we consider a Lorentz gas with charged, moving particles of mass m and charge q. We assume that the particles are subjected to a uniform, constant electric field E. We suppose that the particles travel through the solid having collisions with the scatterers and between collisions move freely subject only to the electric field. We assume that the field is sufficiently weak that the equilibrium expression for the mean free time, t, between collisions is unaffected by the field. A rather elementary argument based on the free times between collisions, with mean time between collisions given by tl , leads to an expression for the average drift velocity for the moving particle given by

v =

qtl E. m

(5.5.1)

If the number density of the moving particles is n, then the average electric current is given by j = nq v = n

q 2 Et = σD E, m

(5.5.2)

where the electrical conductivity, σD = nq 2 t /m. This is the Drude expression for the electrical conductivity, of moving charged particles in a solid. Apart from the

166

The Dilute Lorentz Gas

heuristic nature of this expression, it has a wide application in solid state physics, requiring only some means to estimate the transport mean free time for a fermionic system of electrons in metals. Given this, it is correct for virtually all metals. For a classical Lorentz gas, an estimate for the mean free time can be obtained by using arguments based on the equilibrium properties of the Lorentz gas. If in the absence of the field, the particles move with an average speed v and the scatterers have cross-sectional area πa 2 in three dimensions and are distributed at random with density ns , the equilibrium mean free time can be estimated using dimensional arguments and is t ≈

1 , ns πa 2 v

(5.5.3)

leading to an expression for the electrical conductivity σ ≈

nq 2 . ns πa 2 mv

(5.5.4)

Although we will presently argue that the application of the Lorentz–Boltzmann equation to a system acted on by an electric field must be treated with some care, we will ignore this for the moment and show how the Drude formula can be obtained from this equation. The Lorentz–Boltzmann equation in the presence of a conservative electric or gravitational field ϕ(r) assumes the form 1 ∂f (r,v,t) + v · ∇r f (r,v,t) − ∇ϕ · ∇v f (r,v,t) ∂t m  ˆ ˆ = ns d kB(v, k)[f (r,v,t) − f (r,v,t)] = ns λL f .

(5.5.5)

To simplify the calculation, we assume that the system is spatially homogeneous and is in a stationary state, and that the applied electric field is sufficiently weak that we only need to keep terms to first order in the field when constructing a solution to this equation. The Lorentz-Boltzmann equation then becomes qE · ∇v f = ns λL f . m

(5.5.6)

We suppose that when there is no electric field, the distribution function for the moving particles is given for three-dimensional systems by fo (v) =

n δ(v − v0 ). 4πv2o

(5.5.7)

We assume that the electric field modifies this distribution function by an amount δf, taken to be first order in the electric field. In this case, we can replace Eq. (5.5.6) by

5.5 External Fields

qE · ∇v f0 = ns λL δf , m with solution δf = λL

−1



qE · ∇v f0 , ns m

167

(5.5.8)

(5.5.9)

provided that E · ∇v δ(v − v0 ) is orthogonal to the zero eigenfunctions of the Lorentz–Boltzmann collision operator. Given this result, we can formally obtain an expression for the average current in the gas to first order in the field

 qE −1 j = nq dvvλL (5.5.10) · ∇v f0 . ns m By allowing the collision operator to operate to the left, we can write

 qE j = qn dv · ∇v f0 λL −1 v. ns m

(5.5.11)

This expression can be evaluated easily for the action of the inverse collision operator by writing λL −1 v = αv,

(5.5.12)

where11 α=

v2  , v · λL v

(5.5.13)

where the average is taken with respect to the zeroth-order distribution function Eq. (5.5.7). After an integration by parts, this expression for the current becomes, for a three-dimensional hard-sphere Lorentz gas, for example, where α −1 = − πa 2 v0, j=

nq 2 E = σ E, ns mπa 2 v0

(5.5.14)

in agreement with the heuristic derivation given before. Difficulties in the Theory for Electrical Conduction for a System without Boundaries In order to obtain a satisfactory derivation of Ohm’s law, using the Lorentz– Boltzmann equation, one must relax both the requirement that the system be spatially homogeneous and that it be unbounded, with no heat-absorbing boundaries. Here we briefly illustrate the difficulties that arise with respect to Ohm’s law when one tries to solve the Lorentz–Boltzmann equation for an unbounded,

168

The Dilute Lorentz Gas

spatially homogeneous Lorentz gas system subjected to an external electric field. We will do this by summarizing work by J. Piasecki and E. Wajnryb [527] and by K. Olaussen and P. C. Hemmer [506]. We will limit our discussion to the case of a three-dimensional Lorentz gas with hard-sphere scatterers, since the simplicity of the Lorentz–Boltzmann equation for this case makes it possible to describe a solution in a reasonably direct way. Furthermore, for this model, all collisions are instantaneous so we do not have to worry about the effects of the electric field on the dynamics of collisions. We consider Eq. (5.5.5) with a uniform electric field in the z-direction. We consider the spatially homogeneous case, and we use the hard-sphere collision operator given by the right-hand side of Eq. (5.4.4). We can no longer consider the speed of the moving particles to be constant, due to the acceleration provided by the field. The Lorentz–Boltzmann equation for this case is ∂f (v,t) ∂f (v,t) +E = ns πa 2 v(PL − 1)f (v,t), ∂t ∂vz

(5.5.15)

where we simplify the notation slightly by taking E as the product of the magnitude of the electric field and the charge of the moving particles and setting the mass of the moving particles equal to unity. It is useful to rewrite this equation using a spherical coordinate system for the velocity, where θ is the angle that the velocity makes with respect to the direction of the electric field, φ is an azimuthal angle in the (x,y) plane, and v is the speed. We assume that the system is invariant under rotation about the z-axis so that there is no dependence of the distribution function on the azimuthal angle. In terms of these variables, the equation becomes   ∂f (v,t) sin θ ∂f (v,t) ∂f (v,t) + E cos θ − = ns π a 2 v(PL − 1)f (v,t). ∂t ∂v v ∂θ (5.5.16) Given the rotational symmetry of the system, it is possible to expand the solution in terms of Legendre polynomials f (v,t) =

∞ 

fl (v,t)Pl (cos θ).

(5.5.17)

l=0

Equations for the coefficients, fl (v,t), can be obtained by means of the properties of the Legendre polynomials, one of which includes the result that PL Pl (cos θ) = δl0 . We can substitute the expansion, Eq. (5.5.17), solution in Eq. (5.5.16) and use well-known properties of the Legendre polynomials to obtain equations for the functions fl (v,t). Of interest here are the equations for f0 and f1 , which are ∂f0 (v,t) E ∂f1 (v,t) 2E + + f1 (v,t) = 0, ∂t 3 ∂v 3v

(5.5.18)

5.5 External Fields





6E ∂f1 2 ∂f2 ∂f0 + +E + f2 + πns a 2 vf1 = 0. ∂t 5 ∂v ∂v 5v

169

(5.5.19)

Clearly, the system of equations for the coefficients fm is an open hierarchy where the coefficients of each order in m depend on the coefficients of order m + 1. However, some important simplifications are possible. We can argue that the longtime dependence of the distribution function is determined by the isotropic term, f0 (v,t), since this term is an eigenfunction of the collision operator corresponding to eigenvalue zero. All of the higher-order coefficients would decay exponentially in the absence of an electric field due to the contribution from collisions of the moving particles with the scatterers, as expressed by the the last term on the lefthand side of Eq. (5.5.19). Therefore, one can expect to obtain a good approximation by truncating this hierarchy, neglecting f2 and higher-order coefficients, and solving the remaining equations for f0 and f1 . Higher-order approximations can then be found successively by a process of iteration, where an expression for f2 is obtained by neglecting f3 and higher-order terms and then solving the coupled set of equations for f0,f1 , etc. We will not do that here, since the results of most interest to us can be obtained already from the asymptotic behavior of the isotropic term, f0 , and the first non-isotropic correction to it given by f1 P1 (cos θ). We neglect f2 in Eq. (5.5.19) and suppose that for small enough E, the time derivative of f1 can be neglected in Eq. (5.5.19). One must then check the consistency of these two assumptions a posteriori. We will not do that here but refer to the original paper by K. Olaussen and P. C. Hemmer for the details [506]. We then find that f1 is given by f1 (v,t) = −

E ∂f0 (v,t) , πns a 2 v ∂v

(5.5.20)

and the equation for f0 becomes   1 ∂2 1 ∂ E2 ∂f0 (v,t) f0 (v,t). + = ∂t 3πns a 2 v ∂v2 v2 ∂v

(5.5.21)

This equation has the form of a diffusion equation in velocity space, and the substitution r = v3/2 produces the diffusion equation for a radially symmetric two dimensional system   ∂2 ˜ 1 ∂ ∂ f˜0 = DE + 2 f0, ∂t r ∂r ∂r

(5.5.22)

where the “diffusion coefficient” DE = 3E 2 /(4πns a 2 ), and we use the notation f0 (v = r 2/3,t) ≡ f˜0 (r,t). The Green’s function for this equation, G(r,r ,t), is well known and is

170

The Dilute Lorentz Gas

G(r,r ,t) =

1 e 4πDE t

−

(r − r  )2 4DE t .

The general solution of Eq. (5.5.22) is  f˜0 (r,t) = dr  G(r,r ,t)f˜0 (r ,t = 0).

(5.5.23)

(5.5.24)

Here the integration is over a plane. Given the fact that f0 and f˜0 are functions only of the scalars, v or r, respectively, the angular integration in the integral can be carried out with the result that r2  r 2

∞ − rr  1  ˜   4D t 4D t ˜ E E , I0 dr f0 (r ,0)r e e f0 (r,t) = 2DE t 2DE t 0 −

(5.5.25)

where I0 (x) is a Bessel function of imaginary argument of order zero. Reverting to the velocity arguments, we can see immediately that there is a natural scaling in the variable v/t 1/3 . This is the first indication that the average speed increases with a non-integer power of the time, t. We are particularly interested in the behavior of this function for large times. For fixed values of r, or better, the velocity v, and as the time t gets very large, there is a range of values of r  in the integrand, for which the argument of the Bessel function in the preceding integral approaches zero, and the Bessel function itself approaches unity. This is not a uniform approximation since there are values of r  for which the argument is not small, but it is easy to see that if the initial value of the distribution is sharply peaked about some velocity v0 , there will be a large time for which v30 /(DE t) 1, and this time sets the scale on which the asymptotic results are valid, perhaps with small exponentially decaying corrections. Thus, for DE t v30 , the asymptotic form of f˜0 is f˜0 (v,t) 

−

v3

3 e 4DE t . 16πDE t

(5.5.26)

If we combine this result with Eq. (5.5.20), we obtain an explicit expression for f1 (v,t). Having obtained approximate expressions for the functions f0 (v,t) and f1 (v,t) appearing in the first two terms in the Legendre polynomial expansion of the distribution function of the moving particles, we can examine the interesting asymptotic behaviors of the quantities of interest to us, namely the drift velocity and the mean square velocity of the moving particles. Since the z-axis of the system is taken in the direction of the external electric field, the drift velocity will be the average value of vz = v cos θ. The only non-zero term in the calculation of this average using the

5.6 Uniform Magnetic Fields

171

expansion, Eq. (5.5.17), will be the term f1 (v,t)P1 (cos θ). Thus, the drift velocity, < vz >, is, asymptotically,  1 1 < vz >= dvv cos θf1 (v,t)P1 (cos θ) ∼ BE 3 t − 3 , (5.5.27) The mean square velocity contains no angular dependent terms so that its average is  4 2 2 (5.5.28) < v >= dvv2 f0 (v,t) ∼ CE 3 t 3 , thus growing with time as t 2/3 . These fractional powers can also be obtained by means of more heuristic, random walk arguments [400, 401], and using either method, one demonstrates that, instead of exhibiting normal electrical conduction properties, the Lorentz gas in unbounded space exhibits anomalous behavior with respect to the transport of charged particles in an electric field. There is no steadystate drift velocity, and all the physically important quantities we have discussed depend on fractional powers of the electric field. As we stated earlier, more interesting and more useful physical descriptions of the Lorentz gas in an electric field require that one includes in the model some mechanism for removing the Joule heat. Of these, the most physically relevant is the model of a spatially inhomogeneous, finite system with heat absorbing boundaries. Another way to get a system with normal conduction is to relax the requirement that the scatterers be fixed, and suppose instead that they are a gas of much heavier particles with a Maxwell-Boltzmann distribution at some temperature, T . Then the heavy particles will absorb some of the kinetic energy imparted to the light charged particles by the imposed electric field, and Ohm’s law can be recovered [89]. This system is a binary mixture of heavy and sparsely distributed light particles, a special case of binary mixtures discussed in Chapter 4. The distribution function for the light particles will then satisfy a modified Lorentz-Boltzmann equation. 5.6 Transport of Particles in a Uniform Magnetic Field The solution of the Lorentz–Boltzmann equation for the distribution function for the moving particles in a Lorentz gas placed in a uniform magnetic field depends dramatically on the number of spatial dimensions of the system [48, 112]. Because of the cyclotron orbits of free, charged particles in a magnetic field, this equation takes very different forms if the system is two-dimensional with a magnetic field perpendicular to the plane containing the particles and the scatterers from that of a higher-dimensional Lorentz gas placed a uniform magnetic field. For a two-dimensional system with a field as described, the Stosszahlansatz is certainly

172

The Dilute Lorentz Gas

violated, unless the mean free path is very small compared to the cyclotron radius. Otherwise, the usual Lorentz–Boltzmann equation must be replaced by a modified version that takes into account the effects of the re-collisions due to cyclotron orbits of the moving particles between collisions. For reasons that will presently be clear, the fact that, in three dimensions, free particles move in helical orbits rather than circles in most cases allows the use of the Lorentz–Boltzmann equation without further modifications due to the presence of the field. We first treat the three-dimensional case and then turn to the resolution of the complications that arise for two-dimensional systems caused by the cyclotron orbits for the moving particles.12 5.6.1 Magneto-transport in the Three-Dimensional Lorentz Gas The product of the magnetic field vector, B, and the charge per unit mass of the particles will be represented by the angular velocity vector . The Lorentz force acting on a particle with velocity v is v × . To simplify the model further, we take the scatterers to be hard spheres.13 Then the Lorentz–Boltzmann equation is ∂f (r,v,t) + v · ∇r f (r,v,t) + (v × ) · ∇v f (r,v,t) = πns a 2 v (PL − 1) f (r,v,t). ∂t (5.6.1) The magnetic field vector is taken to be constant in time and uniform in space. We take  to be along the positive z-axis, and we express the particle velocity in spherical coordinates (v,θ,φ). Between collisions, the z-component of the velocity remains constant, while the components of the velocity, vx ,vy , undergo a Larmor rotation about the z-axis. Our goal here is to solve Eq. (5.6.1) in a manner similar to that used for the field free case and then to extract the hydrodynamic, or diffusive, properties by showing that there is a hydrodynamic pole in the complex plane of the appropriate time-Laplace transform variable, denoted in Eq. (5.6.10) by s, to avoid confusion with the z-direction. We begin by expressing the Lorentz–Boltzmann in velocity-spherical coordinates as ∂ ∂f (r,v,t) + v · ∇r f (r,v,t) − ω f (r,v,t) = πns a 2 v (PL − 1) f (r,v,t). ∂t ∂φ (5.6.2) The magnitude of the Larmor angular velocity is denoted by the scalar ω = qB/m. In order to construct the solution to this equation, we will need the properties of free particle motion in this field, , which are readily obtained by solving the equations of motion. The velocity of a free particle changes in time as ˆ v(t) = v cos θ zˆ + v sin θ cos(φ0 − ωt)ˆx + v sin θ sin(φ0 − ωt)y,

(5.6.3)

5.6 Three dimensions - hard spheres

173

ˆ yˆ , z,ˆ form an orthogonal set of unit vectors. With this velocity, the position where x, of a particle at time t is r(t) − r(0) = v cos θt zˆ + +

 v sin θ  − (sin(φ0 − ωt) + sin φ0 ) xˆ ω

v sin θ (cos(φ0 − ωt) − cos φ0 ) yˆ . ω

(5.6.4)

Note that the equation for the position at time t can be expressed in terms of the rotation operator Uz (α) for a rotation about the z-axis by an angle, α, as 1 π [v⊥ (t) − v⊥ (0)] , (5.6.5) r(t) − r(0) = v cos θt zˆ + Uz ω 2 where v⊥ = vx xˆ + vy yˆ , is the projection of the particle’s velocity onto a plane perpendicular to the direction of the magnetic field. We will need the solution of the homogeneous form of Eq. (5.6.2), obtained by setting the right-hand side equal to zero. This solution is easily found by realizing that the homogeneous equation is the one-particle Liouville equation for this system, with solution fh (r,v,t) = f (r(−t),v(−t),0), namely the distribution function at time t = 0, evaluated at the point (r(−t),v(−t)). In terms of the position and velocity at a previous time, the full distribution function at r,v at time t satisfies the integral equation 2 f (r,v,t) = e−πns a vt f (r(−t),v(−t),0)  t 2 2 dτ e−πns a v(t − τ ) [PL f (r(−t + τ ),v,τ )]. + πna v

(5.6.6)

0

To obtain this integral form, we have used the fact that the projection operator on the right-hand side of Eq. (5.6.2) produces a function only of the position and time variables, and the scalar velocity, since the operator is an integration over all possible directions. In order to check the result, one can rewrite Eq. (5.6.2) in the form ∂f (r,v,t) + L1 f (r,v,t) + πns a 2 vf (r,v,t) = πns a 2 vPL f (r,v,t), ∂t

(5.6.7)

where L1 is the one-particle Liouville operator L 1 = v · ∇r − ω

∂ . ∂φ

It is then straightforward to formally integrate Eq. (5.6.7) to obtain Eq. (5.6.6). We use the space–Fourier transform of f (r,v,t)  fk (v,t) = dreik · r f (r,v,t)

174

The Dilute Lorentz Gas

and use Eqs. (5.6.7) and (5.6.5) to obtain 

  fk (v,t) = exp g(θ,t) fk (v(−t),0)+πns a 2 v

t

  dτ exp g(θ,t − τ ) [PL fk (v,τ )],

0

(5.6.8) where g(θ,t − τ ) = −πns a 2 v(t − τ )

1 π [v⊥ (−(t − τ )) − v⊥ (0)] . + ik · v(t − τ ) cos θ zˆ − Uz ω 2 (5.6.9) The convolution structure of the expression for fk (v,t) allows a useful application of the Laplace transform in time. We define the Laplace transform of fk (v,t) by  ∞ dte−ts fk (v,t). (5.6.10) fk (v,s) = 0

We apply the transformation to Eq. (5.6.8), which results in a useful expression for fˆk (v,t) as fk (v,s) = A0,k (v,s) + πns a 2 vBk (v,s)[PL fk (v,s)], where



∞

A0,k (v,s) =

dt exp [h(θ,t) − st] fk (v(−t),0),

(5.6.11)

(5.6.12)

0

and



∞

Bk (v,s) =

dt exp [h(θ,t) − st] ,

(5.6.13)

0

where



1 π [v⊥ (−t) − v⊥ (0)] . h(θ,t) = −πns a 2 vt + ik · vt cos θ zˆ − Uz ω 2 (5.6.14)

As was done in the solution of the field free Lorentz–Boltzmann equation, we apply the projection operator to both sides of this equation in order to obtain an explicit expression for the projected part of the distribution function and use Eq. (5.6.11), for the full distribution function itself. The steps are immediate and we find that fk (v,s) = A0,k (v,s) +

πns a 2 vBk (v,s)[PL A0,k (v,s)] . 1 − πns a 2 v[PL Bk (v,s)]

(5.6.15)

5.6 Three dimensions - hard spheres

175

This is an explicit solution of the Lorentz–Boltzmann equation for this case, expressed in terms of the initial distribution function. As was the case for the Lorentz gas with no external fields, when we invert the Laplace transform we find that the long time behavior of the distribution function is determined by the pole with real part closest to the origin. To find this pole, and to determine the long time behavior of fk (v,t) we need to determine the value of s with smallest real part that solves the equation 1 − πns a 2 v[PL Bk (v,s)] = 0.

(5.6.16)

It is easy to check that when k = 0, the only solution of Eq. (5.6.16) is s = 0. This allows us to consider expanding the hydrodynamic solution, denoted by sD (k), in powers of k about s = 0. We also note that from the structure of Bk (v,s), there may be a difference between particle diffusion in the direction of the field and diffusion in the directions perpendicular to the field. To simplify the analysis, we will suppose that k is a vector in the (x,z) plane, so that ky = 0. Then, with the help of some simple trigonometric identities, Eq. (5.6.16) can be written as  π  2π    πns a 2 v ∞ dt sin θdθ dφ exp q(s,t,θ,φ) = 1, (5.6.17) 4π 0 0 0 where



2ivkx ωt sin θ sin cos(φ + ωt). q(s,t,θ,φ) = −(s + πna v)t + ikz tv cos θ − ω 2 (5.6.18) 2

We find the location of the hydrodynamic pole by solving Eq. (5.6.17) for s as a perturbation expansion for small kx ,kz about the value s = 0. So far we have set ky = 0. We easily find that sD (k) = −Dz kz2 − Dx kx2 + O(k 4 ),

(5.6.19)

where v , 3πns a 2

(5.6.20)

πns a 2 v3  . 3 ω2 + (πns a 2 v)2

(5.6.21)

Dz = and Dx =

This motion is properly described by a diffusion tensor, Dij , with i,j = x,y,z, as given by F. Cornu and J. Piasecki [112]. The tensor elements are Dzz = Dz,Dxx = Dyy = Dx ,Dzy = Dyz = 0, and Dyx = −Dxy = ωDx (πns a 2 v)−1 . Diffusion

176

The Dilute Lorentz Gas

Figure 5.6.1 Some possible trajectories or fragments for the cyclotron motion of charged moving particles in a two-dimensional Lorentz gas with magnetic field perpendicular to the plane of the system. Figure adapted from A. V. Bobylev, F. A. Maaø, A. Hansen, and E. H. Hauge [48]

in the plane perpendicular to the field is slower, due to the circular motion of the particles when their motion is projected on this plane. For zero magnetic field, one obtains the usual results, while for very large fields, the diffusion coefficient in the perpendicular plane decreases as the inverse square of the field strength. The derivation of the Lorentz-Boltzmann equation for the transport of charged particles in a dilute, two dimensional Lorentz gas with a field perpendicular to the plane of the gas provides us with a challenging situation unlike any that we have encountered so far, namely, the role of the Stosszahlansatz in the derivation of the Lorentz-Boltzmann equation for this system. We begin noting that the radius, R of the cyclotron orbits of a particle with speed v is given by R = v/ω where, as above, ω = qB/m, and for convenience we consider the charge to be positive. We refer the reader to the discussion of Bobylev et al. for an extensive discussion of the different possible types of trajectories for the moving particle. Some of them are illustrated in Fig (5.6.1). We will be concerned here with the macroscopic transport of particles in the field, that is to say, we consider only the kinetic equation for particles that make one or more collisions with a given scatterer before moving on to collisions with other scatterers and are not trapped in an orbit about some number of scatterers. It is the distribution function for these particles that is needed for a description of the transport properties of this system. However there is a problem using the Lorentz-Boltzmann equation without any further modifications. The problem is that the Stosszahlansatz cannot be used to describe the probability of all of the collisions between the moving particles and the scatterers in the case where the particle makes more than one collision with a scatterer before moving on. Although the Stosszahlansatz can be applied to the first collision of a moving particle with

5.6 Three dimensions - hard spheres

177

Figure 5.6.2 Successive collisions of a moving particle with a fixed scatterer of radius a. The successive collisions are labeled 1 → 2 → 3. The quantity

denotes the distance from the center of the scatterer to the centers of the cyclotron orbits. The angle subtended by two adjacent incidence points is denoted by 2β. Figure taken from [48]

a scatterer, the second and further successive collisions with the same scatterer are determined only by the mechanics of the system, and stochastic arguments are then used only to calculate the probability that such successive collisions will take place, in other words the probability that the moving particle does not encounter other scatterers which would prevent further collisions with the original scatterer. In this situation, the Stosszahlansatz can be modified so as to allow a generalized version of the Lorentz-Boltzmann equation that applies to this system. In constructing the modified form of the Lorentz-Boltzmann equation, we continue to follow the procedure of Bobylev et al. [48]. First we consider the geometry of one re-collision of a moving particle with a scatterer. Consider a system of one moving particle and one scatterer in which the particle has cyclotron radius R and makes more than one collision with the scatterer, as illustrated in Fig. (5.6.2). At each collision the particle moves from one cyclotron orbit to another one with the same radius and although it collides at a different point on the scattering disc, the angle of incidence is the same as that of the previous collision. The arc length on the scatterer separating the impact points of the two successive collisions is 2aβ where 2β is the angular shift of the impact points on the circumference of the scatterer. One can see that

178

The Dilute Lorentz Gas

y a a

b

a

Figure 5.6.3 The dynamics of the collision of a moving particle with a scatterer in a magnetic field perpendicular to the plane of the system. Here b = a sin α is the impact parameter. The scattering angle is denoted by ψ and α = (π − ψ)/2. This figure is taken from [48]

cos β =

2 − R 2 + a 2 , 2a

where is the distance between the center of the cyclotron orbit and that of the scatterer, and is in the range R − a ≤ ≤ R + a. We will have in mind the limit that a/R → 0, together with the Grad limit, a → 0,ns → ∞ with a finite mean free path,  = (2ns a)−1 . For what is to follow we will need the average number of scatterers within an area of width 2a around a single cyclotron orbit, from which we obtain the probability that the moving particle will not encounter a scatterer. This area, A0, is A0 = (2πR)(2a). One can argue that in the Grad limit this result is correct to lowest order in the radius of the scatterers, a, since the difference between this 2πR and the length of an actual orbit that has to be free of scatterers is of order a 2 . The probability, P (R), that this region is free of scatterers is P (R) = exp(−A0 ns ) = exp(−4πaRns ) = exp(−νT ), where T = 2π/ω is the period of the cyclotron orbit and the collision frequency ν = 2ns va. Thus P (R) is also the probability that the particle will not encounter a scatterer in a cyclotron orbit. Let us now consider the rate of change of the distribution function for particles whose velocity direction makes an angle φ with respect to some fixed axis in space.

5.6 Three dimensions - hard spheres

179

We suppose that in the collision that produced a particle with angle φ, the precollision velocity made an angle φ − ψ with the fixed axis. Suppose further that the particle with velocity angle φ was produced not by one but by a sequence of two or more collisions with the same scatterer. Then the pre-collision velocity direction is shifted by an angle −ψ at each collision, as illustrated in Fig. (5.6.3). Thus if the particle had a sequence of k collisions with the same scatterer, then there must have been k − 1 cyclotron orbits around the same scatterer with no intervening collisions with other scatterers. The probability of a sequence of k − 1 orbits taking place without collisions with other scatterers is [P (R)]k−1 and the pre-collision velocity of the moving particle at the very first collision is φ − kψ. It is this first collision that should be described by the Stosszahlansatz. The subsequent collisions are all determined by the mechanics of the first one and are subject only to the condition that the moving particle not encounter any other scatterers. With this in mind we can write the Lorentz-Boltzmann in terms of the collision probabilities, and the probabilities of not encountering other scatterers. Since T = 2πR/v is the period of a cyclotron orbit of radius R, the Lorentz-Boltzmann equation is given by ∂ ∂f (r,φ,t) + v · ∇r f (r,φ,t) − ω f (r,φ,t) ∂t ∂φ    [t/T ] 2π  ψ ν  k  [P (R)] dψ sin  = 2 k=0 2 0   × f (r,φ − (k + 1)ψ,t − kT ) − f (r,φ − kψ,t − kT )

(5.6.22)

where, as above φ denotes the angle that the velocity vector v makes with some space-fixed direction. The quantity denoted by [t/T ], is defined to be the integer part of number of cyclotron orbits completed by time t. The usual LorentzBoltzmann equation obtains for the initial period where [t/T ] = 0. The transport properties of this system have been studied by a number of authors. Here we will not go into the details of the solution, but only mention that the diffusion process is not isotropic and the diffusion equation with a scalar coefficient of diffusion must be replaced by a diffusion equation with a diffusion tensor with elements that satisfy Dxx = Dyy and Dxy = −Dyx . The off-diagonal terms satisfy the form of the Onsager relations [508, 509] appropriate for systems with an applied magnetic field. Explicit values for these quantities can be found elsewhere [48]. It is possible to extend this analysis beyond the Grad limit, and to study the case where the density of scatterers is sufficiently high that issues of percolation must be considered. We refer to the paper of Bobylev et al. for a more detailed discussion of this interesting phenomenon [48].

180

The Dilute Lorentz Gas

5.7 Chaos in the Lorentz Gas The dilute random Lorentz gas is an example of a chaotic dynamical system [513, 243, 152]. In this section, we will briefly introduce the notion of chaos in a Hamiltonian mechanical system and demonstrate that the Lorentz gas studied in this chapter does indeed exhibit chaos.14 We discuss how to quantify the chaotic properties of this model in the limit of low density of scatterers. In the usual context of statistical mechanics, a chaotic mechanical system has properties that make it a suitable candidate for the application of stochastic analysis, despite the fact that its time evolution, when treated by classical equations of motion, is completely deterministic. The stochastic analysis of a chaotic system can be justified by the following considerations: It is known that a classical dynamical system (e.g. a Hamiltonian one) has the property that, given the complete and precise set of initial conditions, the future development of the system is entirely predictable. Suppose that the system is very sensitive to the exact values of the initial variables that specify the trajectory of the system in phase space, so much so that the slightest change in the initial conditions will eventually lead to very different trajectories in phase space. In such a case, it is pointless to try to predict the time development of such a system if one has imprecise and incomplete knowledge of the initial conditions. Instead, one might try to predict the average behavior of an ensemble of such systems, all with nearly the same values of the variables for which the initial conditions are known. The rate at which two infinitesimally close trajectories separate in phase space sets the time scale after which stochastic methods can be used to describe the average behavior of the ensemble of systems. If any two trajectories, initially infinitesimally close but picked at random from the set of possible trajectories in phase space, separate exponentially rapidly as the time increases toward infinity, the system is said to be chaotic. Chaotic systems are characterized by exponential separations of trajectories, and the coefficients, λ+ , are called positive Lyapunov exponents. The rate of separation can be different for different orientations of the initial separation vector so there may be a spectrum of Lyapunov exponents.15 We note that for a Hamiltonian chaotic system, running the trajectories backward in time also leads to exponential separations, with the same Lyapunov exponents, as one can see from fairly simple considerations based on the reversibility of the equations of motion. Here we will show that the dilute random Lorentz gas with hard-sphere scatterers is chaotic, and we will illustrate the method used to calculate the associated Lyapunov exponents. In this section, we first discuss the chaotic properties of the Lorentz gas satisfying Hamiltonian, hard-sphere dynamics, and later, those of a gas of charged, moving particles in an external electric field, and acted upon by a Gaussian thermostat that keeps the kinetic energy constant for the moving particles. For simplicity, we consider

5.7 Chaos in the Lorentz Gas

181

here only two-dimensional systems and refer to the literature for the calculation of Lyapunov exponents for systems in higher-dimensional systems. For some systems, it is possible to relate their chaotic and transport properties [146, 145].16 Here we will discuss one example of such a connection, namely that the diffusion coefficient of a moving particle can be expressed in terms of the Lyapunov exponents for a driven but thermostatted gas of particles. Many other examples are known and discussed in the books by Gaspard [243]; Dorfman [152], mentioned before; and, more recently, R. Klages [393].

5.7.1 Phase-Space Considerations For understanding the chaotic properties of the Lorentz gas, it is important to have a method to calculate the number of its Lyapunov exponents. This number can be found by examining the properties of phase-space trajectories for this system. Let us first consider the phase space. A Lorentz gas in d-dimensions has 2d degrees of freedom that specify the position and the momentum for a moving particle. Since the motion of each moving particle in a Lorentz gas is independent of the motion of all the others, we may restrict ourselves to a system with one moving particle. Here we consider the Hamiltonian case and postpone the thermostatted case until Section 5.4.7. Since the system is Hamiltonian, Liouville’s theorem applies, and the volume of any region in this phase space stays invariant with time. We consider a pencil of trajectories that all have the same energy so that the number of degrees of freedom is 2d − 1, since all of the trajectories must lie on the same constant energy surface. If two trajectories separate exponentially in time, then this can only “take place in specific directions on the constant energy surface. Conservation of phase space volumes requires that the sum of the negative Lyapunov exponents must be equal in magnitude but opposite in sign to the sum of the positive ones, so the sum of all of them vanishes. In addition it can be shown that the Lyapunov exponents for a Hamiltonian system come in conjugate pairs, each pair summing to zero. This “conjugate pairing rule” strongly depends on the reversibility of the equations of motion, as well as on the conservation of phase-space volumes for Hamiltonian systems17 . There is one direction for the separation vector of two infinitesimally close initial points on the constant energy surface that has a zero Lyapunov exponent. This occurs whenever an initial points on the energy surface is infinitesimally displaced along the same trajectory. The separation of these two points will not grow exponentially, if it grows at all. Another circumstance leading to a zero exponent is mentioned in a.18 Thus, there can be at most 2d − 2 non-zero Lyapunov exponents, with d − 1 of them positive and d − 1, negative. It is worth noting that, typically, the expansion rate of an

182

The Dilute Lorentz Gas

s-dimensional subspace of phase space will be determined by the sum of the s largest Lyapunov exponents. The Lyapunov exponents can be determined by successively calculating the rates of increase of s-dimensional subsets for s = 1,2, . . .. We will see in what follows that for the hard-sphere Lorentz gas, the separation growth between two trajectories is intermittent, linear during free flight, and with jumps at collisions with a scatterer.” 5.7.2 The Radius of Curvature Before calculating the positive Lyapunov exponent for a two dimensional Lorentz gas, we may gain some insight into the mechanism responsible for the exponential separation of two close trajectories by considering a totally equivalent optical example [404, 593, 246, 243, 152]. Suppose we replace the moving particles, at the initial time, by very narrow rays of light and think of the scatterers as mirrors reflecting the light. The convex nature of these mirrors causes a “defocusing” of the beam of rays every time it encounters a scatterer. This defocusing property of the scatterers is clearly the source of chaotic behavior for this system. After a number of reflections, initially very close rays of light become widely separated and eventually begin to strike entirely different mirrors. In order to have two rays be reflected by the same mirrors over some long time interval we have to take rays that are arbitrarily close to each other in the initial beam. We can replace these rays by particle trajectories and observe the same defocusing effects in a Lorentz gas with hard-sphere scatterers. This separation of trajectories is illustrated in Fig. 5.7.1. In order to relate the defocusing properties of the scatterers to Lyapunov exponents, we perform the following thought experiment. We follow one particle as it moves through the system of scatterers. To this particle we attach a ghost particle with the same energy that starts at some point in time, say t = 0, at the same point in space where the particle starts, at the same time, but with a velocity that is infinitesimally close to, but not identical to, that of the real particle. We then follow the separation of the trajectories between the real and the ghost particle.19 If the trajectory has position and velocity coordinates r(t),v(t) at time t, we will take the ghost trajectory to have coordinates r(t) + δr(t),v(t) + δv(t) also at time t. Since both trajectories have the same energy, it follows that δv(t) · v(t) = 0. Now we introduce a new quantity, the radius of curvature, ρ(t), which is a scalar quantity for the two-dimensional Lorentz gases we are treating here, but it is a tensor for higher-dimensional systems. If we follow the two trajectories for a time t0 before the first collision, we see that they form a wedge of a circle with radius ρ(t0 ) = vt0 and arc length δS(t0 ) = ρ(t0 )δθ, where δθ is the angle between v and v + δv. The radius of the circular wedge is called the radius of curvature, and it

5.7 Chaos in the Lorentz Gas

183

Figure 5.7.1 The defocusing effect of the convex scatterers on a small pencil of trajectories of the moving particles. Here ϕ is the angle of incidence of the infinitesimally small pencil of trajectories.

will play a crucial role in the calculation of the Lyapunov exponent. We take t0 to be some very small time after the initial time but before a collision and we take the first such collision to occur at time t0 + τ1 . The arc length of the circular wedge then becomes

vτ1 . (5.7.1) δS(t0 + τ1 ) = v(t0 + τ1 )δθ = δS(t0 ) 1 + ρ(t0 ) Here we express the arc length just before collision in a form chosen for convenience. The collisions of the particle with the scatterers are instantaneous, and the scatterers are circles of radius a. As noted earlier, we can think of the two trajectories as forming an infinitesimal pencil of light with radius of curvature ρ(t0 + τ1 ) and infinitesimal angular width impinging on a mirror with radius a. The formulae of classical optics tell us that the radius of curvature after collision, denoted as ρ+ , is related to that before collision, denoted as ρ− , by

184

The Dilute Lorentz Gas

Figure 5.7.2 The change in the radius of curvature for the collision illustrated in Fig. 5.7.1. Here ρ− and ρ+ denote the radius of curvature before and after the collision, respectively.

1 2 1 = + , ρ+ ρ− a cos φ

(5.7.2)

where φ is the angle of incidence of the pencil on the disk. This is a standard formula of elementary optics (usually for φ = 0, for the locations of objects, ρ− , and images, ρ+ , in front of spherical mirrors), and the proof of the general result is left as an exercise for the reader. The two radii are illustrated in Figure 5.7.2. Let ρ+,1 be the radius of curvature after the first collision of the moving particle with a scatterer. It then follows from the preceding argument that if τ2 is the time between the first and second collisions, then the separation of the two trajectories just before the second collision will be



vτ1 vτ2 δS(t0 + τ1 + τ2 ) = δS(t0 ) 1 + 1+ . (5.7.3) ρ(t0 ) ρ+,1 We have used the fact that, although the radius of curvature changes instantaneously and discontinuously at a collision, the separation of the two trajectories does not change at the instant of a collision; i.e. the separation of trajectories is a continuous function of time. Clearly, if we have a sequence of n collisions, taking place with time intervals τ1,τ2,τ3, . . . ,τn after the initial time t0 , then the separation of the trajectories at some time τ after the nth collision but before the (n + 1)th collision will be ! n

"

# vτi vτ 1+ 1+ . δS(t0 + τ1 + τ2 + · · · + τn + τ ) = δS(t0 ) ρ+,i−1 ρ+,n i=1 (5.7.4) We can now express this separation of trajectories in exponential form, preparatory to getting a final expression for the positive Lyapunov exponent, as

5.7 Chaos in the Lorentz Gas

δS(t0 + T ) = exp T δS(t0 )



1 T

!

n  i=1

vτ vτi + ln 1 + ln 1 + ρ+,i−1 ρ+,n

185

" ,

(5.7.5)

where T + t0 = t0 + τ1 + · · · + τn + τ . We are now going to argue that the quantity in the square brackets in the exponential expression in Eq. (5.7.5) is well behaved as T → ∞, so that, for large T , the right-hand side of Eq. (5.7.5) has the form exp(λT ). We then will identify λ as the positive Lyapunov exponent for this system. To see this, we use a simple identity:

 t 1 vt =v dτ . ln 1 + ρ ρ + vτ 0 This identity allows us to write



n  vτi vτ ln 1 + + ln 1 + ρ+,i−1 ρ+,n i=1  τi  τ n  1 1 v = dt + v dt. 0 ρ+,n + vt 0 ρ+,i−1 + vt i=1

(5.7.6)

Notice that the denominator in each of the integrals is simply the radius of curvature for all times in the interval between the (i − 1)th collision and the ith collision, except for the last integral, where it is simply the radius of curvature in the interval after the last collision in the sequence and some next collision, which occurs after time T . Consequently, the right-hand side of Eq. (5.7.6) can be written as one integral,  t0 +T 1 v dt , ρ(t) t0 and the expression for the separation of trajectories, Eq. (5.7.5), becomes   t0 +T  v 1 δS(T ) = exp T dt . δS(t0 ) T t0 ρ(t)

(5.7.7)

This result allows us to obtain the Lyapunov exponent as the time average of the inverse of the radius of curvature,  v t0 +T 1 dt. (5.7.8) λ+ = lim T →∞ T t ρ(t) 0 Here we have taken the limit that the two nearby trajectories start infinitesimally close to each other, and we let the time interval grow large in order to extract a quantity representing the exponential rate of separation of very nearby trajectories. One might question our identification of this expression with the Lyapunov exponent since we have just examined the spatial separation of two trajectories, while

186

The Dilute Lorentz Gas

the Lyapunov exponent characterizes the total separation of trajectories in phase space. A simple argument shows that these two quantities are the same. To see this, let’s look at the separation of trajectories in phase space 

1/2 δv(t) 2 δr(t) 2 δ(t) = + . (5.7.9) r0 v0 Here, r0 and v0 are scaling constants chosen so that the dimensions of the two terms in the preceding expression will be the same, i.e., dimensionless. Now the quantity |δr(t)| is just δS(t) computed before. The quantity δv(t) is simply the time derivative of δr(t), so the time derivative of δr(t) will also be an exponential function of time, with the same exponential behavior. Thus, both terms in the square root on the right-hand side of Eq. (5.7.9) will grow with the same exponent, λ+ . Equation (5.7.8) was first obtained by Ya. G. Sinai [593]. This expression can be evaluated for a dilute, random Lorentz gas, using the Lorentz–Boltzmann equation. Before doing so, we apply Birkhoff’s ergodic theorem, which asserts that for ergodic systems, one may replace time averages by ensemble averages [290]. If one can prove that the random Lorentz gas is ergodic, then an expression for λ is given by  1 , (5.7.10) λ+ = v ρ eq where the angular brackets denote an average over an equilibrium ensemble. The proof that a two-dimensional Lorentz gas is ergodic was given by Sinai for the case that the scatterers are placed in a periodic arrangement on the plane in such a way that arbitrarily long free paths are not possible. Since this original work, much more has been learned about the chaotic properties of these and related systems. Many of the original papers are included in two books edited by Sinai [593, 592]. More recent work is described in the collection of papers edited by D. Szasz [617]. In the light of the considerable amount of work done on the chaotic properties of the periodic Lorentz gas and on systems of moving hard spheres, it is reasonable to suppose that the motion of a particle in a dilute, random Lorentz gas in a large but finite system is sufficiently chaotic that Birkhoff’s theorem can be applied to it. This allows us to evaluate the positive Lyapunov exponent by calculating the ensemble average indicated in Eq. (5.7.10). 5.7.3 The Extended Lorentz–Boltzmann Equation Since the Lorentz gas is taken to be dilute, an equation for the distribution function needed to evaluate expression Eq. (5.7.10) can be obtained as an extended

5.7 Chaos in the Lorentz Gas

187

Lorentz–Boltzmann equation with the aid of familiar Stosszahlansatz [639, 640].20 That is, we assume that the moving particle is not correlated with any of the scatterers before it collides with them. We will not be able to use the Lorentz– Boltzmann equation, Eq. (5.2.2), as it stands since we have another variable to take into account, namely the radius of curvature ρ. This variable can be incorporated into the equation by making the following construction: The moving particle is described by its position r, its velocity v, and by a radius of curvature ρ. This variable is added by supposing that each trajectory of the moving particle is accompanied by its ghost, which is infinitesimally close to the one under consideration. Then the quantity ρ characterizes the rate of separation of the two trajectories by the radius of curvature for the two trajectories – the reference trajectory of the moving particle and the fictitious ghost trajectory. We know that the radius of curvature grows linearly in time between collisions, ρ(t) = ρ(t0 ) + v(t − t0 ) t ≥ t0,

(5.7.11)

and changes discontinuously at a collision according to Eq. (5.7.2). This is just what we need for a Boltzmann equation. We consider a distribution function f (r,v,ρ,t) and apply the heuristic method of Boltzmann to derive an equation for it. To proceed, we let f (r,v,ρ,t)drdvdρ be the probability of finding the moving particle with position in the region dr about r, with velocity in the region dv about v, and with radius of curvature (with a neighboring trajectory as described above) in the region dρ about ρ, all at time t. Then, following the previous derivation of the Lorentz–Boltzmann equation, we consider that f changes with time due to the free motion of particles between collisions and due to collisions. Then we write an equation for f as   f (r + vdt,v,ρ + vdt,t + dt) − f (r,v,ρ,t) drdvdρ = + − − . (5.7.12) The Taylor expansion of the left-hand side of Eq. (5.7.12) is immediate, of course. The loss term − can easily be computed just by considering the collision cylinders in which particles with v and ρ are lost. That is,  π/2 dφ| cos φ|drdvdρdt − = ns avf (r,v,ρ,t) −π/2

= 2ns avf (r,v,ρ,t)drdvdρdt,

(5.7.13)

where φ is the angle of incidence in the loss collisions. Remember that the collisions under discussion here are those of the moving particle with fixed scatterers. We can compute + in a very similar way. Here we need to consider restituting collisions that are so constructed that a particle with velocity v and radius of curvature ρ  before collision with a scatterer produces a particle with velocity v

188

The Dilute Lorentz Gas

and radius of curvature ρ after collision. The probability per unit volume of finding a particle with velocity v and radius of curvature ρ  in small regions dv and dρ  is given by f (r,v,ρ ,t)dv dρ  , and the probability of finding such a particle in a collision cylinder about a scatterer, characterized by angle of incidence in the range φ to φ + dφ, is, as is discussed in the derivation of the Boltzmann equation for a dilute gas, f (r,v,ρ ,t)dv dρ  ns av cos φ dφdt. ˆ k, ˆ Here, the restituting velocity is given by the usual expression v = v − 2(v · k) ˆ and k is the unit vector in the direction from the center of the scatterer to the point of incidence of the moving particle. We note that v has the same magnitude as v so that the transformation from v to v is just an orthogonal rotation, and dv = dv. Also, ρ  is given by 2 1 1 = −  ρ ρ a cos φ so that 



dρ =

a cos φ a cos φ − 2ρ

2 dρ.

(5.7.14)

It is important to notice that the radius of curvature after collision must satisfy the inequality ρ≤

a cos φ . 2

(5.7.15)

We can now find an expression for + as

2  π/2 a cos φ dφ cos φ (a cos φ − 2ρ) + = ns av a cos φ − 2ρ −π/2

ρ  × f r,v , ,t drdvdρdt. (5.7.16) 1 − (2ρ/a cos φ) This expression for + can be transformed to a slightly simpler-looking expression  π/2  ∞   dφ dρ  δ ρ − y(φ,ρ  ) × f (r,v,ρ ,t) drdvdρdt, + = ns av −π/2

0

(5.7.17) where y(φ,ρ  ) =

a cos φ . 2[1 + (a cos φ/2ρ  )]

(5.7.18)

5.7 Chaos in the Lorentz Gas

189

Now Eqs. (5.7.12), (5.7.13), and (5.7.17) can be combined to yield an extended Lorentz–Boltzmann equation for the probability distribution f (r,v,ρ,t) ∂f ∂f + v · ∇r f + v = ns av ∂t ∂ρ





π/2

dφ −π/2 

∞

  dρ  cos φ δ ρ − y(φ,ρ  )

0 

× f (r,v ,ρ ,t) − 2ns avf (r,v,ρ,t).

(5.7.19)

This is the equation that can now be used to calculate the positive Lyapunov exponent for the dilute Lorentz gas [639]. It is useful to note that if this equation is integrated over all values of the radius of curvature ρ, one then obtains the usual Lorentz–Boltzmann equation ∂f (r,v,t) + v · ∇r f (r,v,t) = ns av ∂t



π/2

−π/2

dφ cos φ[f (r,v,t) − f (r,v,t)], (5.7.20)

 where f (r,v,t) = dρf (r,v,ρ,t) is the ordinary probability distribution function for a moving particle in the Lorentz gas. In order to compute λ+ , we will look for equilibrium solutions of (5.7.19) where f does not depend on position r or time t. This simplifies the solution a bit, but we can make an even more important simplification by realizing that the typical radius of curvature of the trajectories just before a collision with a scatterer will be on the order of the mean free path between collisions, which is of order (ns a)−1 . Then y(φ,ρ  ) is well approximated for low densities by (a cos φ)/2, and we can neglect the ρ  term in the argument of the Dirac delta function in Eq. (5.7.19). We obtain

 π/2 a cos φ ∂f (v,ρ) = ns av f (v ) − 2ns avf (v,ρ), dφ cos φ δ ρ − v ∂ρ 2 −π/2 (5.7.21) where



∞

f (v) =

dρf (v,ρ). 0

In case the gas is in equilibrium, the velocity distribution function F (v) is just a constant on a circle of constant speed, so the possible directions, θ, of motion, are uniformly distributed over the interval 0 ≤ θ ≤ 2π. In equilibrium, the function f0 (v,ρ) may be written as f0 (v,ρ) = (2πv0 )−1 δ(v − v0 )f0 (ρ). Furthermore, we impose the boundary condition f0 (ρ = 0) = f0 (ρ = ∞) = 0,

190

The Dilute Lorentz Gas

since any zero value for the radius of curvature for two diverging trajectories will immediately grow with time and will never be reduced again to zero by collisions. The condition at infinite radius of curvature is required so that f will be properly integrable over all values of ρ. This equation is now easily solved by considering the regions 0 ≤ ρ ≤ a/2 and a/2 ≤ ρ ≤ ∞ separately and by requiring that f be continuous to relative order ns at ρ = a/2. Then one finds ! 2ns a exp(−2ns aρ) for ρ ≥ a/2 $ f0 (ρ) = (5.7.22) 2ns a[1 − 1 − 4ρ 2 /a 2 ] for 0 ≤ ρ ≤ a/2. This expression can now be inserted in Eq. (5.7.10) to obtain  ∞ 1 dρ f0 (ρ) λ+ = v0 ρ 0   %  a/2  ∞ 1 4ρ 2 1 1− 1− 2 dρ dρ exp(−2naρ) + v0 = v0 ρ a ρ 0 a/2 v   0 (5.7.23) − ln(2ns a 2 ) − C + 1 . = 2ns a 2 a Here, C = 0.577 . . . is Euler’s constant, which occurs in the density expansion of the logarithmic integral function appearing on the right-hand side of Eq. (5.7.23). The functional form of the Lyapunov exponent, λ+ ∼ −ns ln ns , was anticipated by N. Krylov [404], who gave simple mean free path arguments for the exponential separation of trajectories, which led to this form.21 Similar calculations can be carried out for obtaining the Lyapunov exponents and other chaotic properties of Lorentz gases in higher dimensions [649]. In Fig. 5.7.3, we show a comparison of this result with that obtained by C. Dellago and H. Posch from computer simulations of dilute, random Lorentz gases in two dimensions over a range of densities [141]. For three-dimensional Lorentz gases, there are two positive and two negative Lyapunov exponents. The comparison of the theoretical results with those obtained from computer simulations are shown in Fig. 5.7.4. For both the two- and threedimensional cases, the agreement is excellent [142, 141]. This is now but one of several applications of kinetic theory to the calculation of Lyapunov exponents for Lorentz gases in two and three dimensions as well as for regular gases of hard spheres or disks [643, 641, 650]. The range of results that can be obtained for Lyapunov exponents and other quantities related to chaotic behavior of hard-sphere systems in two and three dimensions is quite remarkable, as is the agreement of these results with numerical simulations of these systems. In many cases, clever methods must be devised in order to obtain these quantities from kinetic theory [662, 663, 132, 133].

5.7 Chaos in the Lorentz Gas

191

Figure 5.7.3 Lyapunov exponents for a dilute, two-dimensional, random Lorentz gas. The solid line are the theoretical predictions [639, 640, 649], while the data points are obtained by molecular dynamics. The dotted line represents the positive Lyapunov exponent for a Lorentz gas where the scatterers are placed on the sites of a triangular lattice. The √ Lyapunov exponents are given in units of v/a. The scaling density is ρ0 = [2a 2 3]−1 . This figure is taken from the paper of C. Dellago and H. A Posch [141]

5.7.4 The Thermostatted Lorentz Gas The application of computer-simulated molecular dynamics to the study of nonlinear phenomena has caused researchers to consider some non-Hamiltonian systems with very interesting properties [312, 92, 313, 212]. Computer studies of nonequilibrium flows generated by external fields or forces, for example, show that an unconstrained system typically will quickly undergo some kind of heating that makes it difficult to obtain useful results for the study of its time development. This problem can be avoided in the simulations by imposing a constraint on the system that the total energy or the kinetic energy of the system remain constant in time. Such systems are said to be thermostatted. The imposition of the constraints is achieved by changing the equations of motion, along the lines first developed by C. F. Gauss for constrained mechanical systems some two centuries ago [312, 313, 212]. While these constrained systems are no longer Hamiltonian, one can obtain equations of motion for them and study their properties in some detail by using

192

The Dilute Lorentz Gas

Figure 5.7.4 Theoretical and computer results for the two positive Lyapunov exponents for an equilibrium, dilute, three-dimensional, random Lorentz gas as a √ function of the reduced density are shown. Here the scaling density is ρ0 = 2[8a 3 ]−1 . This figure is taken from the paper of C. Dellago and H. A Posch [141]

computer-simulated molecular dynamics, and, wherever possible, using standard analytical methods, as we will do here. The thermostatted system does reach a nonequilibrium stationary state, since the extra force on the particle acts as a kind of Maxwell demon that follows the moving particles and provides the force needed to keep the energy constant. The thermostatting constraint removes some of the problems that occur for systems subjected to external forces. For example, as we show next, electrical conduction in these systems satisfies Ohm’s law in the usual sense of obtaining a stationary current proportional to the driving field. Of special interest in the context of chaotic motion is the connection between the transport properties of these systems and the Lyapunov exponents that characterize their chaotic dynamics [212, 312, 313]. Here we discuss one example of a thermostatted system, the thermostatted Lorentz gas, with charged moving particles and fixed, hard-sphere scatterers,22 constrained to have a constant kinetic energy. We will show that the electrical conductivity is proportional to the sum of the Lyapunov exponents and then show how kinetic theory may be used to calculate the Lyapunov exponents for a hard-sphere Lorentz gas [642, 417].

5.7 Chaos in the Lorentz Gas

193

We begin the analysis by describing the motion of a moving particle in a Lorentz gas subjected to an external field, say an electrical field, but with the constraint that the kinetic energy of the moving particle is held at a constant value. The motion of a moving particle is determined by instantaneous, specular, and energy-conserving collisions with the hard-sphere scatterers and by the constrained motion between collisions. The relation between the incoming and outgoing momenta, v and v , at each of the instantaneous collisions with scatterers, is not affected by the field and is, as usual, given by v = v − 2(v · σˆ )σˆ . The motion between collisions is determined by the external field, including the electric charge, E, and by the constraint that the kinetic energy be held constant. This motion can be described in terms of a Lagrange multiplier, α, in the equation m

dv = E − αmv, dt

(5.7.24)

where we have set the charge on the moving particle equal to unity. The Lagrange multiplier is obtained from the condition that the velocity of the moving particle satisfy the relation v˙ · v = 0. This leads to an expression for the multiplier, α, as α=

E·v . mv 2

(5.7.25)

Note that these equations are time reversible and that α is a function of the momentum of the particle and is not a constant of the motion. Two important and related consequences of this constrained equation of motion between collisions are that the distribution function for the moving particles does not satisfy the standard Liouville’s equation of statistical mechanics, or in this case of hard-sphere scatterers, what we will describe in the following chapter, the pseudo-Liouville equation, and that volumes in the 2d-dimensional phase space are not preserved in time. To see this, we recall that an essential ingredient of the conservation of volumes in phase space is the vanishing of the divergence of the phase-space velocity, i.e. ∇ · V = 0, where  d   ∂ r˙i ∂ v˙i ∇ ·V= ∂ri ∂vi i=1

(5.7.26)

where, in this case, V = (˙r, v˙ ). However, for the thermostatted Lorentz gas, one obtains immediately using Eq. (5.7.25), ∇ · V = −α.

(5.7.27)

194

The Dilute Lorentz Gas

The equation for the conservation of members of an ensemble is then given by23 ∂ρ(r,v,t) + ∇·(Vρ(r,v,t)) = 0, or, ∂t ∂ρ(r,v,t) + V · ∇ρ(r,v,t) = −ρ(r,v,t)∇ · V = αρ(r,v,t). ∂t

(5.7.28)

The left-hand side of the second line of Eq. (5.7.28) is the total time derivative of the ensemble distribution. For a non-thermostatted Lorentz gas with no external forces, this would be equal to zero. We mention in passing that the instantaneous collisions of the moving particles with the hard-sphere scatterers conserve phasespace volumes, since the speed of the particle does not change, and the velocity is only rotated by an amount that depends on the angle of incidence when the particle collides with a scatterer. On the average, we expect that the component of the momentum of the particle in the direction of the applied field will be positive, so that the Lagrange multiplier, α, will on the average be positive, and the phasespace density will increase with time while the volume will decrease with time. We expect that the phase-space volume of an ensemble of particles will approach zero in the long time limit, but this does not mean that something mathematically exceptional is taking place but rather that the volume becomes concentrated with time on a fractal attractor with a dimension lower than the dimension of the phase space. The average contraction of volumes in phase space can be directly connected to the Lyapunov exponents in the following way. We consider an infinitesimal volume in phase space with edges aligned along the expanding, the contracting, and neutral directions in phase space. The edges along the expanding directions will grow exponentially with positive Lyapunov exponents, and those along contracting directions will contract exponentially with negative Lyapunov exponents. Thus, a small volume, δV (r,v,t), around the phase point r,v will change in time as δV (r,v,t) ≈ δV (r(t0 ),v(t0 ),t0 ) ×

d−1 #

i(+) (r,v,t − t0 )i(−) (r,v.t − t0 ), (5.7.29)

i=1

where the quantities i(+) (r,v,t − t0 ), are the factors by which the edges of the small volume stretch in each of the expanding directions, and i(−) (r,v,t − t0 ) are the factors by which the edges along the contracting directions contract. For very long time intervals, t − t0, on the order of several mean free times, the stretching factors approach an exponential form, i(±) (t − t0 ) ≈ exp[λi(±) (t − t0 )], where the coefficients, λi(±) , are Lyapunov exponents. These exponents can be expressed as time averages similar to those for the equilibrium system given by Eq. (5.7.8). If we assume that the system is ergodic,24 so that time averages approach ensemble

5.7 Chaos in the Lorentz Gas

195

averages with an appropriate distribution function for forming the ensemble average, the Lyapunov exponents are independent of the starting point, r(t0 ),p(t0 ). Thus, for the long time limit, in a nonequilibrium steady state, we have  d−1    (+)  (−) λi (t0 ) + λi (t0 ) (t − t0 ) , δV (r,v,t) ≈ δV (r(t0 ),v(t0 ),t0 ) × exp i=1

(5.7.30) where λi(±) (t0 ) ≡ λi(±) (r(t0 ),v(t0 )). For an equilibrium system, the corresponding Lyapunov exponents λi(+) and λi(−) have equal magnitudes but opposite signs so that the exponent in Eq. (5.7.30) is zero and volumes are conserved. For a thermostatted system of the kind we have been discussing, we expect that the sum of the Lyapunov exponents will be negative and phase-space volumes approach zero for long times as the trajectories become concentrated on a fractal attractor. The sum of the Lyapunov exponents can be related to α by using Eq. (5.7.28) written as d ln δV (r,p,t)/dt = −α(r,p). This leads immediately to  ln

δV (r(t),p(t),t) = α(t − t0 ), δV (r(t0 ),p(t0 ),t0 )

(5.7.31)

where the brackets indicate the time average over the trajectory from t0 to t. For large t − t0 , the central limit theorem tells us that α is Gaussian distributed with width approaching zero as t −1/2 . Thus on comparing the exponentiated form of Eq. (5.7.31) to Eq. (5.7.30) we find

α = −

d−1   (+)  λi + λi(−) ,

(5.7.32)

i=1

where the Lyapunov exponents are also average values taken with respect to the steady-state distribution function. Conductivity and Lyapunov Exponents By examining the physical meaning of the average value, α, appearing on the left-hand side of Eq. (5.7.32), we can establish for this model a relation between the electrical conductivity of charged particles in the thermostatted Lorentz gas, and the Lyapunov exponents characterizing the chaotic motion of the particles. To do so, we return to the expression for α given by Eq. (5.7.25) and form its average value. Thus, we find

196

The Dilute Lorentz Gas

α =

1 E · v, 2ξ

(5.7.33)

where we have used the fact that the kinetic energy of the moving particles is held at a constant value, denoted by ξ . The quantity v ≡ J is the average electric current per particle in this system, which, given the existence of a nonequilibrium stationary state, is a constant in time. Now let us suppose that this system obeys Ohm’s law for small values of the electric field – in this case, given as J = σL E, where the scalar, σL , is the electrical conductivity. If all of the quantities appearing in Eq. (5.7.32) are finite, then an explicit expression can be obtained relating the sum of the Lyapunov exponents to the electrical conductivity of this system, namely σL = −

d−1  2ξ   (+) λi (E) + λi(−) (E) , 2 E i=1

(5.7.34)

in the limit as the electric field tends to zero. The diffusion coefficient DL is related to the electrical conductivity through an Einstein formula by the relation [212, 92]  vm 2 . (5.7.35) DL = − lim [v 2 (λ(+) (E) + λ(−) (E))] E→0 E It is known from the work of H. Posch and C. Dellago [142, 141], using computersimulated molecular dynamics, that indeed all of the quantities appearing in Eq. (5.7.34) are finite and well behaved in the limit as the field approaches zero [142, 141]. Our next task is to show how the extended Lorentz–Boltzmann equation can be used to calculate the Lyapunov exponents on the right-hand side of Eq. (5.7.34) and to obtain the electrical conductivity. We consider, for simplicity, only the case of two dimensions, but the method can be extended to three dimensions [417]. We mention that even for thermostatted gases, a modified conjugate pairing rule applies giving the same sum of the positive and the negative Lyapunov exponents of each conjugate pair [147, 492, 515].25 5.7.5 Lyapunov Exponents for the Thermostatted Lorentz Gas With the addition of some terms related to the thermostatted field, the extended Lorentz-Boltzmann equation, Eq. (5.7.19), can be used to calculate both the positive and the negative Lyapunov exponents for the thermostatted Lorentz gas. A simple addition to this equation will allow us to calculate the positive exponent while a somewhat more subtle calculation is required to obtain the negative one. We first consider the easier of the two cases, namely the positive exponent. The collision term on the right-hand side of Eq. (5.7.19) is unaffected by the external field. However, some care must be taken when one adds a term to the

5.7 Chaos in the Lorentz Gas

197

left-hand side of this equation, which accounts for the presence of the thermostatted field. Due to the non-Hamiltonian nature of the thermostatted field, one must write the left-hand side of this equation in a more general form that correctly describes the flow of points in the (r,v,ρ) phase space of this system. That is ∂f (r,v,ρ,t) ∂(ρf ˙ (r,v,ρ,t)) + ∇r · (vf (r,v,ρ,t)) + ∇v (˙vf (r,v,ρ,t)) + ∂t ∂ρ  π/2  ∞   = ns av dφ dρ  cos φ δ ρ − y(φ,ρ  ) f (r,v,ρ ,t) − 2ns avf (r,v,ρ,t). −π/2

0

(5.7.36) To obtain an explicit expression for the left-hand side of this equation, we use the equations of motion for a particle in the thermostatted field, r˙ = v, m˙v = E − αmv, as well as the equation for the time derivative of the radius of curvature in the presence of the thermostatted field. This equation requires some geometrical analysis, which we do not provide here. Instead, we merely quote the equation, which – to second order in the applied field – is [642] ρ˙ = v + (ρ) cos θ +

ρ 22 2 sin θ, v

(5.7.37)

where  = E/vm. We assume that the system is in a spatially homogeneous stationary state so that the distribution function appearing in Eq. (5.7.36) is a function only of the velocity, v, and the radius of curvature, ρ. We will need this function, f (v,ρ), in order to calculate the positive Lyapunov exponent, λ(+) (E), given by   ∞ 1 (+) dρ f (v,ρ). (5.7.38) λ (E) = v dv ρ 0 The equation for f (v,ρ,t) can be expressed in terms of the angle θ and ρ by 



∂ ρE ρ 2E2 2 ∂f E ∂ v+ cos θ + 3 sin θ f = , − (f sin θ) + v ∂θ ∂ρ v v ∂t coll (5.7.39) where the right-hand side is the collision term given by the right-hand side of Eq. (5.7.36). One may solve this equation by assuming that the stationary distribution function can be expanded in powers of the electric field strength, E. That is, we write f (v,ρ) = f0 (v,ρ) + Ef1 (v,ρ) + E 2 f2 (v,ρ) + · · · ,

(5.7.40)

198

The Dilute Lorentz Gas

where f0 (v,ρ) is the distribution function already obtained by solving the field free version of the extended Lorentz–Boltzmann equation, Eq. (5.7.19), while f1 (v,ρ) and f2 (v,ρ), respectively, satisfy ∂f

1 ∂ ∂ ∂ ρ 1 − cos θf0 = (f0 sin θ) + v f1 + v ∂θ ∂ρ ∂ρ v ∂t coll

2 ρ ρ 1 ∂ ∂ ∂ ∂f2 2 f1 cos θ + f0 3 sin θ = − . (f1 sin θ) + v f2 + v ∂θ ∂ρ ∂ρ v v ∂t coll (5.7.41) The solutions of these equations can be obtained by straightforward methods that we do not present here, and the resulting expansion for the positive Lyapunov exponent in powers of the electric field is found to be λ(+) (E) = λ0 −

11 tl E 2 + O(E 4 ). 48 (mv)2

(5.7.42)

To complete this analysis, we must also calculate the negative Lyapunov exponent, λ(−) (E). This requires an analysis of nearby trajectories that asymptotically and exponentially approach each other with time. That is, the negative Lyapunov exponent can only be determined if one can find the manifold of trajectories that approach a given trajectory with time. This manifold is called the stable manifold of the given trajectory, and it excludes all of the other trajectories that will exponentially separate from the given trajectory with time. Constructing the stable manifold is not an easy task since almost all of the trajectories infinitesimally close to a given trajectory will eventually diverge from it. To overcome this difficulty, we consider the time-reversed motion of the moving particle, which allows us to consider diverging trajectories again. Thus, we need to determine the steady-state distribution function, f− (v,ρ), for particles with velocity v and radius of curvature ρ for trajectories on the time reversal of the stable manifold. This distribution function must obey an anti-extended Lorentz–Boltzmann equation (AELB) for the following reason: If the moving particle and the scatterer with which it collides are uncorrelated before collision in the forward motion, then in the time-reversed motion, the moving particle and this scatterer will be uncorrelated after their collision. Therefore, to obtain the AELB, one must use the Stosszahlansatz for the exiting collision cylinders rather than for those before the collision. Thus, the gain term in the collision operator must be proportional to the distribution function, f− (v,ρ  ), for the post-collisional velocity v rather than the pre-collisional velocity. It must also include an integration over all radii of curvature that lead to the radius of curvature ρ after the collision. If we use the fact that the radius of curvature after collision is well approximated by ρ = (a cos φ)/2, where the scattering angle φ is defined in the usual way, we may express the gain term in the collision integral as

5.7 Chaos in the Lorentz Gas



∂f− ∂t



 = ns av

gain



π/2

∞

dφ cos φ −π/2

199

dρ  δ(ρ − a cos φ/2)f− (v,ρ  ). (5.7.43)

0

The loss term is somewhat more complicated to construct. Again, the distribution function for the post-collisional velocities, v, must be used to describe the rate at which particles with velocity v are lost, but due to the fact that the equation also describes the loss of particles with radius of curvature ρ, the loss term has to be constructed in a different way than for the forward extended Lorentz–Boltzmann equation. The rate at which particles with velocity v are lost, independent of their radius of curvature, for scattering angle φ, is given by  ∞ dρ  f− (v,ρ  ). ns av cos φ 0

Now the fraction of those with velocity v that disappear having radius of curvature, ρ, is ∞ 0

f− (v,ρ) . dρ  f− (v,ρ  )

The loss term in the equation for f− (v,ρ) is then  ∞

 π/2 dρ  f− (v,ρ  ) ∂f− 0 × f− (v,ρ). = − ns av dφ cos φ  ∞ ∂t loss −π/2   dρ f− (v,ρ )

(5.7.44)

0

Assembling all of the parts, we obtain the full AELB as 

 ∂ ρE ρ 2E2 2 ∂f− E ∂ v+ cos θ + 3 sin θ f = , − (f sin θ) + v ∂θ ∂ρ v v ∂t coll (5.7.45) where

 π/2 ∂f− = ns av dφ cos φ ∂t −π/2   ∞   dρ δ(ρ − a cos φ/2)f− (v,ρ )coll − H (v)f− (v,ρ) , (5.7.46) × 0

where



∞

H (v) = 0 ∞ 0

dρ  f− (v,ρ  ) . dρ  f− (v,ρ  )

(5.7.47)

200

The Dilute Lorentz Gas

Note that the collision operator has a desired property – namely that upon integration over the radius of curvature, ρ, one obtains the anti–Lorentz–Boltzmann collision operator  π/2 ns av dφ cos φ[f− (v) − f− (v )] −π/2

with a sign opposite to that of the Lorentz–Boltzmann equation. This is as it should be since we are considering the time-reversed motion. The negative Lyapunov exponent is obtained from the integral   ∞ v (−) dρ f− (v,ρ), λ (E) = − dv ρ 0 with f− normalized to unity. One can solve this equation by expanding the solution in powers of the electric field E, and one finds λ(−) (E) = −λ0 −

7 tl E 2 + O(E 4 ). 48 (mv)2

(5.7.48)

We can assemble these results using Eqs. (5.7.34), (5.7.35), (5.7.42), and (5.7.48) to obtain an expression for the diffusion coefficient, D, DL(2) =

3v2 tl, 8

(5.7.49)

which agrees with our previous result in Eq. (5.3.14). The quantities λ(±) (E) can also be computed using computer-simulated molecular dynamics for the stationarystate thermostatted Lorentz gas. These studies have been carried out by H. Posch and C. Dellago [142, 141]. We present their results in Figure 5.7.5, where, for two different values of the reduced density n∗ = ns a 2 = 0.001,0.002, the Lyapunov exponents are plotted as a function of the square of a dimensionless electric field. The solid lines are the results given by Eqs. (5.7.42) and (5.7.48) [417], with results of the computer indicated by squares and circles. In the limit of small electric fields, the agreement between the theoretical and computer results is excellent. In Figure 5.7.6, we present results at two reduced densities for the diffusion coefficients, obtained from Eq. (5.7.35), using determinations of the positive and negative Lyapunov exponents by computer simulations. In order to obtain agreement between the theoretical and computer values of D, it is necessary to add to the low-density result given by Eq. (5.7.49) known higher order density corrections terms of the form n∗ ln n∗ + n∗ [69, 70]. The upper dotted line represents the low-density, zero field value at these two densities, while the dashed (n∗ = 0.001) and solid (n∗ = 0.002) lines represent the zero field values for the diffusion coefficient with the

5.7 Chaos in the Lorentz Gas

201

Figure 5.7.5 The positive and negative Lyapunov exponents, λ+ (ns a 2 ),λ− (ns a 2 ), respectively, at two different reduced densities, plotted as a function of the square of the strength of the electric field. The solid lines represent the predicted, low density, values given by Eqs. (5.7.42,48) [642, 417], and the points are the results of H. Posch and C. Dellago, using molecular dynamics for two densities [142, 141]. Here  = E/vm with E the applied field, , the mean free path, and v,m are the speed and mass of the moving particle. Figure taken from the paper of H. van Beijeren, et al. [642]

logarithmic terms taken into account. The computer results converge well to these corrected values for vanishing field and confirm the presence of the logarithmic terms in the density expansion of the diffusion coefficient and in corrections to the low density values, Eqs. (5.7.42,48), for λ± (E) [417, 663]. These logarithmic terms and related quantities have been the subject of intensive studies over a number of years, and we will discuss them in Chapter 13, to which we refer for further explanations and discussions. 5.7.6 The Quantum Lorentz Gas The Fermi temperature for electrons in metals is typically on the order of 105 K. This implies that electrons in metals should be thought of as quantum particles with a Fermi wavelength, λF = 2π/kF , where kF is the Fermi wave number, which is very large compared to the average spacing of the scatterers, (ns )−1/d . Under these circumstances, collective effects become important. For example, if the scatterers were to form a regular lattice with a translational symmetry, the particles would eventually,26 form Bloch waves, due to constructive interference

202

The Dilute Lorentz Gas

Figure 5.7.6 Theoretical and computed values for diffusion coefficients as a function of the applied electric field for a thermostatted, dilute random Lorentz gas at two densities. Figure taken from the paper of van Beijeren et al. [642]

of wave functions from various scattering trajectories, and then travel ballistically, rather than diffusively, through the lattice.27 On the other hand, if the scatterers are fixed and placed randomly in space, the moving particles may, under certain circumstances, become localized due to destructive interference of wave functions for various scattering trajectories [422, 29, 547]. A quantum Lorentz gas in three dimensions, with fixed scatterers placed at random, has a mobility edge as a function of the degree of disorder that separates a diffusive region from a localized region [159]. In two-dimensional Lorentz gases with random placement of scatterers, the moving particles do not diffuse at all, even though the classical version of the system would be diffusive. For such low-dimensional systems, collective effects are important at all densities, and the quantum Lorentz–Boltzmann equation is not useful for a description of its nonequilibrium properties. In many ways, there are deep connections between the quantum Lorentz gas and a classical dense gas where all the particles move [381, 380, 689, 29, 30]. The theory of localization requires a description of dynamical processes that goes beyond the binary collision approximation and is connected in some ways to long-time-tail phenomena and mode-coupling theory discussed in Chapters 13 and 14.

Notes

203

Notes 1 One can also consider the situation where the fixed particles are placed at the vertices of a periodic lattice, but this case requires a separate analysis not given here. 2 We call special attention to two important and instructive papers treating the Lorentz gas by E. H. Hauge and by A. V. Bobylev, F. A. Maaø, A. Hansen, and E. H. Hauge [299, 48]. These papers discuss and answer the question, “What can one learn from the Lorentz model?.” 3 As usual, we imagine that we are considering the behavior of the Lorentz gas in the Grad limit, ns σ d → 0, with a fixed mean free path, l. 4 We will use the notation kˆ for the unit vector in the direction of the apse line when we do not explicitly specify the interaction potential for particle-scatterer interactions, and we will use σˆ for the apse line unit vector for the special case of hard-sphere interactions. As indicated, Eq. (5.2.4) is the equation in an arbitrary d-dimensional space, with an appropriate form of the unit vector σˆ . 5 Since the particle-scatterer collisions are elastic, the conservation of mass implies the conservation of energy, and there can be only one hydrodynamic equation for the moving particles. 6 The delta function appearing in the definition of the distribution function is to be thought of as a smooth and sharp Gaussian around v = v0 : It’s contribution to the H –function given by Eq. (5.2.5) is unimportant. 7 Equivalently, this follows from the observation, the time derivative of the distribution function in the normal solution method is proportional to the time derivative of the density. However, this time derivative must be at least second order in the gradients. This follows from Eq. (5.7.28) and the fact that the particle current is a vector and must therefore be proportional to the gradient of the density. 8 This is analogous to using a slip length to describe the flow of a fluid past a surface discussed in Chapter 3 This length measures the distance between the physical boundary of a solid and a fictitious surface at which hydrodynamic boundary conditions are satisfied. 9 Notice that, in this case, limk→0 nk vanishes since nk is proportional to δ(k = 0) and, thus, differs drastically from nk=0 , which corresponds to the total number of light particles in the system. This phenomenon is not restricted to this case and, in fact, is very common. 10 The gyromagnetic radius must not become comparable to the scatterer radius, a. 11 Since the Lorentz–Boltzmann operator, λL , is isotropic its action on v must be proportional to v. 12 We follow the presentation of Bobylev et al. [48]. 13 Again, the instantaneous nature of collisions for this model allows us to avoid considerations of the effect of the magnetic field on the collision dynamics. 14 An excellent collection of review articles and papers pertaining to the chaotic properties of hard-sphere systems and of the Lorentz gas, in particular, can be found in the book edited by D. Szasz [617]. 15 Conservation of phase-space volume and time reversibility require that the Lyapunov exponents be paired, such that λ− = −λ+ , as discussed later. See [243, 152], for further discussions of this point. For geometric reasons the number of exponents must be ≤ 2d. 16 Chaos is not necessary for transport and there are simple non-chaotic systems that have good transport properties. For example, systems consisting of polygon particles have vanishing Lyapunov exponents, but may have hydrodynamic properties very close to those of hard disks. 17 See for example, R. Abraham and J. E. Marsden, Foundations of Mechanics, 2nd Ed., American Mathematical Society, Providence, 2008. 18 If one relaxes the condition that the ensemble energy is a constant, one obtains a second zero Lyapunov exponent for a displacement in which the position of the moving particle is not changed, and its velocity is multiplied by a factor 1 +  – that is, for a phase-space displacement in a direction perpendicular to the constant energy surface. 19 If we were to take a finite difference in the velocity directions, then after a number of collisions, the real and the ghost particle will start colliding with different scatterers. Then the separation of trajectories will be governed by diffusion. In order to see the exponential separation over a long

204

20

21 22 23

24 25

26 27

The Dilute Lorentz Gas

time interval, we much imagine that the two initial velocity directions are infinitesimally close to each other. We mention that one can also evaluate this Lyapunov exponent and related quantities using mean free path methods, leading to identical results but not requiring the use of an extended Lorentz–Boltzmann equation. Both methods are useful, and often one can choose between them. This point is well illustrated in the paper by A. Latz, H. van Beijeren, and J. R. Dorfman [649]. These arguments are presented in the book written by Krylov and edited by Ya. G. Sinai [404] and the book written by S. K. Ma [448]. The interaction potential can be generalized to include other short-range forces between the moving particles and the scatterers, but this makes the analysis more complicated and will not be considered here. Strictly speaking, for the case of hard-sphere scatterers, another term must be included that corresponds to the instantaneous change of momentum of a moving particle colliding with a scatterer. Since for our argument here, this term is unimportant, we do not include it, but we consider the precise form, called the pseudo-Liouville equation, in the next chapter. We assume that periodic boundary conditions are placed on the the system, so as to avoid possible problems with ergodicity in an infinite system. The first application of Gaussian thermostats to obtain the transport properties of gases was to a thermostated gas under shear. This method plus the conjugate pairing rule allows for an expression for the shear viscosity in terms of the maximal positive and minimal negative exponents. We refer to the papers listed here for details and further references [312, 313, 212, 210, 211, 105, 516]. There are simple model quantum systems with motion of particles in a lattice with translational symmetry, where – given some initial condition – one can see a crossover from diffusive to ballistic motion after some finite time [683]. In the classical version of this situation, the particles will travel diffusively in any lattice with hard-sphere scatterers and at a density where the particles can move through the lattice, provided that the lattice have a finite horizon. A lattice with a finite horizon is one where the particle must have encountered a scatterer if it has travelled a distance greater than some fixed value. Motion of classical particles in a Lorentz gas with infinite horizon is super-diffusive rather than ballistic.

is a binary mixture of [112]. The tensor elements are

6 Basic Tools of Nonequilibrium Statistical Mechanics

We have, up to now, focused on the application of kinetic theory to dilute, monatomic gases. The principal equation for this, the Boltzmann transport equation, has proved to be enormously successful for treating nonequilibrium processes in dilute gases including mixtures. It is natural to try to generalize this work to theories for nonequilibrium processes in gases at higher densities and, if possible, in more general kinds of fluids such as liquids, plasmas, colloidal suspensions, quantum fluids, and granular matter, among others.1 These generalizations all require an approach to the nonequilibrium theory of many particle systems that is more general than that used by Boltzmann for the derivation of the Boltzmann equation. As we have noted, Boltzmann’s equation and its derivation apply only to dilute gas systems. For hard spheres, it is possible to generalize Boltzmann’s approach to higher densities in an intuitive and approximate way,2 but a systematic theory needs to be based on the fundamental equations of statistical mechanics. For nonequilibrium systems, the most fundamental equation of statistical mechanics is the Liouville equation [631, 549], which describes the time development of a phasespace distribution function for a system of N interacting particles. We will derive this equation, first for systems of particles that interact with smooth, differentiable potential energy functions. Then we turn to the modifications of the Liouville equation that are necessary for describing systems of hard-sphere particles. Since all collisions between such particles are instantaneous and binary, the required modifications of the Liouville equation involve the introduction of binary collision operators in order to correctly account for the singular forces that would arise in the Liouville equation in the limit where the inter-particle interactions approach those of hard spheres. Once the binary collision operators have been defined, we use them to describe the version of the Liouville equation appropriate for hardsphere systems, an equation called the pseudo-Liouville equation [190, 158]. We will add some remarks on how to generalize this equation to the more general case of inter-particle potentials exhibiting one or more jump discontinuities. 205

206

Basic Tools of Nonequilibrium Statistical Mechanics

Having developed the Liouville or pseudo-Liouville equations for systems of N interacting particles, we will turn our attention to an important implication of these equations, namely an exact set of equations for n-particle distribution functions obtained by integrating the Liouville equation. These equations are known as the BBGKY hierarchy equations [55].3 The hierarchical structure of these equations resides in the fact that the equation for an n-particle distribution function, where n = 1,2, . . ., depends on the n + 1–particle distribution function,4 and only the equation for the N-particle distribution function, which is identical to the (pseudo-) Liouville equation, is self-contained. In Chapter 7, we will use the pseudo-Liouville equation as the basis for the description of the (revised) Enskog equation [89, 644, 645], the first important attempt to extend the Boltzmann equation to higher densities. We must mention at the outset the considerable importance of the binary collision operators and the binary collision expansion for hard-sphere particles. We take advantage of the fact that hard-sphere collisions are instantaneous, and the mechanics of binary collisions, being elastic and specular, is particularly simple. The use of a hard-sphere model allows us a method to describe exactly the dynamics of a collection of them if one ignores wall collisions or if one considers periodic systems. Issues related to durations of collisions and the need for computational methods to describe the collision dynamics of particles interacting with, say, Lennard–Jones potentials, make the hard-sphere model particularly well suited for theoretical and computational studies. 6.1 The Liouville Equation The proper formulation of classical statistical mechanics for N particles requires the construction of a 2dN-dimensional Cartesian space, called phase space or -space [625], with one dimension for each of the position coordinates and one for each of the momentum coordinates for each of the N particles. One point in -space provides the location and momentum of each of the particles and as the particles move and interact, the point follows a trajectory in phase space that is governed by the Hamiltonian equations of motion for the system. An ensemble of mechanically identical systems is described by a time-dependent distribution function ρ(r 1,p 1,r 2,p 2, . . . ,r N ,p N ,t), usually abbreviated as ρ(x1,x2,. . . . ,xN ,t) or, more simply, ρ(,t). Here xi = (r 1,p i ).5 This distribution function satisfies a conservation law for the number of members of the ensemble, which, in -space, reads      d N ∂ p˙ i,α ρ(,t) ∂ρ(,t)   ∂ r˙i,α ρ(,t) + + = 0. (6.1.1) ∂t ∂ri,α ∂pi,α i=1 α=1

6.2 Time-Displacement Operators

207

Here the index i denotes any of the N particles, while the index α denotes any of the d coordinate directions. This is the general form of the conservation law, which can also be written in vector notation, as ∂ρ(,t) + ∇ · (V ρ(,t)) = 0, ∂t

(6.1.2)

where we introduced the divergence in -space of the density current, V ()ρ(,t), with V the 2dN-dimensional phase-space velocity vector   (6.1.3) V () = r˙1, r˙2, . . . , r˙N , p˙1, p˙2, . . . , p˙N . The Hamiltonian motion of systems with differentiable potential energies is that of an incompressible fluid in this high-dimensional phase space. That is to say, the divergence of the phase-space velocity vanishes. To see this, write ∇ · V () =

d  N   ∂ ri,α ˙ i=1 α=1

∂ri,α

∂ pi,α ˙ + ∂pi,α



 d  2 N   ∂ 2 H () ∂ H () = − = 0, ∂ri,α ∂pi,α ∂pi,α ∂ri,α i=1 α=1

(6.1.4)

since, for smooth Hamiltonians, the second derivatives of the Hamiltonian appearing in Eq. (6.1.4) are well defined and, of course, don’t depend upon the order in which the derivatives are taken. Under these circumstances, we may write   ∂ρ(,t)   ∂ρ(,t) ∂ρ(,t) ∂ρ(,t) + V · ∇ρ(,t) = + + p˙ i,α r˙1,α ∂t ∂t ∂ri,α ∂pi,α α i ≡

dρ(,t) = 0. dt

(6.1.5)

Equation (6.1.5) is the well-known Liouville equation for classical ensembles, which expresses the fact that the total time derivative of the phase-space distribution function does not change with time. In other words, if one were to sit on and move along with a trajectory of a mechanical system in -space, the phase-space density in the immediate neighborhood would not change with time. 6.2 Time-Displacement Operators Equation (6.1.5) of the previous section can easily be written, at least formally, in terms of a Liouville operator, denoted L, as ∂ρ(,t) + L()ρ(t) = 0. ∂t

(6.2.1)

208

Basic Tools of Nonequilibrium Statistical Mechanics

The Liouville operator can be decomposed into a kinetic part, L0 (), and an interaction potential part LI (), L() = L0 () + LI ().

(6.2.2)

which, for a Hamiltonian of the form HN (x1,x2, . . . ,xN ) =

N   p 2i φ(|r i − r j |), + 2m i< T | +

1 |p >< n|]. n20

(8.3.17)

328

The Boltzmann–Langevin Equation

We mention in passing that in Section 8.4 and in later chapters, we will use representations of the projection operator in terms of hydrodynamic mode eigenfunctions defined in Section 3.8. They are eigenfunctions of the operator Lk having eigenvalues that approach zero in the limit k → 0. We remark that these modes were generalized to higher densities in Chapter 7 for the revised Enskog equation, and they are very important in the general framework of mode-coupling theories [536, 164, 59] discussed in Chapter 13. We return to Eq. (8.3.10) and rewrite it as ∂P φˆ k (p,t) = −P (l) Lk P (r) φˆ k (p,t) − P (l) Lk P⊥ exp[−t Lˆ k ]P⊥ φˆ k (p,t = 0) ∂t  t dτ P (l) Lk P⊥ exp[−(t − τ )Lˆ k ]P⊥ Lk P (r) φˆ k (p,τ ) − 

0 t

−

dτ P (l) Lk P⊥ exp[−(t − τ )Lˆ k ]F˜ k (p,t − τ ).

(8.3.18)

0

Inner products with < nk |, < uk |, and < sk |, respectively, yield the equations ∂n(k,t) + n0 ik · u(k,t) = 0; (8.3.19) ∂t

 t  ∂u(k,t) ik ik 1 p2 + p(k,t) = − · dτ pp − 1 exp(−Lˆ k τ )P⊥ F˜ k (p,t − τ ) ∂t ρ0 m 0 m dm 

 1 β0 k 2 t p2 dτ + pp − 1 · kˆ m 0 m dm 2

p 1 pp − 1 · u(k,t − τ ); × exp(−Lˆ k τ )kˆ · m dm (8.3.20) 

 t p β0 p 2 d + 2 kB ∂s(k,t) dτ = −ik · − exp(−Lˆ k τ )P⊥ F˜ k (p,t − τ ) ∂t m 0 m 2m d 

 2 p kB 2 t β d + 2 p 0 dτ kˆ · + k − m m 2m d 0

p β0 p 2 d + 2 ˆ − τ ). − T (k,t (8.3.21) × exp(−Lˆ k τ )kˆ · m 2m d The angular brackets denote, as usual, an average over a Maxwell–Boltzmann distribution, φ0 (p). The first terms on the right-hand sides of Eqs. (8.3.20) and (8.3.21) can be simplified by observing that for small k, the operator Lˆ k can be

8.3 Linear Hydrodynamic Equations with Fluctuations

329

approximated by −(p), by virtue of the gap in the spectrum of the linearized Boltzmann collision operator and the fact that the exponential operator acts on functions that are orthogonal to the eigenfunctions of  with zero eigenvalues, i.e. the conserved quantities. The width of this gap is on the order of the collision frequency; hence, the operator exp(−τ Lˆ k ) ≈ exp(τ ) leads to functions that decay on the scale of the mean free time. On this time scale, the τ dependence of the functions u(k,t − τ ) and T (k,t − τ ) may be neglected since these functions change on hydrodynamic time scales. Using these approximations, one may obtain the linearized fluctuating hydrodynamic equations for a dilute gas. Expressing the time rate of change of the local entropy per unit mass in terms of the time derivatives of the temperature and density, we recover the same linearized Navier–Stokes equations as obtained in Chapter 3, with the addition of terms representing the fluctuation contributions to the pressure tensor and heat flow vector. In this way, one finds ∂nk (t) (8.3.22) + n0 ik · u(k,t) = 0, ∂t   ˜ kp(k,t) ik · P(k,t) η 2 ∂u(k,t) +i = − k · ku + uk − k · u1 − . (8.3.23) ∂t ρ0 ρ0 d ρ0 dn0 kB ∂T (k,t) ˜ + in0 kB T0 (k · uk (t)) = −λk 2 δTk (t) − ik · q(k,t). 2 ∂t

(8.3.24)

The transport coefficients appearing in these equations, η and λ, are expressed as 1 0p p x y −1 px py  , (8.3.25) η = −β0 n0 m m 2  2



p px p d +2 d +2 −1 px − −  , (8.3.26) λ = −β0 n0 m 2m 2β0 m 2m 2β0 in agreement with the results of Chapter 3. The fluctuating stress and heat currents, ˜ ˜ P(k,t) and q(k,t), respectively, are defined as ˜ P(k,t) = n0





t

dp 0

 ˜ q(k,t) = n0



p2 1 pp − 1 × exp(−τ Lˆ k )P⊥ F˜ k (p,t − τ ), dτ φ0 (p) m dm (8.3.27)

t

dp

dτ φ0 (p) 0



p m



p2 d +2 − 2m 2β0



× exp(−τ Lˆ k )P⊥ F˜ k (p,t − τ ). (8.3.28)

330

The Boltzmann–Langevin Equation

Since the fluctuation average, < F˜ k > vanishes, the same is true for the fluctuation averages of the fluctuating currents. In order to characterize the fluctuation terms in these equations, we need to determine the correlation functions of the fluctuating terms. As an example, we consider the correlation of the fluctuating part of the heat current. This correlation function can be found by using Eqs. (8.2.16), (8.2.23), and (8.3.28) as   q˜i (k1,t1 )q˜j (k2,t2 )H (t1 − t2 )   2    d + 2 p 1,i p22 d + 2 p 2,j p1 2 = n0 dp 1 dp 2 − − φ0 (p 1 )φ0 (p 2 ) 2m 2β0 m 2m 2β0 m  t2  t1 dτ1 dτ2 exp[−Lˆ k1 (p 1 ) (t1 − τ1 )] exp[−Lˆ k2 (p 2 ) (t2 − τ2 )] × 0

0

(2π)d × δ(k1 + k2 ){Lˆ k1 (p1 ) + Lˆ k2 (p 2 )}δ(p 1 − p2 ) n0 φ0 (p1 ) × δ(p 1 − p 2 )δ (τ1 − τ2 ) H (t1 − t2 )  2   p1 d + 2 p1,i − = n0 dp 1 φ0 (p 1 ) 2m 2β0 m  2  p1 d + 2 p 1,j ˆ × exp[−Lk1 (p 1 )(t1 − t2 )] − (2π)d δ(kˆ 1 + kˆ 2 ). (8.3.29) 2m 2β0 m One obtains the second equality by noticing that (Lˆ k1 (p 1 ) + Lˆ k2 (p 2 )) exp[τ2 (Lˆ k1 (p1 ) + Lˆ k2 (p 2 ))] =

d exp[τ2 (Lˆ k1 (p 1 ) + Lˆ k2 (p 2 ))] dτ2

(8.3.30)

and carrying out the τ2 integration. A similar expression is obtained when the Heaviside function H (t2 − t1 ) is used. Combining the two results, we see that the exponential operator appearing in the integrand can be written as exp[−Lˆ k1 (p 1 )|t1 − t2 |]. From this expression, it is clear that the correlation function of the fluctuating heat current is not strictly a delta function of the difference of the two times, but rather it is peaked at t1 = t2 . For small wave numbers, it is a function that decays exponentially on the time scale of the mean free time between collisions. This is in contrast to the correlation function, Fk1 (p1,t1 )Fk2 (p2,t2 ), of the fluctuating term in the linearized Boltzmann–Langevin equation. On a macroscopic time scale, however,

8.4 Light Scattering

331

the decay of the heat current correlation is so rapid that it can be represented by the delta function, leading to   2λ q˜i (k1,t1 )q˜j (k2,t2 ) = (2π)d δ(t1 − t2 )δ(k1 + k2 )δij , (8.3.31) kB β02 which on integration gives the same result as the expression Eq. (8.3.29) for small wave numbers. In situations where the hydrodynamic fields vary on time and/or spatial scales of the same order as the mean free time or mean free path, it would be preferable to use the generalized hydrodynamic equation, Eq. (8.3.21), in combination with the autocorrelation function given by Eq. (8.3.29). In deriving Eq. (8.3.31), we have used the self-adjoint property of the linearized Boltzmann operator, as well as the normal solution expression for the coefficient of thermal conductivity, λ. Consequently, the fluctuation part of the heat flow vector has a simple, white noise correlation function, reflecting that of the fluctuation term in the Boltzmann–Langevin equation. Notice that the strength of the correlation function satisfies a fluctuation-dissipation theorem like the one we obtained for Brownian motion. Thanks to this, the distribution of temperature fluctuations in an equilibrium state is maintained in the form required by equilibrium statistical physics. A very similar calculation can be carried out for the stress tensor. We then obtain the following expression for the correlation of the fluctuating part of stress tensor matrix elements, P˜ij (k,t):

  1 d 2η ˜ ˜ δ(k1 + k2 )[δil δj m + δim δj l ] 1 − δij δ(t1 − t2 ). Pij (k1,t1 )Plm (k2,t2 ) = (2π) β0 d (8.3.32) The results given by Eqs. (8.3.31) and (8.3.32) agree with the correlation functions proposed by Landau and Lifshitz [412] when they are evaluated for a dilute gas. In summary, we have derived the fluctuating hydrodynamic equations of Landau and Lifshitz, starting from a fluctuating Boltzmann equation. Local equilibrium versions of these equations can be used to describe fluctuations about NESS as was shown in [372]. 8.4 Detection of Fluctuations about Equilibrium and Nonequilibrium Stationary States by Light Scattering An important application of the Boltzmann–Langevin equation is the calculation of the spectrum of light scattered by a gas in equilibrium when the wavelength of the light is large compared to the mean free path of particles in the gas. Further on in

332

The Boltzmann–Langevin Equation

this chapter and then in more detail in Chapter 14, we will consider the spectrum of light scattered by a gas in a nonequilibrium stationary state (NESS). The theory for light scattering by a simple fluid, as described in the book by B. Berne and L. Pecora [38], among others, relates the intensity of scattered light as a function of wave number k and frequency ω to the dynamic structure factor S(k,ω), defined by  −1    τ/2  τ/2 1 2 drp (r) dτ1 dτ2 p(r 1 )p(r 2 ) S(k,ω) = dr 1 dr 2 τ V −τ/2 −τ/2 × p(r 1 )p(r 2 )e−i[k·(r 1 −r 2 )+ω(τ1 −τ2 )] × < δ n(r ˜ 1,τ1 )δ n(r ˜ 2,τ2 ) > .

(8.4.1)

Here V is the volume of the whole system, τ is the time over which the measurement takes place, p(r) is a form factor that describes the intensity distribution in the small volume, called the scattering region, in which the fluid is both illuminated and observed by the optics of the scattering system, as illustrated in Fig. 8.4.1. This form factor is taken to be close to unity within the scattering volume and zero outside of it. Its specific form in each case is determined by the details of the light scattering experiment performed. Often the intensity distribution is taken to have a Gaussian form.7 The wave number k = kf − ki is the difference between the wave numbers of the scattered and incident light and determines the momentum transferred to the fluid by the light. Similarly, ω = ωf − ωi is proportional to the energy transferred to the fluid. The density–density correlation function appearing in the integrand of Eq. (8.4.1) is an ensemble average, either equilibrium or nonequilibrium stationary state (NESS), of the product of density fluctuations, δ n(r,t), ˜ at two different locations and times. It is assumed that the mean free path of the particles in the gas is very small compared to the inverse of the wave number, k 1, so that a hydrodynamic description of the gas can be used. Furthermore, this inequality also results from the condition that the Brillouin peaks are separated in frequency by an amount large compared to the widths of the peaks; that is, ck/ s k 2 1. We also assume that the characteristic size of the scattering region, Lsc , is greater than the wavelength of the light, or kLsc 1. This latter condition allows us to use a form factor that is large in the scattering volume and small everywhere else. These two conditions on k,L,Lsc are quite general, but we also need to specify the range of values of the temperature gradient that are allowed in order for us to make a useful calculation of the structure factor. It is remarkable that for a dilute monatomic gas over a large range of temperatures and densities, quite a wide range of values of this gradient can

8.4 Light Scattering

333 T1

Kf Ki

wf

wi

q

Figure 8.4.1 A very schematic representation of a light-scattering experiment where an incoming beam of light with wave vector ki and frequency ωi is scattered by the fluid. One adjusts the detector to observe light scattered into angle θ with wave vector kf and frequency ωf . The scattering volume is the shaded region where the incoming and outgoing beams intersect. For the case of the NESS with a temperature gradient, boundary plates controlling the temperature gradient are maintained at constant temperatures T1 and T2 .

be treated. We introduce a quantity, L∇ , called the gradient length defined by L∇ , which is the characteristic length over which the temperature changes given by the average temperature between the plates, Tav , divided by the gradient in temperature, L ≡ Tav /| T |. The calculation of the structure factor requires the relative change of the temperature over a mean free path to be small. However, one can maintain this condition and still consider situations where (c/L∇ ) = O(1). DT k 2

(8.4.2)

That is, one can treat situations where the time that it takes a sound wave to travel a gradient length is comparable to the lifetime of a temperature fluctuation or of a sound wave.8 8.4.1 Light Scattering by a Gas in Equilibrium To proceed, we must calculate the density–density correlation function, which ˜ 2,τ2 ) >, in the preceding expression for the strucis given by < δ n(r ˜ 1,τ1 )δ n(r ture factor, S(k,ω). The desired correlation function can be obtained from the

334

The Boltzmann–Langevin Equation

correlation function of fluctuations in the single-particle distribution function, ˜ δ f˜(r 1,p 1,t1 )δ f˜(r 2,p 2,t2 ). Here δ f˜(r,p,t) = feq (p)φ(r,p,t) is the fluctuation of the single-particle distribution function about its equilibrium value and is taken to satisfy the Boltzmann–Langevin equation, Eq. (8.2.13). In order to calculate the desired density–density correlation function, we begin by expressing the density fluctuation in terms of frequency rather than time, as has been done before. The Fourier transform of Eq. (8.2.12) is iωφˆ k (p,ω) + i

p · kφˆ k (p,ω) = (p)φˆ k (p,ω) + F˜ k (ω), m

(8.4.3)

with solution

−1 p F˜ k (ω). φˆ k (p,ω) = iω + i · k − (p) m

(8.4.4)

The required density–density correlation function can be obtained directly from φˆ k (p1,ω)φˆ k (p 2,ω ) first by using the correlation of frequency dependent fluctuating forces and then by integration over the momenta, p1,p 2 . We need the average of the fluctuating forces expressed in terms of frequency and easily find 

 F˜ k (p 1,ω)F˜ k (p 2,ω ) = 





dt1 dt2 exp[iωt1 + iω t2 ] × B(p 1,k,p 2,k )δ(t1 − t2 )

= 2πB(p 1,k,p 2,k )δ(ω + ω ).

(8.4.5)

The needed density–density correlation function may be expressed as   δ n˜ k (ω)δ n˜ k (ω ) =



 dp1

  dp 2 φˆ k (p 1,ω)φˆ k (p 2,ω ) .

(8.4.6)

Before integrating over momenta but after averaging over the fluctuating forces, we may use Eq. (8.4.4) to write

−1 −1   p p φˆ k (p1,ω)φˆ k (p 2,ω ) f luc = iω + i 1 · k − (p 1 ) iω + i 2 · k − (p 2 ) m m   × 2πB(p 1,k,p 2,k )δ(ω + ω ). (8.4.7) We wish to consider light scattering at small wave numbers and frequencies, since this will provide insights into the long-wavelength, long-time correlations of density fluctuations in the gas. For small wave numbers and frequencies, the leading

8.4 Light Scattering

335

contributions to the inverse operators above will come from the hydrodynamic p modes of the operator i · k − (p), which we have discussed at some length m in Chapter 3. Therefore, the right-hand side of Eq. (8.4.7) is dominated by the contributions of the hydrodynamic modes, which leads to   φˆ k (p1,ω)φˆ k (p 2,ω ) f luc    ψα (p 1,k)ψβ (p 2,k ) ψα (p 1 )ψβ (p 2 )B(p 1,k,p 2,k ) ≈ 2π α,β

× [iω + ωα (k)]−1 [−iω + ωβ (k )]−1 δ(ω + ω ).

(8.4.8)

Here are the sums are over the d +2 hydrodynamic modes for each operator appearing in Eq. (8.4.7). Using now expression Eq. (8.2.23) and evaluating the momentum integrals appearing in the inner product of the hydrodynamic eigenfunctions with B(p 1,k,p 2,k ) and using the delta function, δ(p1 −p 2 ) appearing in the expression for B(p 1,k,p 2,k ), we obtain 

  2 ψα (p 1 )ψβ (p 2 )B(p 1,k,p 2,k ) = −(2π)d ψβ (p 1,k ) |(p)| ψα (p1,k)  3 + ψα (p 1,k) |(p)| ψβ (p 1,k ) δ(k + k ). (8.4.9)

Because the zeroth order in wave number for the modes is conserved quantities, each of the preceding inner products is at least of order k 2 . From Chapter 3, expressions for the first-order terms in the eigenfunctions are,   ψα,1 (p,k) = −1 ik · p/m − ω1,α ψα,0,

(8.4.10)

which has a useful form for the evaluations of the matrix elements appearing in Eq. (8.4.9). Using the first-order mode eigenfunctions given by Eq. (8.4.10) in the right-hand side of Eq. (8.4.9), we obtain   ψα (p 1 )ψβ (p 2 )B(p 1,k,p 2,k )   = 2(2π)d (ik · v − ω1,α )ψ0,α [−1 (ik · v + ω1,β )ψ0,β ] δ(k + k ).

(8.4.11)

Combining all of the results just obtained, Eqs. (8.4.5)–(8.4.11), including only hydrodynamic mode contributions and keeping track of all of the equilibrium distribution functions that we have not indicated in the preceding expressions, we obtain

336



The Boltzmann–Langevin Equation

     δ n˜ k (ω)δ n˜ k (ω ) = 2(2π)d+1 1|ψ0,α 1|ψ0,β α,β

  × (ik · v − ω1,α )ψ0.α |[(p)−1 (ik · v + ω1,β )ψ0,β ] × [iω + ωα (k)]−1 [−iω + ωβ (k )]−1 δ(ω + ω )δ(k + k ). (8.4.12) Here the prime on the summation indicates that only hydrodynamic mode contributions are to be included in the sum. Since factors of the form 1|ψ0  appear in the preceding expression, one can see immediately that, to lowest order in the wave number, the shear modes do not contribute to the summation since for them this inner product vanishes. These inner products then restrict the summation to include only sound modes and heat modes. We skip the algebra needed to evaluate the righthand side of Eq. (8.4.12). However, it is important to use the isotropic property of the Boltzmann operator. This allows us to conclude that only functions having the same tensorial character are coupled by the inverse operator, −1 , leading to nonzero values for the inner product appearing in the second line of Eq. (8.4.12). The final result for the structure function becomes, for wavelengths large compared to the mean free path length,  γ − 1 2DT k 2 S(k,ω) = SE γ ω2 + DT2 k 4   s k 2 1 s k 2 + + , (8.4.13) γ 2(ω + ck)2 + s2 k 4 /2 2(ω − ck)2 + s2 k 4 /2 where we have given the numerators to leading order in k. Here SE = kB T /c2 is an overall multiplicative constant, which depends only upon the thermodynamic properties of the fluid; c0 is the adiabatic velocity of sound in the gas; γ = cp /cv is the ratio of the specific heat at constant pressure to the specific heat at constant volume, which, for monatomic dilute gases, is d/(d +2); DT = λ/(ncp ) is the thermal diffusivity; an s is the sound damping constant given for a general isotropic, one-component fluid, by   (d−1) (γ − 1)λ ζ + 2 d η . (8.4.14) + s = ncp nm Here ζ is the coefficient of bulk viscosity, which vanishes for dilute gases but, in general, is nonvanishing. There are some small, higher order in the wave number,

8.4 Light Scattering

337

Figure 8.4.2 Rayleigh–Brillouin spectrum of light scattered by liquid argon in equilibrium at 84.97 K. Figure taken from P. A. Fluery and J. P. Boon [219]

corrections to Eq. (8.4.13) for the structure function that we neglect here [59]. When the wave number, k, satisfies the inequalities ck DT k 2 and s k 2,

(8.4.15)

the structure factor consists of three well-separated Lorentzian peaks as a function of frequency ω. There is a central peak, around ω = 0, called the Rayleigh peak, associated with the diffusion of energy in the fluid, due to the presence of entropy fluctuations [38]. There are two side peaks, called Brillouin peaks, due to the presence of long-wavelength sound waves generated by fluctuations of the hydrodynamic fields in the fluid, to leading order in k, pressure and longitudinal velocity. These peaks occur at frequencies shifted away from ω = 0 due to the Doppler shift of the frequency of the light scattered by microscopic sound waves traveling toward or away from the incident direction of the light. In Fig. 8.4.2, we show the general form of the structure factor as a function of frequency for a typical fluid – in this case, liquid argon. The integrated intensity of the Rayleigh peak, IR (k), can be found by taking the integral of the first term on the right-hand side of Eq. (8.4.13) over all frequencies, ω, and dividing by 2π. We find IR (k) = SE

(γ − 1) . γ

(8.4.16)

Similarly, the integrated intensity of each of the Brillouin lines is SE /2γ . Both the structure factor and the integrated intensity can be measured in a light scattering experiment. The results can be used, of course, to provide information about the

338

The Boltzmann–Langevin Equation

thermodynamic and transport properties of the fluid. In the next section, we discuss the modifications of these results when the fluid is not in equilibrium but instead is in a stationary state with a fixed temperature gradient. Under such conditions, the Rayleigh peak can be one or two orders of magnitude larger than in equilibrium. 8.5 Fluctuations in Nonequilibrium Steady States We will discuss the kinetic theory and fluctuating hydrodynamic methods used to obtain an expression for the enhancement of the Rayleigh line in an NESS with a stationary temperature gradient.9 8.5.1 The Nonequilibrium Structure Function Here we will consider a stationary-state system where a temperature gradient in the x-direction is maintained in the gas by placing it between two parallel walls maintained at different temperatures, and where the average velocity vanishes at all points in the gas. We will assume that the relative change in the temperature over distances on the order of a mean free path is very small, as is usually the case. The gas will be maintained in a steady state that is close to local equilibrium. Therefore, all of the local thermodynamic properties and transport coefficients will be functions of position10 due to the imposed temperature gradient in the gas. Although the translation invariance in the x-direction is broken by the temperature gradient, we may assume that for small enough values of this gradient, the system is translationally invariant in the yz-plane. We expect that the structure factor will depend upon the x-coordinate of the location of light scattering region in the fluid. 8.5.2 NESS Boltzmann Equation with Fluctuations The calculations of light scattering by a fluid in equilibrium in Section 8.4 were based upon the linearization of the distribution function about the equilibrium Maxwell–Boltzmann distribution, leading to Eq. (8.2.9). Suppose instead that the system is in a nonequilibrium stationary state (NESS) with a distribution function, fSS (r,p), that satisfies the nonlinear Boltzmann equation but is not time dependent.11 In order for a system to be maintained in a nonequilibrium stationary state, there must be a flow of particles, momentum, or energy into and out of the system at the boundaries. For a system with a stationary temperature gradient, for example, we may suppose that there are reservoirs at different temperatures on

8.5 Fluctuations in Nonequilibrium Steady States

339

opposite sides of the container with which the particles in the system interact and exchange energy in such a way that the steady temperature gradient is maintained. In our discussion, we will consider a region of the gas that is not subjected to external forces and is far enough from the boundaries that the wall collision terms in the Boltzmann equation can be dropped. Then the distribution function, fSS (r,p), satisfies p · ∇fSS (r,p) = J (fSS ,fSS1 ). m

(8.5.1)

For systems not too far from local equilibrium, the stationary-state distribution function may be taken to be the Chapman–Enskog solution to Eq. (8.5.1), expanded about a stationary-state local equilibrium distribution function, floc (r,p), with, in general, a position dependent density, temperature, and mean velocity. Here 

 β(r) d/2 β(r) 2 (8.5.2) (p − mu) exp − floc (r,p) = n(r) 2πm 2m and fSS (r,p) = floc (r,p)[1 + ce (r,p)],

(8.5.3)

where ce (r,p) has a Chapman–Enskog expansion in powers of the gradients of the local hydrodynamic densities, as described in Chapter 3. We explore fluctuations in this system by writing the fluctuating distribution function, f (r,v,t), as f (r,p,t) = fSS (r,p) + δ f˜(r,p,t)

(8.5.4)

and assuming that the correction term δ f˜(r,p,t) satisfies a Boltzmann–Langevin equation, given by p ∂δ f˜(r,p,t) ˜ + · ∇δ f˜(r,p,t) = ss (p)δ f˜ + S(r,p,t), ∂t m

(8.5.5)

where ss is a linearized Boltzmann operator given by  ˜ ss (p 1 )δ f = dx2 T0 (1,2)(1 + P12 )fSS (r 2,p 2 )δ f˜(r 1,p 1,t), where T0 (1,2) is the Boltzmann binary collision operator defined in Chapter 2. For notational convenience, we have replaced (r,p) with (r 1,p 1 ). The operator P12 interchanges particle labels 1 and 2. The fluctuation term in Eq (8.5.5) is taken to satisfy a set of Langevin-like conditions

340

The Boltzmann–Langevin Equation

 ˜ 1,p 1,t) = 0, S(r   ˜ 2,p 2,t2 ) = δ(t1 − t2 )C(r 1,p 1,r 2,p 2 ). ˜ 1,p 1,t1 )S(r S(r 

(8.5.6) (8.5.7)

For the moment, we assume that the fluctuations are Gaussian with the property that all higher cumulants vanish. We use methods similar to those for the equilibrium case in order to determine the correlation coefficient C(r 1,p 1,r 2,p 2 ). That is, we will calculate the equal-time correlation function of the fluctuating parts of the distribution functions, δ f˜, in the NESS, by using properties of the stationary state and then use this result to calculate C(r 1,p 1,r 2,p 2 ) by expressing the equal-time correlation function in terms of the solutions to Eq. (8.5.5). We begin by writing the formal solution to Eq. (8.5.5): δ f˜(r,p,t) = e−t L δ f˜(r,p,0) +

t

˜ ), dτ e−(t−τ )LSS S(r,p,τ

(8.5.8)

0 p where LSS = m ·∇ − SS (p). The time correlation function of the fluctuation part of the distribution function satisfies



 δ f˜(r 1,p 1,t1 )δ f˜(r 2,p 2,t2 ) = e−[t1 LSS (r 1,p1 )+t2 LSS (r 2,p2 )] δ f˜(r 1,p 1,0)δ f˜(r 2,p 2,0)  t∗ + dτ e−[(t1 −τ )(LSS (r 1,p1 )+(t2 −τ )Lss (r 2,p2 )]. C(r 1,p 1,r 2,p 2 ),

(8.5.9)

0

where we have used Eq. (8.5.7) and set t ∗ = min(t1,t2 ). To obtain an explicit expression for the correlation function C(r 1,p 1,r 2,p 2 ), we first consider the equaltime version of Eq. (8.5.9), multiply by (LSS (r 1,p 1 ) + LSS (r 2,p 2 )) integrate by parts, and then take the limit of large t, to obtain   (LSS (r 1,p 1 ) + LSS (r 2,p 2 )) δ f˜(r 1,p 1,t)δ f˜(r 2,p 2,t) = C(r 1,p 1,r 2,p 2 ). (8.5.10) The explicit expression for C(r 1,p 1,r 2,p 2 ) can be obtained by a separate calculation of the stationary-state equaltime correlation function, 

 δ f˜(r 1,p 1,t)δ f˜(r 2,p 2,t) .

This calculation is the steady-state version of the equilibrium calculation done following Eq. (8.2.21). The steady-state analog of Eq. (8.2.21), but without applying the Fourier transform, it is

8.5 Fluctuations in Nonequilibrium Steady States

δ f˜(r,p,t) =

N 

δ(r i (t) − r)δ(p i (t) − p) − fSS (r,p).

341

(8.5.11)

i=1

The stationary-state distribution functions are, of course, independent of time. A straightforward calculation leads to the expression for the equal time correlation function as 

 δ f˜(r 1,p 1,t)δ f˜(r 2,p 2,t) = fSS (r 1,p 1 )δ(r 1 − r 2 )δ(p 1 − p 2 ) + gSS,2 (r 1,p 1,r 2,p 2 ),

(8.5.12)

where gSS,2 (1,2) = fss,2 (1,2) − fss (1)fss (2) is the stationary-state two-particle correlation function. From this result and Eq. (8.5.1), it follows that12 C(r 1,p 1,r 2,p 2 ) = (LSS (r 1,p 1 ) + LSS (r 2,p 2 )) × [fSS (r 1,p 1 )δ(r 1 − r 2 )δ(p 1 − p 2 ) + gSS,2 (r 1,p 1,r 2,p 2 )] = −[SS (p 1 ) + SS (p 2 )]fSS (r 1,p 1 )δ(r 1 − r 2 )δ(p 1 − p 2 )  + δ(r 1 − r 2 )δ(p 1 − p2 ) dx2 T 0 (1,3)fSS (1)fSS (3) + [(LSS (r 1,p 1 ) + LSS (r 2,p 2 ))]gSS,2 (r 1,p 1,r 2,p 2 ). (8.5.13) The last term on the right-hand side of Eq. (8.5.13) can be evaluated using the BBGKY hierarchy equations and cluster expansion methods to be discussed in Chapter 11 and 12. Here we quote the final result, leaving the derivation for later chapters, namely [(LSS (r 1,p 1 ) + LSS (r 2,p 2 ))]gSS,2 (r 1,p 1,r 2,p 2 ) = T 0 (1,2)fSS (1)fSS (2). (8.5.14) Consequently, the correlation function for the fluctuating forces, C(r 1,p 1,r 2,p 2 ), can be expressed entirely as a function of the stationary-state single-particle distribution functions and kinetic operators acting on them. The analysis of the dynamic structure factor based on Eq. (8.5.13) is lengthy and will not be given here. All of the details can be found in the papers of Kirkpatrick, Cohen, and Dorfman, and we quote here only the final result [371, 372, 373, 374]. 8.5.3 Enhancement of the Rayleigh Line Although it is a lengthy calculation, it is relatively straightforward to solve the Eq. (8.5.19) and then calculate the shape and intensity of the Rayleigh peak in the

342

The Boltzmann–Langevin Equation

structure factor, due to entropy fluctuations in the fluid. Here we merely quote the results. The contribution of the Rayleigh peak to the structure factor is given by13  

T 2 ˆ (α T X ) γ − 1 k T ⊥ + S (R) (R0,k,ω) = ρkB T ρχT γ DT (ν + DT )k 4 2DT k 2 (kˆ⊥ αT T XT )2 + ρk T B ω2 + (DT k 2 )2 (DT − ν)(ν + DT )k 4   2νk 2 2DT k 2 × − . (8.5.15) ω2 + (νk 2 )2 ω2 + (DT k 2 )2

×

Here R0 is a point at the center of the scattering volume, and kˆ⊥ is the magnitude of the projection of the unit vector kˆ in the plane perpendicular to the direction , where dT /dx is the gradient of the of the temperature gradient, XT = T −1 dT dx temperature, taken to be in the x-direction. In addition, χT = ρ −1 (dρ/dp)T is the isothermal compressibility of the fluid and αT = −ρ −1 (dρ/dT )p is the coefficient of thermal expansion of the gas. All of the fluid properties are to be evaluated at the point R0 . The intensity of the Rayleigh peak is obtained by an integration over all frequencies,   ∞ ρ(γ − 1) (αT kˆ⊥ T XT )2 dω R S(R0,k,ω) = ρkB T + I (R0, k) = . 2π γ DT (ν + DT )k 4 −∞

(8.5.16) It is important to note that the corrections to the equilibrium results are proportional to the square of the temperature gradient and inversely proportional to the fourth power of the wave number. In accordance with the condition expressed by Eq. (8.4.2), they are not necessarily small. In fact, the nonequilibrium corrections can be orders of magnitude larger than the equilibrium values for the Rayleigh peak in the structure factor and for its integrated intensity. Careful estimates given by Kirkpatrick, Cohen, and Dorfman [371, 372, 374] show that the expressions given earlier are the leading terms and all other corrections are smaller. There are no terms of order k −6,k −8, . . . , for example. The preceding expressions also contain an additional guide to the experimental detection of these nonequilibrium terms. That is, due to the presence of kˆ⊥ in the correction terms, one should organize a light scattering experiment [323] with small angle scattering, so that the wave vector k is, so far as is possible, in a direction perpendicular to the direction of the temperature gradient. Small angle scattering will ensure small values of k, and this together with the perpendicular condition suggests that the incident light be in the direction of the temperature gradient, and the scattered light be in a direction close to that of the incident light.14

8.5 Fluctuations in Nonequilibrium Steady States

343

We have written Eqs. (8.5.15) and (8.5.16) in a form that is well defined for a general fluid, suggesting that they might be applicable beyond a dilute gas. This is, in fact, the case, since these results can be derived using nonequilibrium ensemble methods or directly from the Navier–Stokes equations by linearizing them about values for a nonequilibrium, stationary state, and adding fluctuation terms to the pressure tensor and heat flow vector with correlations satisfying the same conditions [323]. The presence of inverse fourth-power terms in the wave vector can be seen as evidence of very strong long-range correlations in a nonequilibrium fluid, even under circumstances where the state of the fluid is far from the critical point of a liquid–gas phase transition, the only state where a fluid in equilibrium exhibits long-range correlations. The spatial correlations in the fluid in the NESS may be obtained from spatial Fourier transforms of the structure factor and the integrated intensity. In this way, one finds that the nonequilibrium spatial correlations of density fluctuations in the fluid grow linearly with the system size, L and, for fixed system size, decay linearly with the separation of two points where the density fluctuations are measured; see section 7.5 in the book by Ortiz de Zárate and Sengers [323]. This somewhat surprising result shows that temperature gradients in a fluid induce long-range correlations of hydrodynamic variables and that the fluid does not need to be close to a critical point to allow for measuring them.15 8.5.4 The Results of the Light Scattering Experiments The appropriate light scattering experiment has been carried out by J. V. Sengers and co-workers [418, 432] with results that beautifully confirm the theoretical predictions given in the previous subsection. Here we show the results of the smallangle light scattering experiments of Sengers et al. These authors measured the time inverse temporal of Eq. (8.5.15), which, for the central Rayleigh peak, can be expressed as St(R) (R0,k) = SE

 3 γ −12 [1 + AT (k)] exp(−DT k 2 t) − Aν exp −νk 2 t , γ (8.5.17)

where Aν =

2 DT cP (∇T )2 kˆ⊥ . AT = ν T (ν 2 − DT2 )k 4

(8.5.18)

In Fig. 8.5.1, we show the experimental points and the theoretical predictions of Kirkpatrick et al. There are no adjustable parameters at all, and the agreement of

344

The Boltzmann–Langevin Equation

Figure 8.5.1 The coefficients, AT ,Aν , expressing the enhancement of Rayleigh scattering by a fluid with a stationary temperature gradient. Here the fluid is nhexane at 25 C. The solid lines are the theoretical predictions of Kirkpatrick et al., and the data points are the results of the small-angle light-scattering experiments. Figure taken from Li et al. [432] .

theory and experiment is all one could hope for. We refer to the original papers [418, 432] and the book by J. M. Ortiz de Zárate and J. V. Sengers [323] for further details. 8.5.5 The Two-Particle Correlation Function, g SS,2 A remarkable feature of the expression for the enhanced Rayleigh line is the k −4 dependence of the dynamic structure factor and the integrated intensity of the Rayleigh peak. As mentioned, this result is an indication of the presence of long-range correlations in the fluid in the NESS. It is important to understand the

8.5 Fluctuations in Nonequilibrium Steady States

345

processes taking place in the gas that are responsible for these correlations. To do this, we will anticipate some results of Chapter 14 and describe, in a qualitative way, how these long-range correlations appear in the two-particle correlation function, gSS,2 . Unlike in the equilibrium case, the pair correlation function in stationary nonequilibrium states does not vanish to leading order in the density. This is due to the time asymmetry resulting from the external sources and sinks of energy, momentum, and/or mass that maintain the stationary state. Mathematically, this asymmetry appears most clearly in the consequences of the Stosszahlansatz. The required absence of correlations between the velocities of two colliding particles just before a collision implies that right after this collision, their velocities are correlated. In nonequilibrium situations, these correlations next will be shared by other particles with which the original collision pair makes further collisions, and so forth. So far, we have had no need to deal with this, but for the present purpose of determining the form of the function C(r 1,p 1,r 2,p 2 ), these correlations are needed.16 Kirkpatrick, Cohen, and Dorfman [372] were the first to use kinetic theory to obtain expressions for gSS,2 , for a fluid in a NESS with a stationary temperature gradient. A version of this calculation will be presented in some detail in Chapter 14. Here we will only comment on some aspects of the pair correlation functions in the NESS. The most important contribution to this correlation will be shown in the later chapter to be due to correlated collision sequences. The correlated collision sequences have the property that the velocity distribution for the final collision in the sequence depends crucially upon the previous collisions that the colliding particles suffered at earlier times. A simple example of such a sequence is a recollision process: Two particles – say, particles 1 and 2 – collide at some time with the velocities of each of them taken independently from some random distribution. Then, as they separate, at some time later, particle 3 collides with 2 – say, in such a way that particle 2 eventually collides again with particle 1, as illustrated in Fig. 8.5.2. This final collision is not random; the velocities of the colliding pair depend on the previous histories of the two particles. One can appreciate that such collision sequences may not be rare and are responsible for long distance and long time correlations among the particles in the gas. These correlations have been detected experimentally and by means of computersimulated molecular dynamics. The long-time correlations in the gas are responsible for the appearance of long-time algebraic decays of time correlation functions in the gas. These will be discussed at some length in Chapters 13 and 14. The longrange spatial correlations in a NESS, are responsible for a striking enhancement of the Rayleigh peak in the NESS, which can be orders of magnitude larger than in equilibrium [371, 372, 373, 374, 323].

346

The Boltzmann–Langevin Equation

Figure 8.5.2 A simple correlated collision sequence. This is a three-body recollision event, as described in the text.

8.5.6 The Dynamic Structure Factor by Fluctuating Hydrodynamics As mentioned earlier, in order to calculate the stationary-state structure factor using fluctuating hydrodynamics, the nonlinear Navier–Stokes equations one must adapt to the stationary state and postulate that the local equilibrium forms of the Landau– Lifshitz fluctuating forces apply to this system. Then one solves these equations to get expressions for the fluctuating parts of the local fluid density, mean velocity, and temperature. These expressions are used to obtain the density–density correlation function and the structure factor [562, 352]. As expected, the calculations involve a great deal of algebra and estimations of the relative magnitudes of various term. It is remarkable that this procedure leads to the same result for the dramatic enhancement of the Rayleigh peak, as obtained by kinetic theory or by means of nonequilibrium ensembles. The application of fluctuating hydrodynamics to this system is not based on a microscopic theory and can only be justified a posteriori. Nevertheless, it is possible to make some connection to more fundamental methods, as we will discuss later.17 There remain puzzling features of this method that will be discussed in Section 8.6. The book by Ortiz de Zárate and Sengers [323, 584] contains a clear derivation of the structure factor. This derivation is considerably simpler than that using kinetic theory and, as mentioned, gives the same results. To illustrate the method based upon fluctuating hydrodynamic equations, we ˜ ˜ = δ u(r.t);T (r,t) = T0 + δ T˜ (r,t), where write n(r,t) = n0 (r) + δ n(r,t);u(r,t)

8.6 Puzzles

347

the stationary fluid velocity is zero, and n0 and T0 are the stationary-state values of the density and temperature, respectively. Since the steady-state value of the fluid velocity is zero, and we assume there are no external forces, the pressure of the fluid must be uniform; otherwise, the pressure gradient would produce a fluid flow. The fluctuating parts of the temperature, fluid velocity, and density satisfy ∂ ˜ δ n(r,t) ˜ + ∇ · (n0 δ u(r,t)) = 0; ∂t 1 ∂ ˜ 2 ˜ ˜ ˜ ∇ ·q(r,t); δ T (r,t) + δ u·∇T 0 = DT ∇ δ T (r,t) − ∂t n0 mCP   1 ∂ 2 ˜ ˜ + ∇ · P(r,t). mn0 δ u˜ = −∇δ p(r,t) ˜ + η ∇ δ u˜ + ∇(∇ · δ u) ∂t 3

(8.5.19)

Here δ p˜ is the fluctuating part of the local fluid pressure. The transport coefficients and thermodynamic variables in these expressions are taken to depend on position. In this set of equations, an additional term, δ u˜ · ∇T0 , appears in the temperature equation due to the linearization of the Navier–Stokes equations about a nonequilibrium stationary state. The local equilibrium Landau–Lifshitz correlations used to obtain an expression for the Rayleigh peak, presented in real space variables, are generalized versions of Eqs. (8.3.31) and (8.3.32), given by  2λ(r 1 ) δ(t1 − t2 )δ(r 1 − r 2 )δij , q˜i (r 1,t1 )q˜j (r 2,t2 ) = kB β 2 (r 1 )   P˜ij (r 1,t1 )P˜lm (r 2,t2 )

2η(r 1 ) 1 = δ(r 1 − r 2 )[δil δj m + δim δj l ] 1 − δij δ(t1 − t2 ), β(r 1 ) d 

(8.5.20)

(8.5.21)

and are identical with forms postulated by J. Keizer [351, 352] when applied to a gas at low densities. Together with the change in correlations of the fluctuating currents, the additional term in the hydrodynamic equations is responsible for the enhanced Rayleigh peak obtained by using this method. 8.6 Puzzles One cannot avoid an apparent paradox when using fluctuating hydrodynamic equations with local equilibrium form of the Landau–Lifshitz correlations of fluctuating currents. These correlations are purely local in both space and time, but they lead to a derivation of a result that clearly indicates the presence of longrange correlations in a fluid maintained in a NESS with a stationary temperature gradient. In fact, the paradox is so striking that one might wonder if the final results

348

The Boltzmann–Langevin Equation

can be correct, despite the excellent agreement of the theoretical and experimental results. The question is this: how can one reconcile the fact that long-range correlations characterize the fluid in the NESS yet the equations used to derive them contain only short-range correlations, approximated by spatial delta functions in Eqs. (8.5.20) and (8.5.21)? A partial answer to this paradox can be found in appendix A of the paper by Kirkpatrick, Cohen, and Dorfman [372]. There it was shown that the equal-time correlation function needed to obtain these fluctuation results is given by an expression similar to but different from the exact one, Eq. (8.5.13): Ceff (1,2) = −[loc (p 1 ) + loc (p 2 )]δ(p 1 − p 2 )δ(r 1 − r 2 )floc (r 1,p 1 ), (8.6.1) where loc (p) was defined in terms of velocity variables in Eq. (3.3.8). Note that the Boltzmann collision operators and the functions on which they act use only local equilibrium distribution functions, instead of the full steady-state distribution functions and the two-particle correlation function, gSS , is totally absent. In addition, it was shown that Eqs. (8.5.20), (8.5.21), and (8.6.1), together with the fluctuating Boltzmann equation   ∂ (8.6.2) + LSS (r 1,p 1 ) δ f˜ = S˜ ∂t and the correlation of the fluctuating forces given by   ˜ ˜ S(1,t 1 )S(2,t2 ) = Ceff (1,2)δ(t1 − t2 ),

(8.6.3)

are sufficient to reproduce the contributions of the two-particle distribution function to the density–density correlation function in the subspace of the hydrodynamic variables! One can conclude that the success of the fluctuating hydrodynamic equations method is an example of fluctuation renormalization very similar to the derivation of algebraic long-time tails in time correlation functions discussed by D. Forster, D. R. Nelson, and M. J. Stephen some time ago [229, 228]. Nevertheless, a microscopic derivation of the steady-state structure factor is essential because the use of fluctuating hydrodynamics to describe fluctuations in a NESS is based upon an unproven postulate and may not describe all the features of fluctuations in such a fluid. In this context, Kirkpatrick and Dorfman examined the validity of using the local equilibrium form of the Landau–Lifshitz correlation functions to describe fluctuations in a NESS. They showed that there are nonlocal correlations of the normal stresses in a NESS that provide important corrections to the local equilibrium Landau–Lifshitz results for these correlations [386]. Such a derivation, based on kinetic theory, will be presented in Chapter 14.

Notes

349

8.7 Other Approaches to the Linearized Boltzmann Equation with Fluctuations The Langevin method, as applied by M. Bixon and R. Zwanzig [42], as presented in Section 8.2, is one way to arrive at an expression for the correlation of the fluctuation term in the postulated Boltzmann–Langevin equation. Several other methods have been described for obtaining this result. A number of these are discussed at some length by M. H. Ernst and E. G. D. Cohen [185]. Here we mention the method that R. Fox and G. E. Uhlenbeck [230, 231] used to obtain results identical to those of Zwanzig and Bixon. They put this calculation in a broader context by applying the general theory of stationary Gaussian – Markov processes to the description of density fluctuations in position and momentum space in a dilute gas. For general systems, L. Onsager and F. Machlup [510] developed a formalism for describing the time evolution of fluctuations in an equilibrium system. As an application of the method, they derived the equations of nonequilibrium thermodynamics by using the theory of stationary random processes together with results from Hamiltonian dynamics, such as the Liouville theorem. Fox and Uhlenbeck applied Onsager– Machlup theory [510] to the Boltzmann equation with fluctuations. This approach does not differ substantially from the Langevin method just discussed. However, the theory of Gaussian–Markov processes18 provides a general framework that can be widely applied in a variety of circumstances. Many examples can be found in the book by J. Keizer, for example [352]. We refer the reader to the literature for a full discussion of this approach. Notes 1 N. van Kampen discusses the Boltzmann equation with fluctuations as well the serious problems that may result if one adds fluctuation terms to a nonlinear transport equation in section IX.5 [655]. See, in particular section IX.5. 2 For spherical Brownian particles of radius R , the drag coefficient is usually taken to be described by Stokes’ law, giving γ = 6π ηR/m, where m is the mass of the Brownian particle and η is the coefficient of shear viscosity of the surrounding fluid [412]. 3 Here we follow the presentation of N. G. van Kampen. We also point out that there are a number of delicate and important mathematical problems that arise in the theory of Brownian motion and a large literature dealing with these issues. See, for example, E. H. Hauge and A. Martin-Löf [302]. 4 As done previously, we consider the situation where there is no external force. 5 Density fluctuations on hydrodynamic time scales will be considered later in this chapter, since they are important for the theory of light scattering. 6 There are actually two averages being carried out here. First, one averages over an ensemble of fluctuations to obtain the right-hand side of Eq. (8.2.16) and then over an equilibrium ensemble to obtain the equal time correlation of the deviations from equilibrium, given by Eq. (8.2.22). We usually will assume the first average without comment and calculate the second. 7 These issues have been addressed in the literature [371, 372, 373, 374]. 8 Note that for a dilute gas, DT /c is proportional to and on the order of a mean free path.

350

The Boltzmann–Langevin Equation

9 We postpone a presentation of a detailed microscopic derivation of this result based on kinetic theory until Chapter 14. Here we refer the reader to the original papers of T. R. Kirkpatrick, E. G. D. Cohen, and J. R. Dorfman [371, 372, 373, 374]. Identical results have been obtained and discussed by a number of authors [544, 543, 64, 563, 628, 627, 449, 562, 352, 323, 584] using either nonequilibrium ensembles or fluctuating hydrodynamics. 10 We suppose that the physical state of the gas is far from any instability, such as the Benard instability, which would lead to a non-zero average velocity at different points in the gas. 11 A discussion of the kinetic theory for fluctuations in nonequilibrium systems with further references is given in [401]. 12 An analysis of this and several related results can be found in [186]. 13 The modifications of the Brillouin spectrum are more complicated. The Brillouin peaks broaden, and the integrated intensity is much lower than in equilibrium [364, 374]. Since the effect of the temperature gradient on the intensity of the Brillouin peaks is not as pronounced as on the Rayleigh lines, we present only the results for the latter. 14 This result, Eq. (8.5.16), has also been derived by D. Ronis and I. Procaccia [564]. 15 This property of nonequilibrium steady states with temperature gradients will play a crucial role in the attempts to formulate kinetic equations for densities beyond the dilute gas limit. This issue will be discussed at some length in Chapter 14. 16 As far as we know, they have first been investigated by M. N. Barber, J. M. Blatt, and A. H. Opie [23], and they figure prominently in the works of O. Lanford [415, 413, 414] on rigorous derivations of the validity of the Stosszahlansatz in hard-sphere systems, provided it holds at the initial time. 17 This connection was first made by Kirkpatrick, Cohen, and Dorfman in the appendix of paper [372]. 18 Simple and clear discussions of such processes can be found in a number of texts. A useful discussion is given in [407].

9 Granular Gases

9.1 Introduction to Granular Gases1 We have devoted the large part of this book to the kinetic theory of dilute gases, composed of monatomic particles that undergo elastic collisions. In reality, many particles that one encounters in day-to-day life are not monatomic, and they often suffer inelastic collisions with each other or with a boundary. Such gases might be composed of grains of sand, or corn kernels, or tennis balls, or inelastic ball bearings, and so on. The gas might be subjected to or driven by some external force to maintain the motion of the gas particles, or, in the force free case, allowed to approach a motionless state as the initial energy dissipates by means of the inelastic processes. The study of such gases and, more generally, of granular systems has been an active area of research in the physics and engineering communities over several decades. The results obtained from these studies and the formulation of the many questions that remain open are of importance for many industries, for the understanding of astrophysical processes and structures, such as the formation of planets and planetary rings, and for a fundamental understanding of particulate matter in general [330]. Studies of active matter can use ideas and results from studies of various granular systems to provide examples of the phenomena under study [546].2 In this chapter, we will present some of the main features of the kinetic theory of granular gases composed of particles that undergo inelastic collisions,3 specializing to the case of inelastic hard spheres as an example of one of the more tractable granular gases. Our discussion here will be based upon a modification of the Boltzmann equation appropriate for inelastic, binary collision processes [660, 169, 66, 25, 198, 666, 170]. We will see that this model is rich enough to exhibit many of the phenomena encountered in more general systems. Kinetic theory is particularly important for understanding a wide range of phenomena that may, or may not, take place in the gas, depending on the inter-particle interaction model

351

352

Granular Gases

under consideration.4 It is important to mention at the outset that we assume that the energy dissipated in inelastic collisions is released to the environment, or if stored in internal degrees of freedom of the colliding particles, it is not available to be converted to kinetic energy upon collisions with other particles. 9.2 Inelastic Collisions We begin the study of the kinetic theory of granular gases by considering the binary collision events since the description of these events provides the foundation for all of the discussions in this chapter. We can consider either the case when no external forces are present and describe the cooling of the gas as collisions dissipate energy to the environment, or we can consider that the gas is being acted on by some external force that supplies energy to the gas. Next, we suppose that the particles are all mechanically identical,5 and consider two colliding particles an instant before they actually collide. If there are no external forces acting on them or if the forces do not change over the duration of a collision, the total momentum is conserved during the collision. However, the total energy is not conserved, due to the energy loss in the process. As usual, we describe the motion of the particles in the relative coordinate frame. In the lab frame, the particle velocities immediately before the collision are v1,v2, and immediately after it are v∗1,v∗2 . Conservation of momentum requires that6 v1 + v2 = v∗1 + v∗2 .

(9.2.1)

To describe the collision in the relative frame, we imagine that the coordinate system is placed on particle 2, and that g = v1 − v2 is the relative velocity before collision and g ∗ = v∗1 − v∗2 , after collision. The inelastic nature of the collision appears in our discussion when we postulate a relation between these two relative velocities. In general, finding this relation is not easy. One has to know the force between the particles in detail, identify the source of energy loss,7 and then, having solved the two-body dynamics, determine the relation between the initial and final relative velocities. However, much of the physical behavior of more realistic models can be described by using a simple description of the inelastic binary collisions, so we will not get involved with these more complicated issues here, except to mention that a careful analysis of binary collision dynamics of inelastic spheres has been carried out by a number of authors, with interesting but quite complicated results [66]. Our work will be based upon the assumption that inelastic collisions can be described in terms of restitution coefficients with constant values, independent of the velocities of the colliding particles. We denote the apse vector – that is, the unit vector connecting the center of particle 1 to the center of particle 2 at the

9.2 Inelastic Collisions

353

ˆ or for hard spheres, σ, point of closest approach8 – by k, ˆ and consider the normal and tangential components of the relative velocity before and after collision. The restitution coefficient for the normal component is denoted by n and that for the tangential component by t . The inelastic collision dynamics is then described in terms of the restitution coefficients by ˆ g ∗ · kˆ = −n g · k,

(9.2.2)

for the normal component of the relative velocity, where 0 ≤ n ≤ 1, and, in three dimensions, for the tangential component (i)

(i)

g ∗ · kˆ ⊥ = t g · kˆ ⊥ ,

(9.2.3)

ˆ kˆ (1), kˆ (2) form an orthonormal triad of unit vectors, and we assume that where k, ⊥ ⊥ −1 ≤ t ≤ 1. The case where n = t = 1 corresponds to an elastic collision. The case where t = −1 describes a rotating particle whose spin reverses direction on collision.9 In this discussion, we should also take into account the fact that the coefficients of restitution are, in general, complicated functions of the relative velocity and relative angular velocity of the colliding particles. This adds an additional complication in the discussion, one we wish to avoid in order to present the kinetic theory of granular matter in a simple form. In what follows, we will take the tangential restitution coefficient to be t = 1, so as to ignore any changes in the tangential component of the relative velocity upon collision. In addition, we will take the coefficient of the normal component of the relative velocity to be a constant, denoted by , independent of the relative velocity of the colliding particles. Thus, the collision rule is simplified as much as possible and is expressed by ˆ k. ˆ g ∗ = g − (1 + )(g · k)

(9.2.4)

If we assume that the particles are mechanically identical, then combining the preceding collision rule with the conservation of momentum, we obtain the velocities after collision, v∗1,v∗2, as 1 ˆ k, ˆ v∗1 = v1 − (1 + )(g · k) 2 1 ˆ k. ˆ v∗2 = v2 + (1 + )(g · k) 2

(9.2.5) (9.2.6)

An easy calculation shows that the kinetic energy lost in a binary collision is m m ∗2 2 2 2 ˆ 2 (v1 + v∗2 2 − v1 − v2 ) = − (1 −  )(g · k) . 2 4

(9.2.7)

354

Granular Gases

It is important to note that the magnitude of the relative velocity of the collision particles decreases upon collision, so that the velocities of the two particles become more aligned after a collision than they may have been before it. 9.3 The Boltzmann Equation Since our goal is to arrive at the Boltzmann equation for a granular gas, we anticipate that we will need the restituting velocities for a binary collision. These are the velocities of two particles that collide with apse vector, −σˆ , such that after collision one particle has velocity v1 and the other has v2. If we denote the restituting velocities by v1,v2 , then an easy calculation shows that these velocities are related to v1,v2 by

1 1  ˆ k, ˆ 1+ (g · k) (9.3.1) v1 = v1 − 2 

1 1  ˆ k. ˆ v2 = v2 + 1+ (g · k) (9.3.2) 2  We will also need the Jacobian for the change in velocity variables from the precollision, restituting velocities, v1,v2, to the post-collision velocities, v1,v2 . This Jacobian can easily be calculated by going to center of mass velocity, V , and relative velocity coordinates, g , in the restituting frame, and noting that only one component of the relative velocity, the component along the apse line, changes under collision. Thus, dv1 dv2 = dV  dg  = dV  −1 dg =  −1 dv1 dv2 .

(9.3.3)

ˆ This follows from the facts that dV  = dV and g  · kˆ = − −1 g · k. 9.3.1 Inelastic Hard Spheres We suppose that the particles of the gas are inelastic hard spheres with diameter a. This assumption allows us to express the Boltzmann collision integral for this gas in a simple and familiar form. Using all of the familiar arguments for the Stosszahlansatz for the both the direct and the restituting collisions, and using the transformation from the restituting velocities to the velocities after the restituting collision, given by Eq. (9.3.3), we arrive at the granular Boltzmann equation (GBE) for the single particle distribution function, f (r,v,t), for a dilute, inelastic hardsphere gas. The GBE is

9.3 The Boltzmann Equation

355

∂ f (r,v,t) + v · ∇r f (r,v,t) + +F (r,v,t) · ∇v f (r,v,t) ∂t   2 =a dv1 d σˆ H (−σˆ · g)|g · σˆ | × [ −2 f (r,v,t)f (r,v1,t) − f (r,v,t)f (r,v1,t)].

(9.3.4)

Here we have inserted a factor of  −2 in front of the restituting term. One factor of  −1 comes from the Jacobian discussed earlier, and the other factor comes from restituting collisions, where |g  · σˆ | =  −1 |g · σˆ |. Also, we have denoted the external force acting on the particles by F (r,v,t), which may depend on the location and velocity of the moving particles at the time, t, of interest. 9.3.2 Inelastic Soft Spheres We can consider an extension of the Boltzmann collision integral to more general granular gases.10 We begin by expressing the collision integral in a general form, denoted by J (f (v)), where  J (f (v)) =





dv dwdw



ˆ (v,w |v,w; − k)f ˆ (v )f (w ) d k[W

ˆ (v)f (w)], − W (v,w|v,w ; k)f

(9.3.5)

ˆ is the transition probability per unit time for the direct colliwhere W (v,w|v,w ; k) sions, such that a collision between particles with velocities v,w with apse line unit ˆ vector kˆ will lead to velocities v,w after collision. Similarly, W (v,w |v,w; − k) is the transition probability per unit time for restituting collisions to take place with ˆ If we now use the notation a · kˆ = a , and a⊥ = a−a k, ˆ apse line unit vector −k. for an arbitrary vector a, we can express the transition probability per unit time for the direct collisions as ˆ = K(g ,g⊥ )δ (3) (V  − V ) W (v,w|v,w ; k) × δ (2) (g ⊥ − g ⊥ )δ (1) (g  + g )H (−kˆ · g).

(9.3.6)

Here the superscripts on the delta functions denote the dimensionality of each delta function. The function K(g ,g⊥ ) remains to be specified, and it depends on the collision model that is chosen. Similarly, the restituting kernel, W (v,w |v,w; − σˆ ), can be expressed as

356

Granular Gases

g

,g⊥ δ (3) (V  − V )   g H (kˆ · g). × δ (2) (g⊥ − g)δ (1) g  + 

ˆ = 1K W (v,w |v,w; − k) 



.

(9.3.7)

The primes here indicate restituting velocities given by Eqs. (9.3.1) and (9.3.2) with v∗1,v∗2 replaced with v,w , respectively. The factor  −1 in the gain term arises from the Jacobian as in the hard-sphere case. When we insert these two expressions for the transition probabilities per unit time in Eq. (9.3.5) and carry out two velocity integrations using the delta functions, we obtain    1  g

ˆ ,g⊥ f (v )f (w ) J (f (v)) = dw d k H (kˆ · g) K    − K(g ,g ⊥ )H (−kˆ · g)f (v)f (w)    1  g

= dw d kˆ H (−kˆ · g) K ,g⊥ f (v )f (w )    − K(g ,g ⊥ )f (v)f (w) . (9.3.8) In the final step, we have replaced kˆ with −kˆ in the restituting term since the restituting velocities are invariant under this replacement. We can now describe different models by different expressions for the function K. Simple forms for this function can be expressed in terms of the parameter – such that K(g ,g⊥ ) = Aν g ν−1 |g | = Aν g ν | cos θ|,

(9.3.9)

ˆ = g| cos θ|, where Aν is a constant that depends upon the model, |g | = |g · k| 11 and we suppose that ν ≥ 1. The case ν = 1 and  ≤ 1 is that for inelastic hard spheres. The case of ν = 0 corresponds to inelastic Maxwell molecules. Later in this chapter, we will consider the behavior of the gas for a range of values of ν. For the moment, we return to the case of hard inelastic spheres. 9.3.3 A Useful Identity We return to the hard-sphere collision integral, given on the right-hand side of Eq. (9.3.4), and derive a useful identity from it. If we denote this collision integral by JH S (f ), we will be interested in obtaining an expression for I (ψ) where ψ is an arbitrary function of velocity, and I (ψ) is given by  (9.3.10) I (ψ) = dvψ(v)JH S (f ).

9.4 Homogeneous Cooling State

357

We can write I (ψ) as   2 dvdv1 d σˆ ψ(v)H (g · σˆ )|g · σˆ |[ −2 f  f1 − ff1 ] I (ψ) = a a2 = 2



 dvdv1

d σˆ H (g · σˆ )|g · σˆ | (ψ(v) + ψ(v1 )) [ −2 f  f1 − ff1 ]. (9.3.11)

Here we have used the symmetry between the two colliding particles to obtain the second equality. We have also written, f1 = f (v1 ), and so on. Now we consider the restituting collision term. By using the Jacobian of the transformation from pre-collision to post-collision velocities given by Eq. (9.3.3) and the change in the relative velocity in a collision, we can write the term involving the restituting velocities as   a2 dvdv1 d σˆ H (g · σˆ )|g · σˆ | (ψ(v) + ψ(v1 ))  −2 f  f1 2     a2 = (9.3.12) dvdv1 d σˆ H (g · σˆ )|g · σˆ | ψ(v∗ ) + ψ(v∗1 ) ff1 . 2 Here the velocities v∗,v∗1 are the post-collision velocities, given by Eqs. (9.2.5) and (9.2.6). This result enables us to express I (ψ) as   a2 dvdv1 d σˆ H (g · σˆ )|g · σˆ |f (v)f (v1 ) I (ψ) = 2   (9.3.13) × ψ(v$ ) + ψ(v$1 ) − ψ(v) − ψ(v1 ) . 9.4 The Homogeneous Cooling State One of the features of the GBE, as given for hard spheres by Eq. (9.3.4), is the existence of a homogeneous cooling state, as pointed out by A. Goldshtein and M. Shapiro in 1995 [259, 660, 253, 66]. An intuitive argument for the existence of such a solution to the GBE can be given in the case that the collisions are nearly elastic, such that  = 1 − α, where α 1, when there is no forcing and the gas is spatially homogeneous. In such circumstances, an isolated gas will lose energy very slowly. The distribution function satisfies     ∂ f = a 2 dv1 d σˆ H (−σˆ · g)|g · σˆ | ×  −2 f  f1 − ff1 = JI H S (f ) ∂t (9.4.1) and depends only on velocity and time. The subscript in the last term on the right indicates that the model used is that for inelastic hard spheres. One argues that the

358

Granular Gases

time dependence could be parametrized by the slowly varying average speed of the gas particles. This suggests that one might look for solutions of the GBE in terms of a scaling function. That is to say, we look for scaling solutions of the form [659]

n v n f (v,t) = 3 fs = 3 fs (u) , (9.4.2) vT (t) vT (t) vT (t) where vT (t) is the thermal speed of particles in the gas and u = v/vT is the scaled velocity. We then insert this expression in the GBE and use it to determine the form of fs (u). The thermal speed, vT (t), is defined in terms of a temperature, Tg , often called the granular temperature, by mvT2 = kB Tg, 2

(9.4.3)

where the granular temperature is defined in terms of the average kinetic energy by

  1 2 mnvT2 (t) 3n kB Tg (t) = dv mv f (v,t) = duu2 fs (u), (9.4.4) 2 2 2 where we require that the scaling function, fs (u), be normalized such that the integral on the right-hand side of Eq. (9.4.4) has the value 3/2. We obtain an equation for the time rate of change of the temperature by multiplying both sides of Eq. (9.4.1) by mv2 /2 and integrating over the velocity, so as to obtain 2

2 mv ∂Tg (t) = , (9.4.5) I ∂t 3nkB 2 where I (ψ) is as defined previously. We can obtain an explicit expression for the derivative of the temperature by using Eq. (9.3.13) for ψ(v) = mv2 /2. Clearly, the time rate of change of the temperature is determined by the loss of the total energy of two particles at each collision, given by Eq. (9.2.7). The rate of temperature change in the gas is then   ma 2 (1 −  2 ) ∂Tg (t) =− dvdv1 d σˆ H (g · σˆ ) × (g · σˆ )3 f (v)f (v1 ). ∂t 12nkB (9.4.6) At this point, we can insert the scaling form for the distribution function, Eq. (9.4.2), to obtain inhomogeneous cooling state ∂Tg (t) ∂vT (t) m vT (t) = ∂t kB ∂t =−

(1 −  2 )na 2 m 3 vT (t)s , kB

(9.4.7)

9.4 Homogeneous Cooling State

where s is independent of time and is given by   1 dudu1 d σˆ H (g u · σˆ )(g u · σˆ )3 fs (u)fs (u1 ), s = 12

359

(9.4.8)

and g u = u − u1 . We can solve Eq. (9.4.7) for vT (t) and then obtain Tg (t). The result is Tg,0 Tg (t) =  2 , 1 + (1 −  2 )ν0 s t

(9.4.9)

where the initial collision frequency, ν0 = na 2 vT (0), with vT (0) as the initial value of the thermal velocity. This result is known as Haff’s law, and was first obtained by P. K. Haff in 1983 [289].12 The decrease of the temperature with time can be expressed in terms of the decay of the temperature with the number of collisions taking place in the gas, which is in some ways a more natural time scale for systems that are losing energy. To do this, we define a new dimensionless time, τ , that is tied to the frequency of collisions at time t. This time scale is defined by the equation13 ∂τ = na 2 vT (t). ∂t

(9.4.10)

In terms of the new time scale, the temperature decays exponentially with the number of collisions, as   Tg (τ ) = Tg,0 exp − (1 −  2 )2s τ ,

(9.4.11)

where the relation between τ and t is τ=

  1 ln 1 + (1 −  2 )s ν0 t . 2 (1 −  )s

(9.4.12)

It remains for us to determine s , which in turn requires us to obtain an expression for fs (u). Using Eqs. (9.4.1), (9.4.2), and (9.4.7), we find that fs (u) satisfies   s (1 −  2 ) [3 + u · ∇u ] fs (u) = a 2 du1 d σˆ H (−σˆ · g u )  × |g u · σˆ |[ −2 fs fs,1 − fs fs,1 ].

(9.4.13)

To solve this equation, we suppose that the gas is only slightly inelastic, so that , the coefficient of restitution, is very close to unity. Thus the simplest way to solve this equation is by making expansions in powers of α = 1 − . We suppose that

360

Granular Gases

under these circumstances, the function fs (u) can be expanded about an equilibrium Maxwell–Boltzmann distribution, φ0 (u), as ⎡ ⎤ ∞  α j hj (u)⎦ , fs (u) = φ0 (u) ⎣1 + (9.4.14) j =1

where14 φ0 (u) = π −3/2 e−u . 2

(9.4.15)

We then write 1 −  2 = 2α − α 2,  −2 = (1 − α)−2 ; we expand all terms in powers of α and equate equal powers of α on each side of Eq. (9.4.13). We give only the first two equations here. To order α 0 , we consistently have fs = φ0 (u), and the lowest-order value for  is 2 s(0) = . (9.4.16) 3(2π)1/2 One can take the functions hj (u) to be Sonine polynomials, as done by Goldshtein and Shapiro [259]. Here we give only the first-order correction obtained by T. van Noije and M. H. Ernst [660], and refer interested readers to their papers. Their result is

3 (0) (9.4.17) 1 − α + ··· . s = s 64 It is important to point out that, when the coefficient of restitution becomes too small, there are serious convergence problems with the expansion methods used to determine the form of the distribution function for the cooling state discussed here. As we will show in the next section, for small enough , there is an overpopulation of high-energy particles in the cooling state compared to the population as described by the Maxwell–Boltzmann distribution function. This situation leads to the result that the coefficients of the higher-order Sonine polynomials do not decrease significantly rapidly as the order of the polynomial increases, thus calling the convergence of the expansion into question. The Homogeneous Cooling State for Visco-elastic Particles In our treatment so far, we have been concerned with the homogeneous cooling state for inelastic hard spheres, with a constant coefficient of restitution. The assumption of constant coefficient of restitution is a very strong one. A study of more realistic collision models shows that the coefficient of restitution is not at all constant. Instead, a better approximation for this coefficient is to suppose that it is a function of the relative velocity of the colliding particles. Pöschel and co-workers

9.4 Homogeneous Cooling State

361

analyzed collisions of what they called visco-elastic particles [66, 579, 495]. These are deformable spheres with a frictional mechanism for dissipating energy, and with an additional assumption that the particles retain their spherical shape after collision in order to apply the same model to subsequent collisions. For this model, they find that a useful expression for the coefficient of restitution takes the form (g) = 1 − ag 1/5 + bg 2/5 + . . . . Here g is the magnitude of the relative velocity before the collision of the two particles, and a,b, . . . are functions of the deformation parameters of the colliding particles, the friction parameters, and, if the particles are not mechanically equivalent, of the reduced mass, μ, and reduced radius, rred of the colliding particles, where μ = m1 m2 /(m1 + m2 ) and rred = r1 r2 /(r1 + r2 ). It is possible to repeat the discussion of the homogeneous cooling state for this kind of visco-elastic particles, and in some respects, the results differ from those obtained for the case of constant coefficient of restitution. Since the model is not entirely tractable analytically, a computational analysis is also required. Here we mention some of the main differences between the case of hard-sphere particles with constant  and that for viscoelastic particles. In the visco-elastic case, the time dependence of the temperature in the homogeneous cooling state has an asymptotic long time dependence of the form

t −5/3 Tg (t) ∼ 1+ , (9.4.18) Tg,0 τ0 which differs from the inverse quadratic time dependence for the case of constant . Here τ0 is a relaxation time, typically much longer than the initial mean time between collisions. While the results of the temperature dependence on time for inelastic hard spheres and visco-elastic spheres are only quantitatively different, we will argue in the next section that the homogeneous cooling state is unstable for inelastic hard spheres and not for visco-elastic spheres. 9.4.1 Distribution Function for High-Energy Particles Equation (9.4.13) can be used to determine the high-energy behavior of the distribution function in the homogeneous cooling state, with the result that for large values of the scaled velocity u, the distribution function decays exponentially with the first power of u, rather than quadratically, as is the case with the Maxwell–Boltzmann distribution [660, 66]. To see this, consider the kinetic equation for large values of u. We may replace |g u · σˆ | with |u · σˆ |, assuming that the fast particles collide with slower ones that make up the bulk of the gas. For similar reasons, we may neglect the gain term, since this depends upon collisions between two fast particles, which

362

Granular Gases

are rare compared to collisions between a fast and a slow particle.15 With these approximations, Eq. (9.4.13) becomes  2 s (1 −  ) [3 + u · ∇u ] fs (u) = −ufs (u) d σˆ |uˆ · σˆ |H (uˆ · σˆ ), (9.4.19) where we suppose that the distribution functions are normalized to unity. The angular integral is equal to π, and we can neglect the 3 on the left-hand side compared to the derivative term for large values of u. Next, assuming that the distribution function is isotropic in u, this equation becomes π dfs (u) fs (u), =− du (1 −  2 )s which has the solution

(9.4.20)



 πu fs (u) = C exp − . (1 −  2 )s

(9.4.21)

We see that for high energies, the distribution function is no longer Gaussian in the scaled velocity but is instead a linear exponential function of the velocity. This is larger than the simple, scaled Gaussian function, ∼ exp(−au2 ), for u ≥  π/ a(1 −  2 ) . In order to see that this is a consistent solution of the Boltzmann equation, we still must consider the restituting term and show that it is small compared to the direct term when Eq. (9.4.21) is used for fs (u). First, we consider the magnitude of the restituting velocities, u∗,u∗1 , when u is large. Using Eqs. (9.3.1) and (9.3.2), as well as the approximation, g u ≈ u, we obtain



 1/2 1 1 1 ∗ 2 1+ 3− (uˆ · σˆ ) , |u | ≈ u 1 − 4  

1 u ∗ 1+ |uˆ · σˆ |. |u1 | ≈ 2  Here we suppose that the magnitude of u1 can be neglected compared to the magnitude of u. Using these approximations, we can compare the direct and restituting contributions to the collision integral, if one were to use the exponential form, Eq. (9.4.21). We see that, under these circumstances,     ∗ π fs (u∗ )fs (u∗1 ) ∗ ∝ exp − u + u1 − u . fs (u)fs (u1 ) (1 −  2 )s With the exception of collision regions that are close to grazing – that is, close to |u · σˆ | = 0 – the exponent is negative, and the ratio is small. The region close to grazing requires a slightly more complicated argument, but the result is the same. The neglect of the restituting term compared to the direct term for the high-energy region is consistent with the exponential form of the solution. In Fig. 9.4.1, we

9.4 Homogeneous Cooling State

363

Figure 9.4.1 The velocity distribution, ρ(c), of inelastic, hard-sphere particles in a homogeneous cooling state, for different values of the restitution coefficient, n . Here c = v/vT is the velocity scaled by the thermal velocity. This figure is taken from the paper of Huthmann, Orza, and Brito [317]

present the results of M. Huthman, J. A. Orza, and R. Brito for the overpopulation of high-energy particles in the homogeneous cooling state for inelastic hard spheres for a range of values of the restitution coefficient [317]. The overpopulation of high-energy particles above the Gaussian form is evident in these simulations. This picture has been experimentally verified by P. Yu, M. Schröter, and M. Sperl, who studied the properties homogeneous cooling state of ferromagnetic spheres is a small cell placed in an arrangement of magnets designed to reduce the effects of gravity on the spheres [693]. The experimental results are illustrated in Fig. 9.4.2 and are closely fit by an exponential form like that given by Eq. (9.4.21). 9.4.2 Stability of the Homogeneous Cooling State Assuming that the relevant expansions converge, we see that the field free GBE has the homogeneous cooling state as a solution. It has the property that the temperature of the gas approaches zero exponentially with the number of collisions.16 However, this solution may be unstable to density fluctuations, and spatial correlations produced by the collisions might lead to structures and patterns in the gas. To understand why this might be so, consider what would happen if a region of higher-thanaverage density were to form in the gas. In such a region, the collision frequency would be larger than average, and, as a consequence, the energy dissipation would

364

Granular Gases

Figure 9.4.2 Experimental results for the scaled velocity distribution in a homogeneous cooling state of ferromagnetic spheres in microgravity produced by arranging magnets around the sample cell. The data clearly shows the overpopulation of high-energy particles, with a velocity distribution that closely fits an exponential decay as the first power of the velocity. Figure taken from the paper of Yu, Schröter and Sperl [693]

be faster than average. The gas pressure and temperature in this region would be lower than in the surrounding gas so that more particles would enter the region, and the nucleus of a cluster would be formed. Another mechanism that might produce spatial patterns in the gas can be discerned in the fundamental inelastic hard-sphere binary collision dynamics. As a result of an inelastic collisions, the relative velocity of the two colliding particles decreases after a collision, and the velocities of the two particles become more aligned. One can imagine that after some time, regions form in the gas where a substantial number of particles are moving more or less in the same direction. This suggests that spatial patterns would form in the gas, and perhaps vortices in the average velocity would occur. This is, in fact, what is found in computer simulations by S. McNamara and W. Young [479, 480], as well as by I. Goldhirsch and G. Zanetti [254] in 1993. This work stimulated many other workers, and considerably more is known now about instabilities and pattern formation in granular gases. Here we will describe an elementary model for inelastic collapse and then turn our attention to the equations of granular hydrodynamics. We can then use a simplified version of these equations to argue that the homogeneous cooling state is unstable for inelastic hard spheres but not for visco-elastic spheres. This result is in agreement with the results of more extensive treatments [66].

9.4 Homogeneous Cooling State

365

Granular Hydrodynamics At first sight, it is not at all obvious that one can apply hydrodynamics to granular systems, even dilute granular gases. We derived the Navier–Stokes equations for a dilute gas of elastic particles, where particles, momentum, and energy were conserved. We assumed that the gas was in local equilibrium throughout the container – except, perhaps, near the walls – and that the gradients of the density, mean velocity, and temperature were small. When the gas is composed of inelastic particles, energy is no longer conserved, but a fraction of the total energy is dissipated at each collision. It is not clear that, in general, a local equilibrium state exists and that the relevant gradients will be small. We have just given plausible arguments that indicate that clustering and pattern formation are likely to take place in a gas of inelastic particles, and one can ask if these processes can be described in terms of hydrodynamic equations. An additional phenomenon, closely related to clustering, can take place with inelastic particles; this pheomenon also raises questions about the applicability of hydrodynamics to such systems [65, 330, 252, 169, 170]. This is the phenomenon of inelastic collapse where particles can suffer an infinite number of collisions in a finite time, thereby dissipating all of their energy and coming to rest. A simple example of inelastic collapse is the motion of an inelastic ball bouncing vertically on a table. To see how the collapse comes about, consider dropping the ball at height H0 from the surface of the table and letting it fall due to gravity. We take  < 1 to be the coefficient of restitution. The time that it takes to hit the surface is t0 = (2H0 /g)1/2, and the ratio of the velocity of the next collision to that of the previous collision is . If we denote by tn the time interval between the nth and the n + 1 collision with the surface, then an easy calculation shows that tn = 

n

8H0 g

1/2 .

Then let T denote the time that it takes for the ball to come to rest, given by 1+ T = t0 + t1 + t2 + . . . = 1−



2H0 g

1/2 .

Thus, an infinite number of collisions with the surface take place in a finite time, and the ball comes to rest. Inelastic collapse in a granular gas would lead to some form of clustering and strong correlations between the colliding particles. Inelastic collapse in a granular gas appears when there is strong clustering so that strong correlations between colliding particles are present. Moreover, that description of

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Granular Gases

the properties of the gas in terms of the number of collisions per particle, τ , fails. It also has a strong impact in molecular dynamics simulations where the eventdriven technique is used. Nevertheless, we may suppose that there are some situations in which a hydrodynamic description of the gas is possible, at least for some period of time. Such an example would be a system near a spatially homogeneous cooling state – which, as we have seen, has a velocity distribution close to a Maxwell– Boltzmann distribution and decays either algebraically or exponentially with time, depending on the time scale used. Since we are concerned with the stability of the homogeneous cooling state, we will assume that a hydrodynamic description of small deviations from it is possible and use the machinery of the Chapman–Enskog method to derive linearized hydrodynamic equations, for the case of inelastic hard spheres. We begin by defining the hydrodynamic fields, n(r,t),u(r,t),T (r,t), as is customary, by ⎛ ⎞ ⎞ ⎛ n(r,t) 1  ⎜ ⎟ ⎟ ⎜ (9.4.22) v ⎝ n(r,t)u(r,t) ⎠ = dv ⎝ ⎠ f (r,v,t), 3 1 2 k n(r,t)Tg (r,t) m(v − u) 2 B 2 where f (r,v,t) satisfies the hard-sphere GBE, Eq. (9.3.4). By taking time derivatives and carrying out the required integrations, we find five hydrodynamic equations: ∂n + ∇ · (n)u = 0, ∂t

1 ∂ +u·∇ u+ ∇ · P = 0, ∂t mn(r,t)

2 ∂ + u · ∇ + ζ Tg + (P : ∇u + ∇ · q) = 0. ∂t 3n Here the pressure tensor, P(r,t) is defined by  P(r,t) = m dvV V f (r,v,t), where V = v − u, and

2 Pij = pδij − η ∇i uj + ∇j ui − δij (∇ · u) 3

(9.4.23) (9.4.24) (9.4.25)

(9.4.26)



with shear viscosity η. The heat flow vector, q is given by [65, 66]  m q = dv V 2 V f (q,r,t) = −κ∇T − μ∇n, 2

(9.4.27)

(9.4.28)

9.4 Homogeneous Cooling State

367

where a term, −μ∇n, not present in the heat flow vector for a molecular fluid, appears because of the energy loss in binary collisions, with μ ∝ (1 −  2 ). Because energy is not conserved, the temperature equation has a contribution, ζ Tg , which is of zeroth order in the gradients and not present in the temperature equation for a gas of elastic particles. The quantity ζ (r,t) is called the cooling coefficient. For inelastic hard spheres, the cooling coefficient is given by    ma 2 (1 −  2 ) dv dv1 d σˆ H (σˆ · g) × (σˆ · g)3 f (v,r,t)f (v1,r,t) ζ (r,t) = 12nkB Tg   ma 2 (1 −  2 ) dv dv1 |v − v1 |3 f (r,v,t)f (r,v1,t). (9.4.29) = 24nkB Tg We can obtain hydrodynamic equations for various model systems by employing a variation of the Chapman–Enskog normal solution method, adjusted for obtaining solutions of the GBE that are near that for the homogeneous cooling state. We will not carry out this procedure here but instead refer the reader to the book by N. V. Brilliantov and T. Pöschel for the details [66]. For our purposes, we need only linearized forms of the hydrodynamic equations, expressed in such a way that they would apply to any granular gas that is close to an inhomogeneous cooling state. It is possible to carry out the stability analysis using a simplified form of the hydrodynamic equations and to apply them to gases differing only in the temperature dependence of the cooling coefficient, ζ . We expand the distribution function in powers of the gradients of the hydrodynamic fields with the zeroth-order distribution given by the homogeneous cooling solution, Eq. (9.4.2), of Eq. (9.4.1), and write f (r,v,t) = f (0) (r,v,t) + f (1) (r,v,t) + . . . , where f

(0)

n(r,t) (r,v,t) = 3 fs vT (t)



v . vT (t)

(9.4.30)

(9.4.31)

In the Chapman–Enskog solution, the time derivatives of the hydrodynamic fields are expanded in powers of the gradients, corresponding to the Euler, Navier–Stokes, . . ., equations. We apply the same method here, with a zeroth-order equation ∂ (0) f (0) = JI H S (f (0) ). ∂t

(9.4.32)

By carrying out the integrals with respect to 1,v,mv 2 /2, we obtain the zeroth-order hydrodynamic equations ∂ (0) u ∂ (0) Tg ∂ (0) n = 0, = 0, = −ζ (0) Tg, ∂t ∂t ∂t

(9.4.33)

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Granular Gases

where the zeroth-order cooling coefficient is   ma 2 (1 −  2 ) (0) ζ = dv dv1 |v − v1 |3 f (0) f1(0) 24nTg

kB Tg 1/2 = 2na 2 (1 −  2 ) s m

(9.4.34)

where the constant s is given by Eq. (9.4.8). The first-order equations for the hydrodynamic fields follow from the conservation and temperature equations, Eqs. (9.4.23)–(9.4.25) and are ∂ (1) n = −∇ · (nu), ∂t 1 ∂ (1) u = −(u · ∇)u − ∇(nTg ), ∂t nm ∂ (1) Tg 2 = −(u · ∇)Tg − ζ (1) Tg − T ∇ · u, ∂t 3

(9.4.35) (9.4.36) (9.4.37)

where we have used f (0) to evaluate the pressure tensor, to obtain p = nTg 1. The first-order cooling factor, ζ (1), is obtained by inserting the gradient expansion for the distribution function, Eq. (9.4.30) in Eq. (9.4.7) and keeping only terms of first order in the gradients. By now, this procedure is very familiar. We will skip the detailed derivations and present a simplified version of the hydrodynamic equations, keeping only the terms that are essential for our analysis of the stability of the homogeneous cooling state. As we will now argue, the homogeneous cooling state for inelastic hard spheres is unstable under density fluctuations, while that for visco-elastic spheres is stable. These results agree, qualitatively, with those from more detailed analyses [66]. The simplified hydrodynamic equations used here are versions of the “inelastic” Navier–Stokes equations where we keep only those terms that capture the essential features of the stability analysis. We will avoid scaling the velocity variables with the thermal velocity, vT , so as to work only with laboratory variables, and the physical time t. The linear equations are ∂δn(r,t) = −∇ · u(r,t), ∂t ∂u(r,t) = −Tg ∇δn(r,t) + ν(Tg )∇ 2 u(r,t), ∂t ∂Tg (r,t) = −ζ (Tg )Tg (r,t). ∂t

(9.4.38) (9.4.39) (9.4.40)

9.4 Homogeneous Cooling State

369

Here ν(T ),ζ (T ), are the coefficient of shear viscosity and cooling coefficient, respectively, and, for simplicity, we have not included any multiplicative constants in these equations. The crucial features of our analyses are the temperature dependences of the coefficients, ν(T ) and ζ (T ). For inelastic hard spheres, both of these coefficients are proportional to T 1/2, and we write νihs = ν0 T 1/2 , and ζihs = ζ0 T 1/2 . We solve Eq. (9.4.40) for the time-dependent temperature of the gas and obtain, in accordance with Haff’s law, Eq. (9.4.9), Tg (t) = 

Tg,0

, 1/2  2 1 + ζ0 Tg,0 t /2 

(9.4.41)

where we assume that at t = 0,Tg = Tg,0 . The velocity equation is linear, and one can easily obtain an equation for the spatial Fourier transform of the velocity. The equation for the transverse velocity field is then 1/2

−ν0 k 2 Tg,0 ∂u⊥ (k,t) u (k,t). = −ν0 Tg1/2 k 2 u⊥ (k,t) =   1/2   ⊥ ∂t 1 + ζ0 Tg,0 t /2

(9.4.42)

The solution for the velocity field is then u⊥ (k,t) = 

u⊥ (k,0) ,  1/2  γ (k) 1 + ζ0 Tg,0 t /2

(9.4.43)

where γ (k) = 2ν0 k 2 /ζ0 . Thus, for non-zero wave number, the transverse velocity decays for large time as an inverse power of the time, t, with the power proportional to k 2 . Now let’s examine the density fluctuations in this approximation. To obtain a useful equation for the wave number–dependent density fluctuations, δn(k,t), we take the time derivative of Eq. (9.4.38), and use the Euler term in Eq. (9.4.39) to obtain k 2 Tg,0 ∂ 2 δn(k,t) = − δn(k,t).   1/2  2 ∂t 2 1 + ζ0 T t /2

(9.4.44)

g,0

For long times, t, the density fluctuation equation becomes k2 ∂ 2 δn(k,t) = −A δn(k,t), ∂t 2 t2

(9.4.45)

where A is a constant. We look for power law solutions of this equation of the form δn(k,t) ∼ t a . This leads to a simple quadratic equation for the exponent a with two solutions: a± =

 1 1 ± [1 − 4Ak 2 ]1/2 ≈ 1,Ak 2 > 0. 2

370

Granular Gases

Figure 9.4.3 Vortex patterns and clustering in a gas composed of identical of inelastic hard disks, as obtained from molecular dynamics. The gas is prepared in an initially homogeneous state. At about 80 collision times per particle, vortex patterns and spatial inhomogeneities appear in the gas, as illustrated on the left. At about 160 collision times, clustering of the particles is observed, as shown on the right. The system was composed of 50,000 inelastic hard disks with  = 0.9 and at density π nσ 2 /4 = 0.4. Here σ is the diameter of the disks. This figure is taken from the paper of T. C. P. van Noije and M. H. Ernst [658]

Thus, both solutions lead to growth in the density fluctuations with time, with leading term linear in time for large times. This growth of the density fluctuations for inelastic hard spheres indicates that the homogeneous cooling state is unstable to density fluctuations for these particles. We obtain the same result if we include the effect of viscous damping in the equation for the density fluctuations. That is, we include the viscous term in Eq. (9.4.39) to obtain another term in the equation for the density fluctuations, namely ∂δn(k,t) ∂ 2 δn(k,t) = −k 2 Tg δn(k,t) − ν(Tg )k 2 . 2 ∂t ∂t

(9.4.46)

Again, for inelastic hard spheres where for long times, Tg ∼ t −2, and ν ∼ t −1 , we also obtain expressions for the density fluctuations that grow with time. Numerical studies by R. Brito and M. H. Ernst [658] illustrating this behavior are shown in Fig. (9.4.3). By way of contrast, we carry out a similar analysis for visco-elastic spheres using Eq. (9.4.18) for the time dependence of the temperature in the homogeneous cooling state, and the fact that for this model as well, the viscosity is proportional 1/2 to Tg [66]. For this system, the analog of Eq. (9.4.42) is

9.4 Homogeneous Cooling State

371

1/2

−ν0 k 2 Tg,0 ∂u⊥ (k,t) 1/2 2 u (k,t), = −ν0 T k u⊥ (k,t) =   1/2  5/6 ⊥ ∂t 1 + ζ0 Tg,0 t /2

(9.4.47)

and the transverse velocity decays as u⊥ (k,t) ∼ exp[−k 2 t 1/6 ].

(9.4.48)

2 2 ∂ 2 δn(k,t)  k  k ∂δn(k,t) = −A δn(k,t) − B . ∂t 2 t 5/3 t 5/6 ∂t

(9.4.49)

The analog of Eq. (9.4.46) is

For this equation, we look for solutions of the form δn(k,t) ∼ exp[−at b ]. Some elementary algebra shows that the leading terms have the form   δn(k,t) ∼ exp ±ikαt 1/6 − βk 2 t 1/6 , where α,β are constants. These terms oscillate in time with a slow exponential decay. There is no indication that there is an instability for visco-elastic spheres. This is in accord with already-known computational results. In this case, computer simulations show that apparent structure formation is a transient effect and disappears for very long times. Thus, the visco-elastic gas eventually returns to the homogeneous cooling state [53].17 Because of the slow power law decay of the transverse velocity fluctuations in the inelastic hard-sphere case, the convective nonlinear term, (u · ∇) u, is very important in a nonlinear stability analysis. An iterative solution suggests that transverse velocity fluctuations are also unstable with a characteristic wave √ number k ∗ ∝ ζ0 /ν0 . This in turn is consistent with the vortex structure shown in Fig. 9.4.3. Additional studies of the instabilities and their connection with vortex and pattern formation can be found in the papers of Goldhirsch and Zanetti and of Ernst and van Noije [254, 658]. Although the GBE does not have an associated  H -theorem, we may examine the behavior of an entropy function, S(t) = −kB dvf ln f . If we use the distribution function for the cooling state, Eq. (9.4.2), we find that for long times, the entropy is decreasing as S(t) ∼ − ln t. The decreasing of entropy with time is a well-known phenomenon in dynamical systems theory that occurs whenever a chaotic system is approaching a strange attractor and the phase space available to the system is contracting [152]. That the

372

Granular Gases

homogeneous cooling state has a negative rate of entropy production can be taken as an indication that the system is approaching an attractor; in this case, the attractor is simply the delta function, δ(v). . 9.4.3 Other Models: BGK, Pseudo-Maxwell Models, and Soft-Sphere Models In the previous section, we outlined how one might generalize the binary collision models from inelastic hard-sphere models to inelastic soft spheres. The GBE was generalized in Eq. (9.3.8), and a possible form for the collision kernel was given by Eq. (9.3.9), in terms of parameter ν. The models are general enough that one could allow for a coefficient of restitution that depends on the relative velocity of the colliding pair, but we will not examine this case here. Instead, we shall consider the properties of the homogeneous cooling state for various values of the parameter ν, including the case ν = 0. This is the case of pseudo-Maxwell particles, since the kernel of the collision integral does not depend upon the relative velocity, similar to Maxwell molecules, but the effective collision cross section is finite, and there are no molecular interactions that would lead to such a kernel [46, 183, 198, 47]. The BGK model We begin by considering a simple model collision operator, the Bhatnagar–Gross– Krook (BGK) model that we discussed earlier in the context of elastic systems. The BGK equation is a linear equation, and for a spatially homogeneous system, it is ∂f (v,t) (9.4.50) = −ων (t)[f (v,t) − f0 (v,t)], ∂t where ων (t) is the mean collision frequency taken to depend upon time, through a time-dependent thermal velocity, vT (t), as well as upon a parameter, ν, such that ων (t) = vTν (t). We suppose that the function, f0 (v,t), to which the distribution function is to relax has the form of a Maxwell–Boltzmann homogeneous cooling form, namely   −3  exp − (v/(vT (t)))2 , fh (v,t) = αvT (t)π 1/2 = (αvT (t))−3 φ0 (u/α),

(9.4.51)

where u = v/vT , and φ0 (u/α) is the Maxwell–Boltzmann distribution given by Eq. (9.4.15). Here α is a parameter that will be related to the inelasticity of the model, such that the limit α → 1 is the elastic limit. If we define the thermal

9.4 Homogeneous Cooling State

373

velocity, vT , and granular temperature, Tg (t), as in Eq. (9.4.4), then the BGK equation, Eq. (9.4.50), leads to an equation for vT (t) as dvT (t) = −γ vTν+1 (t), dt

(9.4.52)

where γ = (1 − α 2 )/2. Then we obtain a homogeneous cooling solution vT (t) =

vT (0) . [1 + νγ vTν (0)t]1/ν

(9.4.53)

From this, we see that α = 1 corresponds to the elastic limit and for α < 1, and for long times, Tg (t) ∝ t −2/ν . Haff’s law corresponds to ν = 1. The distribution function in the homogeneous cooling state can be determined and is given by

v −3 , f (v,t) = (vT (t)) fh vT (t) where



a

fh (u) = u−(3+a) A +  3 απ 1/2





u

x 2+a exp[−x 2 /α 2 ]dx ,

(9.4.54)

0

where A is a normalization constant, and a = γ −1 = 2/(1 − α 2 ). We see that for this model, there is also an overpopulation of high-energy particles, but unlike the exponential tail in the case of inelastic hard spheres, Eq. (9.4.21), the distribution function has an algebraic high-energy tail, proportional to v −(3+a), obtained by taking the limit u → ∞ in the integral in Eq. (9.4.54). Pseudo-Maxwell and Soft-Sphere Models We return to the case of soft-sphere models, with GBE given by Eq. (9.3.8) and with collision kernel given by Eq. (9.3.9) for ν ≥ 0. We consider the homogeneous cooling states for these systems and first derive an equation for the change in the thermal velocity, vT (t), or the granular temperature, related by kB Tg (t) = mvT2 /2, with 3kB Tg = m < v2 >. Following exactly the same steps as for the inelastic hard spheres, we immediately obtain an equation for the thermal velocity, with the same form as found for the BGK model, namely vT (t) =

vT (0) , [1 + νγ vTν (0)t]1/ν

(9.4.55)

for ν > 0 and with γ ∼ (1 −  2 ). In the case of pseudo-Maxwell molecules, the thermal velocity decays exponentially with time, ν = 0, the thermal velocity decays exponentially with time.

374

Granular Gases

The Fourier Transform Method The pseudo-Maxwell case, for which ν = 0, is of special interest because an exact solution is available for the homogeneous, nonlinear GBE. This case has been studied in detail by A. Baldassarri [17], and co-workers, by E. Ben-Naim and P. L. Krapivsky [36], by A. V. Bobylev [46, 47] and co-workers, as well as by M. H. Ernst and co-workers18 [661, 660, 183, 25, 197, 198]. Exact solutions can be obtained by using the Bobylev Fourier transform method, discussed earlier in connection with the solutions to the nonlinear Boltzmann equation for elastic collisions. This model also exhibits an overpopulation of high-energy particles, with an algebraic decay of the velocity distribution function at these energies. To show this, we consider the GBE for the pseudo-Maxwell model, for the spatially homogenous case. This is     1 ∂f (v,t)   = dv1 d σˆ H (−g · σˆ ) × f (v ,t)f (v1,t) − f (v,t)f (v1,t) . ∂t  (9.4.56) Now define the function (k,t) by  (k,t) = dvf (v,t) exp(ik · v)

(9.4.57)

and take the velocity Fourier transform of Eq. (9.4.56). This leads to     1 1 ∂(k,t)   = dvdv1 d σˆ exp(ik · v) × f (v ,t)f (v1,t) − f (v,t)f (v1,t) . ∂t 2  (9.4.58) Here we use the fact that the integrand on the right-hand side of Eq. (9.4.56) is invariant under the transformation σˆ → −σˆ , so that with the additional factor of 1/2, the integral over the hemisphere, H (−g · σˆ ), can be replaced by an integral over a full sphere, and the implicit dependence of the σˆ integral on the relative velocity g can be removed. Then, because both velocity integrals over v,v1 are being performed, the restituting term can be transformed as     1 1   dvdv1 d σˆ exp(ik · v)f f1 = dvdv1 d σˆ exp(ik · v∗ )ff1, 2 2 where v∗ = v − 12 (1 + )[(v − v1 ) · σˆ ]σˆ . Then it follows immediately that the restituting term can be written in terms of the transform function (k,t) as  1 d σˆ (k1,t)(k2,t), 2

9.4 Homogeneous Cooling State

375

where 1 k1 = k − (1 + )(k · σˆ )σˆ 2 1 k2 = (1 + )(k · σˆ )σˆ . 2 Then we can include the factor of 1/2 by a simple redefinition of the time variable, so that Eq. (9.4.58) becomes  ∂(k,t) (9.4.59) = d σˆ (k1,t)(k2,t) − 4πn(k,t). ∂t We require that the distribution functions are normalized to the number density, n or, equivalently, that (k = 0,t) = n. We can remove all of the constants by ˜ and τ = 4πnt. We obtain writing  = 4πn ˜ ∂ = ∂τ



˜ 1,t)(k ˜ 2,t) − (k,t). ˜ d σˆ (k

(9.4.60)

This equation has been studied in some detail. Here we will show how to use it to find the high-energy tails in the distribution function for the homogeneous cooling state. Another Useful Identity In order to say something about the large-velocity behavior of the distribution function in the homogeneous cooling state, we will study the small-wave-number behavior of its Fourier transform. To see why this might be useful, let us suppose that there is a high-energy tail in the velocity distribution function of the form f (v) ∼ v −b for large velocities. Such a distribution would lead to a contribution to the Fourier transform as  v 2 v −b exp[ikv cos θ]dv. By changing the variables from v to kv, this integral becomes  k b−3 x 2−b exp[ix cos θ]dx. The range of the x- integration is finite if we suppose that, in dimensionless units, kv = x for a range of x values. Thus, for asymptotically large v, we consider the transform for small k. If we then look for solutions of Eq. (9.4.60) that have a term of the form k b−3 , for small k, we conclude that the velocity distribution function is proportional to v −b for large19 v.

376

Granular Gases

High Energy Tails for Pseudo-Maxwell Models20 ˜ With this preliminary observation, we can examine the behavior of (k) in the homogeneous cooling state. For this state, the velocity distribution function has the form

v n , g f (v,τ ) = 3 vT (τ ) vT (τ ) where we use the rescaled time, τ . The function g(c) is an isotropic function of its argument, c = v/vT (τ ). Thus,  1 ˜ (κ) = dc exp[iκ · c]g(c) 4π  ∞ sin(κc) g(c)dc, (9.4.61) c2 =2 κc 0 where κ = vT (τ )k. We can then rewrite Eq. (9.4.60) as  ˜ d (κ) ˜ 2 ), ˜ ˜ 1 )(κ −γ κ + (κ) = d σˆ (κ dκ

(9.4.62)

where κi = |ki |vT (τ ). We have also assumed that vT (τ ) = vT (0) exp(−γ τ ). Since ˜ we are interested in the small κ behavior of the function (κ), we can proceed to make the following ansatz ˜ 4π (κ) =1−

1 < (κ · c)2 > +Aκ a + · · · , 2

(9.4.63)

for small κ. We insert this expression in Eq. (9.4.62) and equate equal powers of κ. We assume that a = 2. If we equate the terms of order κ 2, we obtain an expression for γ , or, equivalently, the constant c0 appearing in Eq. (9.4.56). We obtain    1 d σˆ < (κ · c)2 > − < (κ1 · c)2 > − < (κ2 · c)2 > . 2γ < (κ · c)2 >= 4π (9.4.64) By carrying out all of the angular integrals, we find γ =

1 (1 −  2 ), 12

(9.4.65)

as expected. When we equate the terms of order κ a , we obtain a transcendental equation for the power a, of the form  

1 1+ a 12 × 1− − L(a) , (9.4.66) a= 1 − 2 3 2

9.5 Driven Systems

377

Figure 9.4.4 The numerical solution of Eq. (9.4.66) for the exponent, a, that determines the high-energy tail as a function of the coefficient of restitution, . This figure is taken from the paper of M. H. Ernst and R. Brito [181]

where

 L(a) =

1

 a/2 dx 1 − qx 2 ,

(9.4.67)

0

and q = (1 + )(3 − )/4. When this equation is solved for a, we know that the homogeneous cooling state has a high-energy tail of the form v −(a+3) . Numerical solution of Eq. (9.4.66) as a function of  is illustrated in Fig. 9.4.4, and it is seen that a is a monotonically increasing function of the coefficient of restitution, , ranging from about a ≈ 5.5 for  = 0, and grows to infinity as  → 1 [182, 183]. This limit is what one should expect from the fact that in the elastic limit, the distribution function is a Maxwell–Boltzmann distribution at all velocities. 9.5 Driven Systems So far, we have only discussed the behavior of granular gases that are not being subjected to external forces. This is, of course, not the typical situation in which such systems are encountered. Most often, they are subjected to mechanical forces, to gravitational forces, to electromagnetic fields, or, in laboratory experiments on sand or on ball bearings, for example, often to vibrating or shaking platforms. These external forcing mechanisms provide a source of energy that can replace

378

Granular Gases

the energy dissipated by collisions, and if there is an energy balance, the system may eventually reach a steady state. Here we will consider solutions of the GBE with some kind of external force and examine steady-state solutions for those situations in which a spatially homogeneous, steady-state solution is possible. One interesting example is the Lorentz model, with inelastic scattering, charged moving particles, and placed in an electric field. Since the scattering dissipates energy while the field adds energy, one might ask if the system will reach a stationary state. A one-dimensional example was studied by T. Biben, Ph. A. Martin, and J. Piasecki [457] and independently by L. Nasser [501]; they show that the system arrives at a steady state for any uniform electric field, provided the coefficient of restitution,  < 1. Martin and Piasecki also use simple scaling arguments to show that the electric current for inelastic hard-sphere Lorentz models is proportional to the square root of the electric field, E 1/2 . An interesting feature of these models is that the velocity distribution in the steady state is not a Gaussian and can be proportional to exp[−av 4 ].

9.5.1 Driven Systems in Three Dimensions Many experiments on granular systems are carried out under circumstances such that it is not possible to specify the exact force experienced by any particular particle in the system at any given time. This is typical of experiments in which a collection of granular particles are placed on a vibrating platform, and the platform transfers energy to the system by means of collisions with particles at the boundary of the platform. Another example is provided by a situation in which the natural cooling of a granular gas is countered by a uniform heating of particles in the gas due to some external force.21 Instead of a well-defined force on a particle – as would be case, for example, if the system were placed in an electromagnetic or gravitational field – such forces are described by stochastic mechanisms, very similar to that used to describe the effect on the distribution function for a dilute gas, due to fluctuations around the predictions of the Stosszahlansatz for the Boltzmann equation, as described earlier, in Chapter 8. In this section, we study the properties of a stationary state when the granular gas is subjected to such a stochastic force [661, 514, 197, 131]. We will also make the simplifying assumption that the gas is spatially homogeneous. A reasonable way to introduce the effect of an stochastic forcing mechanism on the distribution function is to suppose that in the absence of the collisions between the particles, the distribution function would obey a Fokker–Planck equation, with a form determined by the stochastic forces. Alternately, one might suppose that the single-particle distribution function, f (r,v,t) is the average of the delta function

9.5 Driven Systems

379

δ(x − x(t)) taken with respect to an appropriate ensemble distribution function. In such a case, we may write f (x,t) = δ(x − x(t)),

(9.5.1)

where the over-line indicates an average with an ensemble distribution, and x = (r,v) and f (r,v,t) ≡ f (x,t). Then f (x,t + δt) − f (x,t) = δ(x − x(t) − δx(t)) − δ(x − x(t)) = −δx(t)

∂ ∂2 1 f (x,t) + · · · . f (x,t) + δxi (t)δxj (t) ∂x 2 ∂xi ∂xj (9.5.2)

In order to specify the ensemble averages, we suppose that the time interval δt is longer than the correlation time of the fluctuations due to external forces. To calculate the averages needed above, we suppose that in time interval δt, the position and velocity of a particle change as δxi (t) = αi δt + δwi ,

(9.5.3)

where αi = (vi ,ai ), with ai as the ith component of the acceleration of a particle due to any non-fluctuating fields acting on a particle, and δwi = (0,δui ) with  t+δt dτ ξi , (9.5.4) δui = t

with ξi representing the ith component of the random acceleration. We assume that these velocity fluctuations, δui , have zero average, with correlations of the Gaussian form, due to noise fluctuations of acceleration. If so, δui δuj = ξ02 δij δt,

(9.5.5)

where characterizes the strength of the fluctuations in the acceleration of the particles due to the random forces. Combining these results, we obtain a Fokker–Planck equation for the distribution function 1 ∂f (r,v,t) = −v · ∇r f − a · ∇v f + ξ02 ∇v2 f + · · · . ∂t 2

(9.5.6)

The last explicit term on the right-hand side of Eq. (9.5.6) represents a diffusion in velocity space produced by the random forces. All of this ignores the collisions between the particles. We can include them now by supposing that the collisions produce a change in the distribution function is the familiar way, so as to write a

380

Granular Gases

combined equation for the distribution function in the spatially homogeneous, field free case for a heated or shaken system as 1 ∂f (v,t) = ξ02 ∇v2 f + JGBE (f ), ∂t 2

(9.5.7)

where JGBE (f ) is the binary collision operator appearing in the GBE, whose structure depends on the model taken for the binary collisions of the granular particles. Here we will consider stationary-state solutions to Eq. (9.5.7) for inelastic hard spheres, as an example of the method that applies to the other cases we considered, including the BGK model, Maxwell models, and inelastic soft spheres. Before we specialize our calculation to steady states, we can use Eq. (9.5.7) to derive an equation for the granular temperature using the relation 3kB Tg = m < v2 >. The result is  mξ02 m(1 −  2 )a 2 ∂Tg (t) dvdv1 d σˆ H (g · σˆ ) × |g · σˆ |3 f (v,t)f (v1,t), − = ∂t 3kB 12nkB (9.5.8) where we have used Eqs. (9.3.13) and (9.4.6). Next, we assume that there is a stationary state solution of the Boltzmann equation that has the scaling form,

n v f (v) = 3 Fsc . (9.5.9) vT vT In this case, the left-hand side of Eq. (9.5.7) vanishes, and we use this equation to obtain an expression for vT with the result vT3 = where

4ξ02 , (1 −  2 )na 2 h

(9.5.10)

 h =

dudu1 d σˆ H (g u · σˆ )|g u · σˆ |3 Fsc (u)Fsc (u1 ),

(9.5.11)

where u = v/vT , and g u = u − u1 . Using this result, we can express the steadystate temperature as   2/3 m duFsc (u)u2 4ξ02 Tss = . 3kB (1 −  2 )na 2 h

(9.5.12)

In order to complete this analysis, we must determine the scaling function, Fss (u). This can be accomplished by converting the steady-state Boltzmann equation for

9.5 Driven Systems

381

f (v) to an equation for Fsc (u). Using Eqs. (9.5.7) and (9.5.9), we can obtain an equation for Fsc (u) as   (1 −  2 )h 2 ∇u Fsc (u) = − du1 d σˆ H (σˆ · g u )|g u · σˆ | × [ −2 F  F1 − F F1 ]. 8 (9.5.13) As was the case for the homogeneous cooling state, we can find a simple approximate solution if we assume that the coefficient of restitution is close to unity and we expand both sides of Eq. (9.5.13) in powers of α = 1 − . The zeroth-order solution is the Maxwell Boltzmann distribution function, φ0 (u), given by Eq. (9.4.15), and the first-order term can be obtained by the same method as used in the earlier case. We leave the details to the reader. The steady-state temperature is found to be  2/3 3ξ02 . (9.5.14) Tss = (1 −  2 )4π 1/2 na 2 The elastic, unforced limit in this case only makes sense if one takes the limit ξ0 → 0 together with  → 1, so that the expression remains finite in this limit. Similar analyses can be made for other interaction models, and we refer the reader to the literature for the details. We conclude this section by pointing out that a stationary state exists for any value of the forcing parameter, ξ0, and, as we see in 4/3 the nearly elastic case, the steady state grows as ξ0 . High-Energy Tails for Driven Gases We saw previously that at very high energies, the homogeneous cooling solution decays exponentially or algebraically, depending on the interaction model. Here we explore the same issue for a granular gas with the same kind of stochastic forcing as used earlier [659, 661, 514, 660]. We proceed by using the linear approximation, Eq. (9.4.19), for the binary collision operator, and we assume that the distribution function is isotropic in velocity space. Under these assumptions, Eq. (9.5.13) becomes

2 2 d 8πu d Fsc (u) = + Fsc (u). (9.5.15) du2 u du (1 −  2 )h A trial solution of the form Fsc (u) ∼ exp[−AuB ]

(9.5.16)

would have the property of vanishing as u → ∞. When this trial solution is inserted in Eq. (9.5.15), one finds that B = 3/2,

(9.5.17)

382

and

Granular Gases

 1/2 2 8π . A= 3 (1 −  2 )h

Thus, the driven granular gas also has an overpopulation of particles in the highenergy region with a fractional power of the velocity in the exponent. The various approximations made in the Boltzmann collision operator can be justified a posteriori using an argument close to that used for the homogeneous cooling state. 9.6 Planetary Rings The ideas discussed in this chapter provide the basis for an understanding of many of the physical properties of planetary rings, such as spectacular ones about the planet Saturn and those recently discovered by space probes about the three other large planets in the solar system – Jupiter, Neptune, and Uranus.22 The connection between the study of the rings of Saturn and the kinetic theory of gases goes back to the early work of J. C. Maxwell [466], who developed his approach to kinetic theory in least in part in order to explain some properties of Saturn’s rings.23 Since the work of Maxwell, and especially stimulated by results obtained from space probes, the structures and physics of these rings have been the subject of much research [262, 207, 208]. Of special importance to us here is the realization by Maxwell and later workers that planetary rings are composed of particulate matter and that inelastic collisions of the particles with each other are essential for maintaining both the flat structures and the stability of these rings. Although it is a vast oversimplification to ignore the complex structures found in planetary rings, here we shall consider only a few simple features of these rings that can be understood from the ideas described in this chapter, and we will leave further consideration of the many intriguing properties to the experts in the field of astrophysics [601, 498]. We begin by mentioning that there is a particular limiting distance from a planet, called the Roche limit, such that a solid body that, in isolation, would be held together by gravitational forces is instead shredded by the gradient of the gravitational force exerted on the body by the planet [39, 208]. That is, the gravitational force of attraction between any two parts of the body is smaller than the tidal forces exerted by the planet on the body. Thus, any self-gravitating body inside the Roche limit will be broken up into particulate matter, and planetary rings are almost always found within the Roche limit. New rings can be formed from the debris produced when a large body, or small moon, orbiting the planet is destroyed by bombardment from particles it encounters is the space around the planet. The particles in the rings of Saturn are composed mainly of ice particles but other rings may be composed

9.6 Planetary Rings

383

of silicates and other matter similar in composition to that of nearby moons. To get an idea of the size of such rings, we note that the rings of Saturn are about 80,000 kilometers wide and about 100 meters or less thick. They may be among the flattest celestial objects. We imagine that the rings are composed of particles in circular or nearly circular orbits around the central planet. If we were to imagine a cloud of particles close to the planet, the cloud would eventually flatten into a disklike structure, with the angular velocities of the particles decreasing with distance from the planet. The flattening of the cloud into a disk is the result of two effects. One is the equatorial bulging of the planet caused by its rotation about a central axis, leading to a quadrupole contribution to the gravitational field of the planet, which in turn tends to orient the angular momentum of the particles in the direction of the spin axis of the planet and forces the particles around it into a narrow band about equatorial plane. The other process contributing to the flatness is the inelastic collisions of the particles orbiting the planet. To see this, imagine two particles rotating around the planet that collide with each other. They are most likely to collide if the planes of their rotation about the planet make a small angle with each other. Then their orbital velocities around the planet are almost identical, but their relative velocity is essentially perpendicular to the planes of their orbital motion. Due to the inelastic collisions of the particles with each other their relative velocity in the perpendicular direction will be reduced, leaving their orbital velocity around the planet essentially unchanged. After a sufficient number of collisions among nearby particles orbiting the planet, the velocities of particles in the direction perpendicular to the plane of the disk will become exponentially small with the number of collisions, as noted earlier in our study of the cooling state, and the particles will eventually be found in thin rings in the equatorial plane of the planet. While moving more or less in the plane of the disk, the particles in a ring will continue to collide with each other, and the kinetic energy of the colliding pair diminishes by an amount determined by the coefficient of restitution and their relative velocity before collision. We can make some qualitative remarks about the processes taking place in the planetary rings by supposing that the ring is in a stationary state with the energy loss by inelastic collisions is balanced by the energy gained produced by viscous heating due to the differential change in the angular velocity with distance from the planet. This change produces a shear flow leading to viscous heating. We use the Keplerian description of the motion of the particles around the planet. That is, the angular velocity, ωrot , of a particle at a distance r from the center of the planet is ωrot =

GM r3

1/2 ,

(9.6.1)

384

Granular Gases

and the collision frequency of a particle is roughly ωcoll ∼ a 2 nV ,

(9.6.2)

where a is the characteristic size of a particle, n is the number density per unit volume, and V is the magnitude of a typical velocity fluctuation of a particle about the orbital velocity, vrot = ωrot r. The viscous heating occurs because of the transfer of orbital momentum in the radial direction from the inner regions of the ring, which are rotating more quickly, to the outer regions, which are rotating more slowly. Thus, the rate of viscous heating per unit mass, e˙visc , is given by

∂vrot 2 2 ≈ νωrot , (9.6.3) e˙visc = ν ∂r where ν = η/(nm) is the coefficient of kinematic viscosity for the particles in the ring, and η is the coefficient of shear viscosity. The rate of energy loss per unit mass due to inelastic collisions of the particles, e˙coll is roughly e˙coll ≈ (1 −  2 )V 2 ωcoll . The stationary state requires a balance of these two rates, so that 2 ≈ (1 −  2 )V 2 ωcoll . νωrot

(9.6.4)

It is usual in this field to express the collision frequency in terms of the optical depth, given by τo = a 2 σ/m, where σ is the mass density per unit area of the ring, so that nmH = σ, where H is the thickness of the ring. Thus, the optical depth is proportional to the total cross section for the scattering of light by the particles, per unit area of the ring. Thus, ωcoll ∼ τo V /H . One needs to know the velocity distribution function in order to obtain values for the kinematic viscosity, ν, and the average velocity fluctuation, V , which can be obtained by solving a kinetic equation appropriate for the particles in the ring. 9.6.1 Kinetic Theory for Particles in the Rings Many studies of transport processes in planetary rings assume that the particles are inelastic hard spheres, with a distribution function that satisfies the GBE for such particles, Eq. (9.3.4). The coefficient of restitution is assumed to be a decreasing function of the relative velocity of the colliding pair that is determined either experimentally, by studying the collisions of ice particles, or theoretically and numerically, by assuming some model for the elastic properties of ice particles. Important early work on the kinetic theory of ring particles was carried out by P. Goldreich and S. Tremaine [257, 258], using Eq. (9.3.4) and considering the description of

9.6 Planetary Rings

385

the motion in a cylindrical coordinate system, (r,θ,z), with polar coordinates, r,θ, in the plane of the ring, and the cylindrical axis, z, perpendicular to the ring plane. Their work was extended by S. Araki and Tremaine [14], who used the standard Enskog theory, since the mean free path of the particles is on the order of their size. The Enskog theory was used to take into account excluded volume effects and the effects of collisional transfer of momentum and energy on the dynamics of particles in the rings. These authors also included the effects of the roughness of the particles, which allows a transfer of spin angular momentum at binary collisions. An important finite size effect results from the fact that the angular velocities of each of a colliding pair differ due to their slightly different distances from the planetary center. Even if the total angular momentum were conserved in a collision, one particle will lose some angular momentum and fall toward the planet while the particle that gains angular momentum will more further away. This process leads to the spreading of the ring, such that there is a transfer of angular momentum outward and a transfer of mass inward. The application of kinetic theory methods leads to explicit expressions for the spatial and velocity dependence of the distribution function, as well as expressions for the transport coefficients including cooling rates. Of particular importance is determining if the rings are stable under fluctuations or if clusters form, as they would for some cases considered earlier. The kinetic theory results indicate that clusters do not form, but other phenomena, such as density waves and a variety of other instabilities, can occur. Although it is not possible to present the details here, a useful summary of this work can be found in a 2006 paper by F. Spahn and J. Schmidt [603]. A rather clear and useful description of some issues in the kinetic theory of dense planetary rings can be found in the papers of F. Spahn, J.-M. Hertzsch, and N. V. Brilliantov [602], and in the paper by J. T. Jenkins, B. P. Lawney, and J. A. Burns [326]. These authors provide arguments for determining the velocity fluctuations and the volume fraction of particles normal to the plane of the ring. Their results suggest that some rings may be monolayers of particles, driven into collisions by the radial shearing due the planet’s gravitational field. The gap structure of the ring system can be explained to a large extent by considering the perturbation on the motion due to the gravitational fields of the planet’s moons. The effect of these perturbations on the orbital motion of the particles can be described by the Kolmogorov–Arnold– Moser (KAM) theorem from dynamical systems theory, which shows that sections of the ring system, at well-defined distances from the planet, are dynamically unstable. The unstable regions can be then identified with the gaps in the ring system. A very nice analysis of this application of the KAM theorem can be found in a review paper by M. Berry [39].

386

Granular Gases Notes

1 This chapter was written by TRK and JRD. 2 S. Ramaswami has pointed out in a review article that vibrated monolayers of macroscopic grains provide useful examples of active matter [546]. We will say a few more words about active matter in Chapter 15 and provide further references. 3 In order to simplify the descriptions, we will refer to gases composed of particles that undergo inelastic collisions as “granular gases.” 4 As we will see further on, an important instability in a granular gas occurs for some models and not for others. 5 Although they are not discussed here, granular gas mixtures exhibit many interesting properties – such as lack of equipartition among the different components, size segregation due to different masses, sizes, or restitution coefficients including the “Brazil nut effect.” See, for example, [25, 665, 242]. 6 In this chapter, we will use the notation v∗1,v∗2 to indicate the velocities that result from a collision of particles with velocities v1,v2 . We will use the notation v1,v2 to denote the restituting velocities needed to produce v1,v2 after a collision. Occasionally, when there is no danger of confusion, we will use primes to denote both resulting and restituting collisions. See, for example, Eq. (9.3.5). 7 One can imagine that collisions produce sound waves that are radiated into the surrounding space, or that energy is transferred to internal degrees of freedom of the particles, producing a decrease of kinetic energy, for example. 8 We are about to specialize our discussion to the case of inelastic hard spheres. For that reason, ˆ as was done in our first discussion of the we denote the apse line vector by σˆ rather than by k, Boltzmann equation. 9 A “super ball” can be thought of a having a very high value for n, close to 1, and a value for t very close to −1. The compliance of the material of the ball is also essential for its functioning. It is not simply an inelastic hard sphere. See [241]. 10 We follow the analysis of A. Barrat, E. Trizac, and M. H. Ernst [25]. 11 Barrat et al. [25] have demonstrated that if σ = 1, there will be pre-collisional correlations which emphasize grazing or head-on collisions between colliding particles, in contrast to the assumptions made when applying the Stosszahlansatz. 12 Extensions to Haff’s law for longer times have been given by M. H. Ernst and R. Brito [68]. 13 In general, it is not useful to represent a power law decay in time – say, by an exponential decay by changing the time scale. Here it is useful to do so, since the number of collisions in a certain interval of time has some physical meaning for such systems. 14 To get the given form, just use mv 2 /(2kB Tg ) = mvT2 u2 /(2kB Tg ) = u2 . 15 This can be substantiated by more careful arguments, as given in the paper by T. van Noije and M. H. Ernst [659]. 16 Although the decay is algebraic in real time. 17 See, for example, fig. 26.2 of the book of Brilliantov and Pöschel [66] and references therein. 18 A clear review paper of mathematical methods used to describe granular materials, with an extensive bibliography, is written by C. Villani [666]. 19 This is for three-dimensional systems. In d dimensions, we would look for behavior like k b−d . 20 We follow the method of M. H. Ernst and R. Brito [181]. 21 A very nice early (1982) study of a gas of particles falling under gravity and colliding inelastically with a fixed set of larger particles was carried out by D. R. Wilkinson and S. F. Edwards [681]. 22 Recent reviews of these findings can be found in the papers by M. S. Tiscareno [621] and that by S. Charnoz, L. Dones, L. W. Esposito, P. R. Estrada, and M. Hedman [91]. 23 This work of Maxwell is covered in some detail in the book [73]. See also G. I. Ogilvie [505].

10 Quantum Gases

10.1 Introduction1 Much of this book is devoted to a presentation of the kinetic theory of classical gases. Such gases are assumed to be composed of distinguishable particles with a continuous energy spectrum, interacting with short-ranged central potentials, and the spin or angular momentum state of the particles does not need to be taken into account. However, when the de Broglie wavelength of the particles is larger than their average separation, the particles are no longer distinguishable and a quantum treatment is required. Such a treatment would require a specification of the type of statistics that the particles obey, i.e. Fermi–Dirac (FD) or Bose–Einstein (BE) statistics or, if the system is multicomponent, the statistics of each species. Often, especially at low temperatures, the discreteness of the energy spectrum of the particles must be taken into account.2 A rough estimate of the temperature below which the particles become indistinguishable can be found quite easily. The thermal de Broglie wavelength, λq , of particles of mass m at temperature T is given by λq =

h2 β m

1/2 .

(10.1.1)

The condition that quantum statistics be taken into account, for a gas with number density n, is that nλ3q ≥ 1, since the average inter-particle spacing is on the order of n−1/3 . We consider dilute quantum gases for which na 3 1, and a/λq 1, where a is a typical size of the particles, often taken to be the s-wave scattering length. For electrons in a metal, the temperature below which the FD statistics must be taken into account can be on the order of a hundred thousand degrees Kelvin, while for helium atoms, with BE or FD statistics depending on the isotope, this temperature is on the order of a few degrees Kelvin. In any case, a quantum description of the collisions of the particles is required whenever the particles become indistinguishable. We note that there are dimensionless quantities connected with the quantum region, 387

388

Quantum Gases

which satisfy the chain of inequalities nλ3q ≥ nλ2q a ≥ nλq a 2 ≥ na 3 , for a dilute quantum gas. The dimensionless quantity nλ3q is called the degeneracy parameter. In this chapter, we first discuss the general formalism for distribution functions for quantum mechanical systems. The quantum equivalent of the classical phasespace distribution function was first defined by E. Wigner [680], and we show that this function satisfies a Liouville-like equation. We then present the quantum Boltzmann equation, the Uehling–Uhlenbeck (U–U) equation [360, 630, 331], that applies to dilute gases of particles interacting with short-range forces, under conditions where the particle statistics must be taken into account. The U–U equation will appear in various forms in all of the discussions, so we first present it as a phenomenological equation and describe its properties. We show that the U–U equation has an entropy function that satisfies an H -theorem; it has equilibrium solutions that are the ideal quantum gas distribution functions appropriate for FD or BE statistics. We also mention that the Chapman–Enskog, normal solution method can be applied to the U–U equation leading to the usual Navier–Stokes equations with modified expressions for the coefficients of shear viscosity and thermal conductivity. Next we consider a topic that of considerable interest, the theory for dilute Bose gases at temperatures below the superfluid phase transition point [115, 430, 444, 431, 524]. Since the major experimental demonstration by several groups that condensed, dilute BE gases can be produced and studied in the laboratory [13, 118], there has been much theoretical work on the properties of these systems. Here we will discuss the kinetic theory of weakly interacting BE gases in the vicinity of and below the critical point for the phase transition [697, 115, 444, 524, 278, 537]. Of particular interest is the derivation of the Landau–Khalatnikov two-fluid equations for the coexisting superfluid and normal fluid [623, 622, 624, 411, 412, 359, 545]. The hydrodynamic equations for this two-fluid system are considerably more complex than the usual Navier–Stokes equations and have, among other properties, a hydrodynamic equation for the superfluid and a total of six transport coefficients. Apart from the coefficients of thermal conduction and shear viscosity, there are four bulk viscosity coefficients. Although we will not discuss it here due to the space needed for an adequate presentation, the quantum version of the Lorentz gas of non-interacting electrons moving at low temperatures in an amorphous solid is of considerable importance for solid-state systems [381, 29, 689, 690, 370]. Among other reasons, it is important because the phenomenon of localization is possible for such a system, leading, in three dimensions, to a metal–insulator phase transition, the Anderson phase transition [422, 1, 3]. We will make some remarks on the quantum Lorentz gas in Chapter 13 when we discuss long-time-tail phenomena. We assume that the reader has a grasp of quantum mechanics and is familiar with density matrices and at least the rudiments of the properties of quantum field

10.2 Density Matrices and the Wigner Function

389

operators, especially of boson operators. All of this material can be found in any modern text on quantum mechanics [570]. 10.2 Density Matrices and the Wigner Function Since quantum theory is a description of nature very different from that provided by classical mechanics, many of the ideas of classical statistical mechanics must be adapted or even abandoned when constructing a statistical mechanics of ensembles of quantum systems. The notions of phase-space and space-space trajectories for a classical Hamiltonian system are of course not appropriate for quantum ensembles. Instead quantum ensembles are properly described by density matrices. A typical quantum ensemble may be described by a density matrix, ρ, expressed in terms of a complete set of quantum states |i > by  pi |i,t >< i,t|. (10.2.1) ρ(t) = i

Here the quantity pi is the probability that a member of the ensemble will be found in state |i > at the initial time. The states |i > are assumed to form a complete, orthonormal set. The density matrix is normalized so that its trace is unity, i.e.  pi = 1. (10.2.2) T rρ = i

The time-dependent states, |i,t >, are expressed in terms of the basic states, |i >, and the Hamiltonian operator for the system, H, assumed here to be independent of time. Thus, |i,t >= e−itH/h¯ |i > ,

(10.2.3)

so that the time dependent density matrix is given by ρ(t) = e−itH/h¯ ρ(0)eitH/h¯ ,

(10.2.4)

or, equivalently, the time derivative of ρ(t) satisfies i i ∂ ρ(t) = − (Hρ(t) − ρ(t)H) = − [H,ρ(t)], ∂t h¯ h¯

(10.2.5)

which is known as the von Neumann equation for the density matrix, and it plays a role analogous to the Liouville equation in classical statistical mechanics. Finally, the ensemble average at time t of some operator A is given by < A(t) >= T rρ(t)A.

(10.2.6)

390

Quantum Gases

We suppose that our system is composed of N particles, so that the states |i,t > describe quantum states for the N-particle system and that the Hamiltonian operator H is the quantum version of the classical Hamiltonian function HN(cl) =

N N   pi2 + φ(|r i − r j |). 2m it = dx (10.2.11) provided that the Weyl representation of the position and momentum operators appearing in A(xN ,p N ) is used.3

10.3 The Uehling–Uhlenbeck Equation

391

The Wigner function also satisfies a quantum version of the Liouville equation, which is  ∂fW  pi θij fW = 0, + · ∇r i f W + ∂t m i. We will take the thermodynamic limit  → ∞, < N >→ ∞, < N > /  = n, ˆ where n is the number density of the gas.10 The density matrix, ρ(t), satisfies the equation

with formal solution

ˆ −i ˆ d ρ(t) ˆ = [H, ρ(t)], dt h¯

(10.4.7)



i ˆ i ˆ ˆ ˆ exp Ht . ρ(t) = exp − Ht ρ(0) h¯ h¯

(10.4.8)

The first important question about this system is: does the gas undergo a BoseEinstein phase transition, and if so, how is it related to the superfluid phase transition and what is the transition temperature? It is, of course, well known that an ideal Bose gas composed of particles of mass m at number density n undergoes a phase transition at the temperature, T0 , where  2/3 2π n , (10.4.9) T0 = m ζ (3/2) where ζ (3/2) is the Riemann zeta function with argument 3/2. Once interactions are introduced the calculation of the critical temperature becomes much harder, and the literature for this calculation is quite extensive [695]. Here we only mention some recent results. It is convenient to expand the deviation from the ideal gas temperature, T0 , in terms of the dimensionless gas parameter, γ = n1/3 a. Then the change in the critical temperature due to interactions, for small γ , is given by [16] Tc − T0 ≈ c1 γ + (c2 + c2 ln γ )γ 2 + o(γ 2 ). T0

(10.4.10)

The coefficient c2 is known exactly, but only approximate values for the other coefficients, c1,c2 , are available. The higher terms in this expansion are not known.11 Our interest here is in the nonequilibrium, superfluid behavior of the dilute Bose gas; that is, we wish to determine the hydrodynamic behavior of an inhomogeneous Bose gas for very low temperatures. For a gas at temperatures sufficiently above the Bose–Einstein phase transition temperature, the transport properties can be described by the boson form of the Uehling–Uhlenbeck equation, discussed in previous sections.

402

Quantum Gases

The condensate, the Superfluid, and the Normal Fluid The behavior of the gas at absolute zero is, like that of many quantum systems, highly nontrivial. In contrast to the case of an ideal Bose gas, the particles are not all in the single-particle ground state. Physically, one can say that for an interacting system, the many-particle ground state is not uniform due to, for example, excluded volume effects that prevent particles from occupying the same or very nearby positions in real space. We will soon calculate the depletion of the ground state, namely the fraction of the fluid at T = 0 that consists of single-particle excitations above the non-interacting ground state. These single-particle excitations plus the condensate comprise the superfluid.12 At somewhat higher temperatures, the fluid is a mixture of superfluid and thermal excitations, which compose the normal fluid. At T = 0, the fluid is entirely superfluid, but at higher temperatures but below the λ-point the fluid is a mixture of normal and superfluids, while at the λ-point the superfluid component no longer exists, except for local fluctuations. We will see that the excitation spectrum describes phonon-like, or quasiparticle, excitations at low momenta changing to particle-like at high momenta. The nonequilibrium behavior of the system at very low temperatures requires a distribution function for the thermal excitations and the change of this distribution function caused by quasiparticle–condensate and quasiparticle–quasiparticle interactions. Once the proper kinetic equation and the equation for the superfluid velocity have been obtained, it will be possible to derive the Landau–Khalatnikov two-fluid hydrodynamic equations, in much the same way as one derives the Navier–Stokes equations from the Boltzmann equation for a classical gas. However, due to the presence of the condensate and the existence of a broken gauge symmetry for the gas, to be discussed later, the conservation laws upon which the equations of hydrodynamics are based have a somewhat different form than those encountered so far. The derivations of the kinetic and hydrodynamic equations given later in this chapter are somewhat long. We will skip some of the steps and use physical intuition to arrive at the desired kinetic equation. More detailed presentations can be found elsewhere [385, 382, 384]. The Condensate Wave Function We begin with the observation that if the condensate is macroscopically occupied, fluctuations in the number of particles in the condensate are small, and there is a macroscopic wave function, ψ0 (r,t), of the form ψ0 (r,t) = n1/2 c (r,t) exp[iφ(r,t)],

(10.4.11)

where nc is the number density of the condensate, and φ(r,t) is its phase.13 We can immediately anticipate the presence of a Goldstone mode. Such modes are

10.4 Transport in a Condensed, Dilute Bose Gas

403

low-level excitations that result from a broken symmetry. In this case, there is a U (1) gauge symmetry of the phase angle that is broken whenever the phase angle varies in space.14 This variation is characterized by its gradient, ∇φ, and from this, one can construct a function with the dimensions of a velocity, denoted by Vs = h¯ ∇φ/m.

(10.4.12)

Later on, we will identify this velocity with the velocity of the superfluid. We should expect to find a new excitation involving this velocity, arising from the broken gauge symmetry – that is, in equilibrium Vs = 0, due to the constant phase angle that can take on any value without any changes in the description of the system, while for small deviations from equilibrium, the gradient of the phase angle should be small and should be included in the description of low-lying or long-wavelength excitations. That, in turn, suggests that in addition to the usual hydrodynamic fields describing mass or particle density, momentum density, and energy density, there must be an additional hydrodynamic variable related to the phase angle. Thus, we can expect six hydrodynamic equations instead of the usual five, and in addition to the coefficients of shear viscosity and thermal conductivity, there are four coefficients of bulk viscosity. The Field Theory for Bogoliubov Excitations We begin our description of the boson gas by introducing the Fourier transforms of the field operators and using them to describe the excitations of the boson gas, namely the Bogoliubov excitations. We will eventually use a local equilibrium version of the transformation of the field operators, presented here, to operators for the Bogoliubov excitations. The theory for these excitations was first developed by N. N. Bogoliubov in 1947 [54, 2, 359].15 We introduce the Fourier transforms of the field operators, ψˆ and ψˆ † , in Eq. (10.4.1) for the Hamiltonian operator. Their Fourier transforms for systems in a volume  are given by  ˆ eik·r aˆ k, (10.4.13) ψ(r) = −1/2 k

ψˆ † (r) = −1/2



e−ik·r aˆ †k .

(10.4.14)

k

The boson commutation relations for the field operators aˆ k, aˆ †k are [ˆak1 , aˆ k2 ] = 0, [ˆa†k1 , aˆ †k2 ] = 0, [ˆak1 , aˆ †k2 ] = δKr (k1 − k2 ).

(10.4.15)

Since we are using Fourier sums, we use Kronecker deltas, δKr , in the commutation relations. We insert the field operators into the expression for the non-trapping part

404

Quantum Gases

ˆ 0 + V, ˆ and use the pseudo-potential form of the interaction of the Hamiltonian, H potential, Eq. (10.4.6), to obtain  † U0     ˆ = H k ak ak + δKr (k1 + k2 − k3 − k4 )ˆa†k1 aˆ †k2 aˆ k3 aˆ k4 , 2 k k k k k 1

2

3

4

(10.4.16) where k = h¯ 2 k 2 /2m. The single-particle ground state contributions are those at zero wave number, a0, and we take advantage of the fact that at very low temperatures, the ground state is occupied by a macroscopic number of particles. We then treat the zero wave number contributions separately to write  U0 † †  ˆ = aˆ aˆ aˆ 0 aˆ 0 k aˆ †k aˆ k + H 2 0 0 k + +

U0   † † [4ˆa0 aˆ k aˆ 0 aˆ −k + aˆ †k aˆ †−k aˆ 0 aˆ 0 + aˆ †0 aˆ †0 aˆ k aˆ −k ] 2 k

U0       [δKr (k1 − k2 − k3 )ˆa†0 aˆ †k1 aˆ k2 aˆ k3  k k k 1

2

3

+ δKr (k1 + k2 − k3 )ˆa†k1 aˆ †k2 aˆ k3 aˆ 0 ] U0         δKr (k1 + k2 − k3 − k4 ) + 2 k k k k 1

×

2

3

4

aˆ †k1 aˆ †k2 aˆ k3 aˆ k4 .

(10.4.17)

Here the primes on each sum indicate that the value ki = 0 is not included. The commutation relations, Eq. (10.4.15), were used to obtain its expression. In the approximation where we can neglect number fluctuations, we can replace the number operator Nˆ with N = N, where N is the total number of particles. Due to the macroscopic occupation of the ground state, we may replace each of the field operators aˆ †0, aˆ 0 by c-numbers, which, for our purposes here, may be taken to be 1/2 N0 for combinations of two operators whenever the phase angles cancel, and N0 is the number of particles in the ground state. In fact, we may write  †  (10.4.18) aˆ k aˆ k = N. aˆ †0 aˆ 0 + k

That allows us to replace aˆ †0 aˆ 0 with N0 and aˆ †0 aˆ †0 aˆ 0 aˆ 0 in Eq. (10.4.17) by N02 and then write  2  † U0 2 U0   † U0 2  N = N − U0 n (10.4.19) aˆ k aˆ k + aˆ aˆ k . 2 0 2 2 k k k

10.4 Transport in a Condensed, Dilute Bose Gas

405

This replacement leads to some simplification of the above expression for the ˆ q , which is up to quadratic Hamiltonian. We focus on the part of the Hamiltonian, H order in the non-condensate field operators. Explicitly,   U0 n    †  ˆ q = U0 N02 + k aˆ †k aˆ k + 4ˆak aˆ k + aˆ †k aˆ †−k + aˆ k aˆ −k . H 2 2 k k

(10.4.20)

When we eliminate the condensate number of particles, N0, in favor of the total number of particles, using Eq. (10.4.19), we obtain a quadratic Hamiltonian, which in a symmetrized form is   1 1 † † † †  ˆq = [k + U0 n](ˆak aˆ k + aˆ −k aˆ −k ) + nU0 (ˆak aˆ −k + aˆ k aˆ −k ) . H 2 2 k (10.4.21) Note that now all of the field operators refer to excitations above the ground state of the gas. The quadratic part of the Hamiltonian can be diagonalized by means of the Bogoliubov transformation of the field operators. so that it can be written in standard form as   ˆq = E(k)bˆ †k bˆ k, (10.4.22) H k

where the E(k) are the eigenvalues of the quadratic form [2, 359]. Such a diagonalization is appropriate at very low temperatures where the ground state is macroscopically occupied, and the contributions of the cubic and quartic terms in the field operators may be treated as perturbations of the quadratic terms. The field operators, bˆ k, bˆ †k that diagonalize Hˆ q are the excitations of the fluid, which we take to be boson operators satisfying the usual commutation rules. These field operators are defined by writing aˆ k = uk bˆ k − vk bˆ †−k, aˆ †k = uk bˆ †k − vk bˆ −k .

(10.4.23)

The quantities uk,vk will be taken to be real. We require that the boson commutation rules lead to u2k − vk2 = 1.

(10.4.24)

The diagonalization of the quadratic part of the Hamiltonian is fixed by the requirements that the coefficients of the non-number operators, bˆ k bˆ −k and bˆ †k bˆ †−k , vanish

406

Quantum Gases

when the quadratic terms in the Hamiltonian are expressed in terms of the field operators for the excitations. This requirement leads to the condition that nU0 (u2k + vk2 ) − 2(k + nU0 )uk vk = 0,

(10.4.25)

in which case the Hamiltonian Hˆ q becomes   ˆq = 1 [(k + nU0 )(u2k + vk2 ) − 2nU0 uk vk ] × (bˆ †k bˆ k + bˆ †−k bˆ −k ). (10.4.26) H 2 k The energy of the excitations, E(k), is then E(k) = (k + nU0 )(u2k + vk2 ) − 2nU0 uk vk .

(10.4.27)

One can easily see that the expressions uk = cosh γk, vk = sinh γk will satisfy Eq. (10.4.24), and that the quantity γk is the solution of the equation nU0 . (10.4.28) k + nU0 The energy of these excitations is obtained by solving this equation for γk , and we find that the energy of an excitation at wave number k is tanh 2γk =

E(k) = [k2 + 2nU0 k ]1/2 .

(10.4.29)

For small values of the wave number, the modes are sound-like with frequency ω ≈ ck, and speed of sound, c = (nc U0 /m)1/2 . At large wave numbers, the modes are particle-like with energy k . Depletion of the Ground State We can use these results to describe the depletion of the ground state mentioned earlier. For that, we need to evaluate the the expression for the number of particles, not in the ground state. The total particle number operator is given by Eq. (10.4.18), which can be expressed in terms of the field operators for the Bogoliubov excitations. When expressed in terms of densities, we obtain 1  2 1  2 1  n = nc + vk + (uk + vk2 )bˆ †k bˆ k − uk vk (b†k b†−k + bk b−k ),  k  k  k (10.4.30) where we have used the boson commutation relations for the field operators for the excitations. We can interpret this result in a very simple way. The sum of the first two terms on the right-hand side of Eq. (10.4.30) represents the density of the superfluid, and the sum of the last two terms is the density of the quasiparticle excitations. For states with a fixed number of excitations, this interpretation becomes clearer since the expectation value of the operators bˆ †k bˆ †−k + bˆ k bˆ −k vanishes and we may write effectively that

10.4 Transport in a Condensed, Dilute Bose Gas

1  2 1  2 vk + (uk + vk2 )bˆ †k bˆ k  k  k



1   k + nU0 1   k + nU0 ˆ † ˆ = nc + −1 + bk b k . 2 k E(k)  k E(k)

407

n = nc +

(10.4.31)

The density of single-particle excitations at T = 0 due to the repulsive interactions of the particles is then given by

 1/2 1  2 1   1 k + nU0 − 1 ∼ na 3 vk = . (10.4.32) n − nc =  k  k 2 E(k) The sum on the right-hand side of Eq. (10.4.32) can be evaluated by converting it to an integral, with the result showing that the depletion of the zero momentum state of a weakly interacting, low-density boson gas is very small but not zero. We mention that at T = 0; the normal fluid density vanishes and the total density is superfluid. If one ignores the contributions of terms in the Hamiltonian that are cubic and quartic in powers of the field operators, the Hamiltonian operator for the excitations, Eq. (10.4.22), is that of an ideal Bose gas with bosons that are non-interacting Bogoliubov excitations. The higher-order terms in the field operators then describe phenomena where the excitations interact with the condensate or with each other and are needed to derive the Boltzmann equation for the distribution function of the excitations. 10.4.2 The Equations for Bogoliubov Excitations and Superfluid Velocity We turn to the case of interest, an inhomogeneous Bose gas far below its condensation, or λ-point temperature, when the ground state is macroscopically occupied. The macroscopic quantities, such as the condensate density, the velocity of the condensate, and related quantities will be treated as local quantities, varying with position and time on scales large compared to the relevant microscopic scales. As usual, the density operator, n(r), ˆ and current operator, ˆj(r), are defined by ˆ n(r) ˆ = ψˆ † (r)ψ(r),

(10.4.33)

ˆj(r) = i h¯ [(∇ ψˆ † )ψˆ − ψˆ † ∇ ψ]. ˆ 2

(10.4.34)

The time-dependent density matrix is given by Eq. (10.4.8). To describe the condensate with a macroscopic occupation of particles at temperatures below the λ-point and to select a small range of values for the phase angle, we follow the procedure described by P. C. Hohenberg and P. Martin [310] that

408

Quantum Gases

involves the use of a restricted ensemble, the η-ensemble, which can be obtained by adding a fictitious external source to the Hamiltonian of the form  ˆ ˆ Hη = dr[η(r,t)ψˆ † (r,t) + η∗ (r,t)ψ(r,t)]. (10.4.35) The external source η(r,t) selects values for the phase of the condensate wave function to lie within a small range about an average value and produces a condensate where the fluctuations about a mean condensate density are small. That is to say, in this ensemble, the average of the quantum field operators are non-zero due to the selection of nearby phases for all systems in the ensemble. Further,  the average  1/2 1/2 value of the fluctuations (ψˆ − nc exp(iφ))(ψˆ † − nc exp(−iφ)) is small and contains no contribution from the condensate [310].16 In the η-ensemble, the nonvanishing of the average values of the field operators, ˆ ψ, ψˆ † , is an indication of the presence of a condensate. The averages are taken ˆ and the averages of the field operators can be with respect to the density matrix ρ, 17 written in the form given in Eq. (10.4.11). That is, we assume that the expectation value of the field operator is a macroscopic wave function iφ(r,t) ˆ = n1/2 , < ψˆ > = T r(ρˆ ψ) c (r,t)e −iφ(r,t) . < ψˆ † > = T r(ρˆ ψˆ † ) = n1/2 c (r,t)e

(10.4.36) (10.4.37)

Here nc (r,t) is the space- and time-dependent density of the condensate, and φ(r,t) is a phase, taken to be real. This phase plays an important role in our considerations because its gradient will be shown to be proportional to the average velocity of the superfluid, Vs (r,t) = h¯ ∇φ/m. The construction of the distribution function for the excitations will be done through a three-step process: (1) We apply a unitary transformation to change the coordinate system to one in which the superfluid is at rest. This is a convenient transformation since the quasiparticle excitations are most easily described in this frame [2]. (2) The second transformation replaces the boson field operators in the Hamiltonian by new operators that have no contribution form the condensate. We assume that for a slightly inhomogeneous boson gas, the condensate field operators can be replaced by c-numbers. Then each field operator can be written as the sum of a c-number and a non-condensate field operator, and the second transformation removes the condensate contribution. (3) We apply a third unitary transformation, which is a local Bogoliubov transformation that changes the particle operators to operators for the quasiparticle excitations. The three transformations can be done ‘by hand’. When these transformations are carried out, we construct the Wigner distribution function for the quasiparticle excitations that we will use together with the equation for the superfluid velocity to derive the LK hydrodynamic equations.

10.4 Transport in a Condensed, Dilute Bose Gas

409

10.4.3 Step 1: Change of Reference Frame We first move to a frame in which the phases, ±iφ(r,t), of the average field operators vanish, so that the local superfluid velocity is zero [310, 522]. This implies that our equations will describe the gas in the local rest frame of the superfluid and simplify some of the calculations to follow. To remove the dependence of the averages of the field operators, we introduce unitary operator Uˆ 1 (φ(t)) by    † ˆ ˆ ˆ (10.4.38) U1 (φ(t)) = exp −i drφ(r,t)ψ (r)ψ(r) , and its conjugate

   ˆ . Uˆ 1† (φ(t)) = exp i drφ(r,t)ψˆ † (r)ψ(r)

(10.4.39)

We can use these operators to transform the density matrix ρˆ to a new one, ρˆ1 , given by ρˆ 1 = Uˆ 1 (φ(t))ρˆ Uˆ 1† (φ(t)).

(10.4.40)

Under the action of this unitary operator, the field operators transform as18 Uˆ 1† ψˆ Uˆ1 = ψˆ exp[−iφ(r,t)], Uˆ 1† ψˆ † Uˆ1

= ψˆ † exp[iφ(r,t)].

(10.4.41) (10.4.42)

If we use the transformed density matrix to compute the mean value of the field ˆ for example, we find operator, ψ, < ψˆ >1 = T r ψˆ ρˆ1 = T r ψˆ Uˆ 1 (φ(t))ρˆ Uˆ 1† (φ(t)) = T r Uˆ 1† ψˆ Uˆ 1 ρˆ = n1/2 c (r,t).

(10.4.43)

Similarly, the transformed current is expressed in a frame at rest with respect to the condensate, < ˆj(r,t) >1 = T r ˆj(r)ρˆ 1 = J(r,t) − mn(r,t)Vs (r,t).

(10.4.44)

Here n(r,t) = T r ρˆ n(r) ˆ is the local number density of the fluid, and ˆj(r) in T r ˆj(r)ρˆ is the momentum current operator in the lab frame. The transformed distribution function satisfies the the von Neumann equation, Eq. (10.2.5), with an adjusted Hamiltonian operator ∂ ρˆ ∂ ρˆ 1 ∂ Uˆ 1 ˆ † ∂ Uˆ † = ρˆ U1 + Uˆ 1 Uˆ 1† + Uˆ 1 ρˆ 1 ∂t ∂t ∂t ∂t −i ˆ = [H1, ρˆ 1 (t)], h¯

(10.4.45)

410

Quantum Gases

where the new Hamiltonian operator is ˆ ˆ Uˆ † + i h¯ ∂ U1 Uˆ † . ˆ 1 = Uˆ 1 H H 1 ∂t 1

(10.4.46)

We used the fact that Uˆ 1 is a unitary operator, which implies that ∂ Uˆ 1 ˆ † ∂ Uˆ † U1 + Uˆ 1 1 = 0. ∂t ∂t When the expression for Hˆ 1 is evaluated, we obtain   ˆ + drVs (r,t) · ˆj(r) − m drh(r,t)n(r). ˆ1 =H ˆ H

(10.4.47)

Here h(r,t) is given by h(r,t) = −

Vs2 (r,t) h¯ ∂φ(r,t) − . 2 m ∂t

(10.4.48)

One can easily see that the time derivative of the phase factor φ(r,t) appearing in Eq. (10.4.48) comes from the second term on the right-hand side of Eq. (10.4.46), while the terms depending on the local superfluid velocity, Vs (r,t), result from the ˆ 0 . By taking the spatial graevaluation of the transformed kinetic energy operator, H dient of the expression for h(r,t) and using Vs = h¯ ∇φ/m, we obtain an equation for the time derivative of the condensate velocity, namely   Vs2 (r,t) ∂Vs = −∇ h(r,t) + . (10.4.49) ∂t 2 We will see presently that h(r,t) can be identified with the Euler equation value nonequilibrium chemical potential [359]. We can also obtain an expression for the time derivative of the condensate density by combining Eqs. (10.4.12), (10.4.45), and (10.4.47) with the facts that ˆ 1 ψˆ n1/2 c = ReT r ρ

(10.4.50)

ˆ 0 = I mT r ρˆ 1 ψ.

(10.4.51)

and

10.4.4 Step 2: Removal of the Condensate Now we perform another transformation that is designed to remove the contribution of the condensate to the field operators [310, 522]. Recall that we are working in a restricted ensemble where the fluctuations in the value of the local condensate

10.4 Transport in a Condensed, Dilute Bose Gas

411

density are small and in a local frame where the superfluid velocity is zero. Under these circumstances, we replace the contribution of the condensate to the field 1/2 operators by a c-number, namely the square root of the local density, nc (r,t). The removal of the condensate is accomplished by introducing a new unitary transformation,   † 1/2 ˆ ˆ ˆ dr[ψ(r) − ψ (r)]nc (r,t) , (10.4.52) U2 [nc (t)] = exp and its conjugate,    † † 1/2 ˆ ˆ ˆ U2 [nc (t)] = exp − dr[ψ(r) − ψ (r)]nc (r,t) .

(10.4.53)

From these definitions, it follows that ˆ Uˆ 2 [nc (t)] = ψ(r) ˆ Uˆ 2† [nc (t)]ψ(r) − n1/2 c (t).

(10.4.54)

The transformed density matrix, ρˆ2 (2), defined by ρˆ 2 (t) = Uˆ 2 [nc (t)]ρˆ 1 (t)Uˆ 2† [nc (t)] = Uˆ 2 [nc (t)]Uˆ 1 (φ(t))ρˆ Uˆ 1† (φ(t))Uˆ 2† [nc (t)],

(10.4.55)

satisfies the von Neumann equation −i ˆ d ρˆ 2 (t) = [H2, ρˆ 2 (t)], dt h¯

(10.4.56)

where the transformed Hamiltonian operator is ˆ ˆ 1 Uˆ † [nc (t)] + i h¯ ∂ U2 Uˆ† . ˆ 2 = Uˆ 2 [nc (t)]H H 2 ∂t 2

(10.4.57)

The explicit form of this Hamiltonian operator can be obtained after some straightforward but lengthy manipulations of the various field operators, and here we give only the final result. It is   ˆ ˆ ˆ ˆ H2 = H + drVs (r,t)·j (r) − m drh2 (r,t)n(r)   ˆ ˆ − ψˆ † (r)] − drLh (r,t)[ψ(r) + ψˆ † (r)] + i h¯ drLnc (r,t)[ψ(r)  1 ˆ ˆ ψ(r)] ˆ + ψ(r) + U0 drnc (r,t) × [ψˆ † (r)ψˆ † (r) + 2ψˆ † (r)ψ(r) 2  ˆ† ˆ† ˆ ˆ† ˆ ˆ (10.4.58) + U0 drn1/2 c (r,t)[ψ (r)ψ (r)ψ(r) + ψ (r)ψ(r)ψ(r)].

412

Quantum Gases

Here the coefficient U0 is defined in Eq. (10.4.6) for the interaction potential, and the quantities h2 and Lh are defined by h2 (r,t) = h(r,t) − U0 nc (r,t) and

Lh =

mn1/2 c h2 (r,t)

+

h¯ 2 ∇ 2 n1/2 c (r,t). 2m

(10.4.59)

(10.4.60)

Digression on Time-Dependent Averages In the following analysis, we will need to evaluate averages of products of field operators taken with respect to time-dependent density matrices, such as ρˆ 2 (t). It is quite simple to express the same averages as being evaluated with respect to the ˆ initial density matrix, ρ(0), but with time-dependent, transformed field operators [385]. As a general example, consider ˆ ψˆ † ) =< Fˆ (ψ, ˆ ψˆ † ) >ρˆ , T r ρˆ 2 (t)Fˆ (ψ, 2 where Fˆ is some algebraic function of products of the field operators. This average can be transformed by means of the following steps, starting with Eq. (10.4.55), ˆ ψˆ † )Uˆ 2 Uˆ 1 ˆ ˆ Uˆ 1† Uˆ 2† Fˆ (ψ, < Fˆ (ψ(r), ψˆ † (r  )) >ρˆ2 (t) = T r ρ(t) † ˆ† ˆ† ˆ ˆ ˆ† ˆ ˆ ˆ ˆ = T r ρ(0)S t U1 U2 F (ψ, ψ )U2 U1 St

ˆ Fˆ (χˆ (r,t|r,t), χˆ † (r ,t|r ,t)). = T r ρ(0)

(10.4.61)

  ˆ is the time-displacement operator with respect to the Here Sˆt = exp − hi¯ Ht Hamiltonian, Eq. (10.4.1). We have used the definitions and properties of the uniˆ χˆ † by tary operators, Uˆ 1, Uˆ 2, to define the field operators χ, χˆ (r,t|r,t) = Sˆt† Uˆ † 1 Uˆ † 2 ψˆ Uˆ 2 Uˆ 1 Sˆt 1/2 ˆ = ψ(r,t) exp[−iφ(r,t)] − nc(t) (r),

(10.4.62)

χˆ † (r,t|r,t) = Sˆt† Uˆ † 1 Uˆ † 2 ψˆ † Uˆ 2 Uˆ 1 Sˆt = ψˆ † (r,t) exp[iφ(r,t)] − nc(t) (r). 1/2

(10.4.63)

ˆ can be expressed as a sum of condenWe now suppose that the field operators, ψ, ˆ sate part, denoted by !, and a non-condensate part denoted by ψˆ  (r,t) so that ˆ ˆ ψ(r,t) = ψˆ  (r,t) + !(r,t),

(10.4.64)

and similarly for the adjoint operator. Further, in view of Eq. (10.4.36), we have 1/2 ˆ < !(r,t) >= nc(t) (r) exp[iφ(r,t)]

(10.4.65)

10.4 Transport in a Condensed, Dilute Bose Gas

413

and < ψˆ  (r,t) >= 0.

(10.4.66)

A further simplification is possible for temperatures below the transition point, when the ground state is macroscopically occupied, if we assume that in the restricted ensemble and for large systems at low temperatures, the condensate field operator can be replaced by its average value.19 Under these circumstances, we can express the field operators χ, ˆ χˆ † as 

χˆ (r,t) = ψˆ (r,t) exp[−iφ(r,t)], χˆ † (r,t) = ψˆ † (r,t) exp[iφ(r,t)].

(10.4.67) (10.4.68)

These field operators depend on the properties of the condensate only through the phase factor, φ(r,t). These transformations will allow us to express the dynamical properties of the system in terms of the non-condensate field operators, ψˆ , ψˆ † . To ˆ ψˆ † ) >ρˆ2 . Using Eq. (10.4.56), we see this, consider the time derivative of < Fˆ (ψ, can write ˆ ψˆ † ) >ρˆ2 (t) d < Fˆ (ψ, i ˆ 2, ρˆ 2 (t)]Fˆ (ψ, ˆ ψˆ † ) = − T r[H dt h¯ i ˆ 2, Uˆ 2 Uˆ 1 ρˆ Uˆ † Uˆ † ]Fˆ (ψ, ˆ ψˆ † ) = − T r[H 1 2 h¯ i ˆ 2 ]Uˆ 2 Uˆ 1 ) ˆ H ˆ = − T r ρ(t)( Uˆ 1† Uˆ 2† [F, h¯ i ˆ 2 ]Uˆ 2 Uˆ 1 Sˆt ) ˆ H ˆ − − T r ρ(0)( Sˆt† Uˆ 1† Uˆ 2† [F, h¯ i ˆ 2 (χˆ (t), χˆ † (t))]. (10.4.69) ˆ = − T r ρ(0)[ Fˆ (χˆ (t), χˆ † (t)), H h¯ Using expressions (10.4.67) and (10.4.68), we can express this time derivative in terms of the non-condensate field operators and the phase factors. This equation, written out in full, is ˆ ψˆ † ) >ρˆ2 (t) d < Fˆ (ψ, dt  i  ˆ Fˆ (ψˆ (r,t) exp{−iφ(r,t)}, ψˆ † (r,t) exp{iφ(r,t)}), = − T r ρ(0) h¯   ˆ (10.4.70) H2 (ψˆ (r,t) exp{−iφ(r,t)}, ψˆ † (r,t) exp{iφ(r,t)}) . We would like to rewrite Eq. (10.4.70) in such a way as to remove the time dependence of the arguments of the operator Fˆ and to put this time dependence into the

414

Quantum Gases

density matrix. This can most easily be accomplished by defining a unitary operator, Uˆ 1 , and its adjoint, exactly as in the definitions, Eqs. (10.4.38) and (10.4.39), ˆ ψˆ †, are replaced by ψˆ , ψˆ † . We note that, in analexcept that the field operators, ψ, ogy to Eqs. (10.4.41) and (10.4.42), it follows that the commutator in Eq. (10.4.70) can be expressed as ˆ †t [Fˆ (ψˆ , ψˆ † ), H ˆ 2 (ψˆ , ψˆ †,t)]Y ˆ t Uˆ 1 , Uˆ 1† Y

(10.4.71)

ˆ t, Y ˆ †t are the unitary operators that replace ψˆ  (r), ψˆ † (r) with their values where Y at time t, and the unitary operators, Uˆ 1, Uˆ 1† , that produce a change in the phase factor, so that ˆ ˆ ˆ Uˆ † 1 ψ (r,t)U1 = ψ (r,t) exp[−iφ(r,t)]

(10.4.72)

ˆ ˆ ˆ † Uˆ † 1 ψ (r,t)U 1 = ψ (r,t) exp[iφ(r,t)].

(10.4.73)

ˆ 2 (ψˆ , ψˆ † ), is identical to that in Eq. (10.4.58) except that the The Hamiltonian, H field operators are replaced by the field operators, ψˆ , ψˆ †, for the non-condensate component of the gas. Given the properties of these operators, Eq. (10.4.70) can be transformed to ˆ ψˆ † ) >ρˆ2 (t) 3 i 2 ˆ d < Fˆ (ψ, ˆ  ˆ † ˆ 1 ] × Fˆ (ψˆ , ψˆ † ) = − T r [H 2 (ψ , ψ ,t), ρ dt h¯ ∂ (10.4.74) = T r ρˆ 1 (t)Fˆ (ψˆ , ψˆ † ), ∂t where we have defined ρˆ 1 (t) as the solution of the equation20 i ˆ ∂  ˆ  ˆ † ˆ 1 ]. ρˆ 1 = − [H 2 (ψ , ψ ,t), ρ ∂t h¯

(10.4.75)

10.4.5 Step 3: The Wigner Distribution for Bogoliubov Excitations Having obtained an equation for the density matrix, ρˆ1 , for the non-condensate part of the Bose gas, we now turn our attention to the kinetic equation for the distribution function of the excitations. We begin by defining the Wigner operator and then the Wigner distribution function [680]. Referring back to the formal definition of the Wigner function, given by Eq. (10.2.7), we define the Wigner operator for the gas as  ˆ f (R,k) ≡ dr 1 [exp(ik · r 1 )]ψˆ † (R + r 1 /2)ψˆ  (R − r 1 /2). (10.4.76) The Wigner distribution function, expressed in terms of the field operators and the density matrix, is simply the average of the Wigner operator with respect to an

10.4 Transport in a Condensed, Dilute Bose Gas

415

appropriate density matrix to be defined later. As a first step in the transformation to excitation operators, we express the field operators, ψˆ †, ψˆ  , in a Fourier representation as   ψˆ  = −1/2 exp[ik · r]ˆak, (10.4.77) k

ˆ †

−1/2

ψ =





exp[−ik · r]ˆa†k,

(10.4.78)

k

where the prime on the summations indicates that the wave numbers in the vicinity of k = 0 are to be excluded from the sums. From Eqs. (10.4.76), (10.4.77), and (10.4.78), it follows that     (10.4.79) exp (iq · R) a†k−q/2 ak+q/2 . fˆ(R,k) = q

In order to express the Wigner function in terms of the Bogoliubov excitations, we first define these excitations by making transformation of the Fourier components of the field operators aˆ k, aˆ †k so as to diagonalize the quadratic terms in ˆ 2 (ψˆ , ψˆ † ), when expressed in terms of these Fourier the Hamiltonian operator, H ˆ 2 , we see components. By inspecting Eq. (10.4.58) for the Hamiltonian operator H ˆ 2 (ψˆ , ψˆ † ) is a sum of terms that range from linear to quartic in powers of the that H field operators for the non-condensate. We wish to express the Hamiltonian as well as the density matrix in terms of local Bogoliubov excitations, in much the same ˆ q , Eq. (10.4.20), in the spatially homogeneous way as we treated the Hamiltonian, H case discussed before. We will then be able to describe nonequilibrium processes in the non-condensate part of the gas in terms of the dynamics of excitations as they interact with each other and with the condensate. To accomplish this transformation to variable describing the excitations, we introduce a unitary transformation, Uˆ 3 , by     † †  , (10.4.80) Uˆ 3 (γ (R)) = exp i γk (R) aˆ k aˆ −k − aˆ aˆ k −k

k

with the properties that21 Uˆ 3† (γ )ˆak Uˆ 3 = aˆ k uk (R,t) − aˆ †−k vk (R,t), Uˆ 3† (γ )ˆa†k Uˆ 3 = aˆ †k uk (R,t) − aˆ vk (R,t).

(10.4.81)

Here uk (R,t) = cosh γk (R,t); vk (R,t) = sinh γk (R,t), tanh 2γk (R) =

nc (t)(R)U0 , k + nc (t)(R)U0

(10.4.82)

416

Quantum Gases

 1/2 E(k,R) = k2 + 2nc (t)(R)U0 k , 1/2  1 1 uk = E(k)−1 (k + nc U0 ) + , 2 2   1 1 1/2 −1 . E(k) (k + nc U0 ) − vk = 2 2

(10.4.83) (10.4.84) (10.4.85)

Here all the variables, whether so indicated or not, are dependent on space, R, and time, t; as in Eqs. (10.4.28) and (10.4.29). If we compare Eqs. (10.4.81), and the corresponding expression for the transformations of aˆ −k, aˆ† −k , with Eqs. (10.4.23), we see that the unitary transformation Uˆ 3 coverts the initial creation and annihilation operators into those for the annihilation and creation operators for the Bogoliubov excitations, that diagonalize the quadratic part of the Hamiltonian operator. By means of the usual procedure, we can formulate averages in terms of a transformed density matrix, ρˆ3, given by ρˆ 3 = Uˆ 3 ρˆ 1 Uˆ 3†,

(10.4.86)

satisfying −i ˆ ∂ ρˆ 3 = [H3, ρˆ 3 ]. ∂t h¯ The new, local, Hamiltonian operator is ˆ ˆ 2 Uˆ † + i h¯ ∂ U3 Uˆ † . ˆ 3 = Uˆ 3 H H 3 ∂t 3

(10.4.87)

(10.4.88)

ˆ 3 (r,t) is quite lengthy and will not be needed here. The full expression for H 10.5 The Spatially Inhomogeneous Bose Gas at Low Temperatures 10.5.1 The Ordering Scheme Before we describe the transformation of Eq. (10.4.76) to a kinetic equation for the distribution function for the excitations, we return to an issue discussed at the beginning of this section – namely setting the dimensionless parameters that characterize the state of the Bose gas near T = 0. This very-low-temperature region is characterized by the conditions that nλ3q ≥ 1, nλ2q a ≥ 1, na 3 1, and a/λq 1. Here we will choose to set up an ordering scheme by considering a small parameter,  < 1, and take nλ2q a to be of order  0 . If we assign the weight  2 to the inter-particle potential V , or, equivalently to the scattering length, a, and take n ≈ nc to be of order  −2, and λq of order  0, then nλ2q a is of order  0,

10.5 The Spatially Inhomogeneous Bose Gas at Low Temperatures

417

while a/λq is of order  2, and na 3 is of order  4 . This scheme effectively preserves the correct order of magnitudes of the various parameters. In the discussion of the kinetic equation, we will treat all quantities of the first and higher powers of  as small quantities. Since our goal is to describe the derivation of the two-fluid hydrodynamic equations, we will also expand all of the fluid variables, such as nc and Vc, appearing in ˆ 3 about their values at the spatial point r and at time t, the Hamiltonian operator, H in order to derive equations for the fluid variables in the form of gradient expansions similar to the Navier–Stokes equations. In this way, and after considerable algebra, we find that Hˆ 3 can be expressed as an expansion of the form ˆ3 = H

∞ 

ˆ (i,j ), H 3

(10.5.1)

i,j =0

where the superscript i refers the power of  and j refers to the power in the gradients of the macroscopic fluid variables. The first few terms in the expansion ˆ 3 are found to be of H ˆ (0,0) = H 3

  [E(k) + V c · h¯ k]ˆa†k aˆ k,

(10.5.2)

k  1/2  ˆ (1,0) = nc U0 δKr (k1 − k2 − k3 ) × {(u1 v2 v3 − u2 u3 v1 ) H 3 1/2 k ,k ,k 1

×

2

3

(ˆa†−k1 aˆ †k2 aˆ †k3

+ aˆ −k1 aˆ k2 aˆ k3 ) + (u1 u2 u3 − v1 v2 v3 − v3 u1 u2

+ u3 v1 v2 − u1 u3 v2 + u2 v1 v3 ) × (ˆa†k1 aˆ k2 aˆ k3 + aˆ †k3 aˆ †k2 aˆ k1 )},

(10.5.3)

with ui ≡ u(ki ), and vi ≡ v(ki ) are defined by Eqs. (10.4.84) and (10.4.85). All quantities appearing in these equations are to be evaluated as local equilibrium ˆ (0,1) , the first term in the quantities depending on R and t. The full expression for H 3 Hamiltonian operator of first order in the gradients of the macroscopic quantities, is very lengthy and is given in [385]. 10.5.2 The Kinetic Equation below the Transition Temperature So far, our analysis of the equation for the distribution function for the excitations of the non-condensate has been very formal, apart from the expansion of the Hamiltonian given earlier. It is now time to consider Eq. (10.4.76) as the basis for the derivation of a kinetic equation for the distribution function, f (R,k,t), for the Bogoliubov excitations. We should expect that this equation should be similar to that for the time derivative of the classical distribution function, that is, it should

418

Quantum Gases

be expressed, at least in some well-defined limit, as the sum of a free motion term and a collision term. The free motion term can be obtained immediately based on the calculations we have carried out so far. The derivation of the collision term is more complicated and is essentially the quantum analog of the classical derivation of the Boltzmann equation from the Liouville equation or, equivalently, from the BBGKY hierarchy equations. These derivations are beyond the scope of this book, and are given elsewhere. However we will be able to infer the structure of the collision term using heuristic arguments, and the more formal derivations are needed at the moment only to fix some parameters that describe the encounters of the excitations with each other and with the condensate. These parameters will be presented without derivation, but the expressions are more or less what one might expect from a more detailed derivation [382, 385, 384, 383, 697, 698, 278, 284, 550, 286, 285]. From Eqs. (10.4.79) and (10.4.87), it follows that the equation for the time dependence of the Wigner distribution function, defined by f (R,k,t) =T r fˆ(R,k)ρˆ 3,

(10.5.4)

is ∂f (R,k,t) ∂ ρˆ −i ˆ 3, ρˆ 3 ]. = T r fˆ(R,k) 3 = T r fˆ(R,k)[H ∂t ∂t h¯

(10.5.5)

ˆ (0,0) in the right-hand side of Eq. (10.5.5) and use Eq. (10.4.79) so as We insert H 3 to obtain a term labeled A. That is,22 −i   ˆ (0,0) (r,t)]} A= [exp(iq · r)]T r{ρˆ 3 (r,t)[ˆa†k−q/2 aˆ k+q/2, H 3 h¯ q =

−i   [exp(iq · r)](E(|k + q/2|) − E(|k − q/2|) + h¯ q · V s ) h¯ q × T r{ρˆ 3 aˆ †k−q/2 aˆ k+q/2 }.

(10.5.6)

We expand the energy function about E(k), keeping only the first power of q, which corresponds to a gradient with respect to r. In this way, we obtain  1 A = − ∇k (E(k) + h¯ k · V s ) · ∇f (r,k,t) h¯  1  [exp(iq · r)] + − ∇k (E(k) + h¯ k · V s ) · ∇r nc (r) h¯ q   ∂γk−q/2 ∂γk+q/2 × < aˆ k−q/2 aˆ k−q/2 >r,t + < aˆ †k+q/2 aˆ †k+q/2 >r,t + · · · , ∂nc ∂nc (10.5.7)

10.5 The Spatially Inhomogeneous Bose Gas at Low Temperatures

419

where the angular brackets denote the average < X >r,t = T r ρˆ 3 X. The third and fourth lines of Eq. (10.5.7) are the contribution from the spatial derivative of the density matrix, ρˆ 3 , and it can be shown that the average values of the forms, ˆaaˆ  and ˆa† aˆ † , appearing in these lines are of O( 2 ). We drop these terms, keeping only the first term on the right-hand side of Eq. (10.5.7). Although ˆ (0,1) , we will not work through the details here, the insertion of the the operator, H 3 into the right-hand side of Eq. (10.5.5) leads to the following contribution to the equation for f (r,k,t), denoted as B, given by B=

−i   ˆ (0,1) (r,t)]} [exp(iq · r)] × T r{ρˆ 3 (r,t)[ˆa†k−q/2 aˆ k+q/2, H 3 h¯ q

1 = [∇r (E(k) + h¯ k · V s )] · ∇k f (r,k,t) + · · · . h¯

(10.5.8)

If we now combine these results, we can express the kinetic equation for the Wigner distribution function for the excitations of the non-condensate as  ∂ 1 + ∇k (E(k) + h¯ k · V s ) · ∇r ∂t h¯ 

1 ∂f 2 2 − [∇r (E(k) + h¯ k · V s )] · ∇k + O(∇r , ∇r ) f (r,k,t) = . ∂t coll h¯ (10.5.9) The right-hand side of this equation is the change in the distribution function due to encounters between the excitations and with the condensate and contains the irreversible part of the kinetic equation, responsible for the dissipative hydrodynamic flows in the Bose gas. The left-hand side has a natural, free-streaming form for the spatial and momentum gradients. The collision term on the right-hand side, for very low temperatures, when naλ2 1, must involve only the lowest-lying quasiparticle excitations. Quasiparticles with wave vector k can be produced by the collision of two quasiparticles with wave vectors k1,k2 , which leads to only one excitation of wave number k, the other excitation having been absorbed by the condensate. They may also be produced when another quasiparticle interacts with the condensate to generate two new quasiparticles, one of which has wave vector k. On the other hand, the reversed processes can remove quasiparticles with wave vector k. The various processes can be illustrated by simple Feynman diagrams with one vertex, as shown in Fig. 10.5.1. This contribution to the collision integral Ccoll will be denoted by C12 , indicating that by means of interactions with the condensate, two quasiparticle excitations can

420

Quantum Gases

Figure 10.5.1 Representative Feynman graphs for the interactions of quasiparticles with each other or with the condensate that are accounted for in the collision term C12 , Eq. (10.5.10). Figures (a) and (d) represent processes in which a quasiparticle with wave vector k is produced, while figures (b) and (c) represent processes in which a quasiparticle with wave vector k is removed from the system. Incoming lines with wave vector ki are associated with factors f (r,ki ,t) in the collision term C12 , while outgoing lines are associated with factors of (1 + f (r,ki ,t)) in this collision term.

be replaced by one quasiparticle, or one quasiparticle excitation can interact with the condensate to produce two new excitations. The collision integral should be proportional to the condensate density, nc , since the presence of the condensate is essential for these processes, and to U02 , according to the Fermi “golden rule” for the cross section of collision processes produced by quasiparticles interacting with the pseudo-potential given by Eq, (10.4.6). Thus, we would expect the collision term on the right-hand side of Eq. (10.5.9) be expressed in the large system limit as nc πU02 C12 (k) = h¯ 



dk1 (2π)3



dk2 (2π)3



dk3 2 σ (k2,k3 ;k1 )δ(k1 − k2 − k3 ) (2π)3

× δ(E(k1 ) − E(k2 ) − E(k3 ))[δ(k − k1 ) − δ(k − k2 ) − δ(k − k3 )] × {f (r,k2,t)f (r,k3,t)[1 + f (r,k1,t)] − f (r,k1,t)[1 + f (r,k2,t)][1 + f (r,k3,t)]}.

(10.5.10)

Here we have inserted the correct factors that are obtained in a complete derivation of this collision term. It is to be understood that the local equilibrium quantities appearing below depend on r and t. Here we have inserted

10.5 The Spatially Inhomogeneous Bose Gas at Low Temperatures

421

σ (k2,k3 ;k1 ) = (u3 − v3 )(u1 u2 + v1 v2 ) + (u2 − v2 )(u1 u3 + v1 v3 ) − (u1 − v1 )(u2 u3 + v2 v3 ).

(10.5.11)

There are two delta functions in the integrand representing the conservation of momentum and the conservation of energy. In these delta functions, the condensate is taken to have zero momentum or zero energy. Each of the sum of three delta functions correspond to one or another of the diagrams illustrated in Fig. 10.5.1.23 It is important to note that when all of the wave numbers, k1 k2,k3 , are small, the function σ (k1,k2,k3 ) satisfies σ (k1,k2,k3 ) ∝ (k1 k2 k3 )1/2 .

(10.5.12)

The numerical coefficient can be found in the literature [172, 278]. This is the general result expected for three phonon processes.24 The collision operator, C12 satisfies an H -theorem with an entropy function defined by Eq. (10.3.6) for the value θ = +1. That is, we define the entropy function as  dk s = kB [(1 + f (k)) ln(1 + f (k)) − f (k) ln f (k)], (10.5.13) (2π)3 where we have used a simplified notation. Using this function, we can easily calculate the rate of change of this entropy due to the dynamics of the excitations described by C12 . We find that this kinetic equation satisfies an H -theorem, namely

   nc πU02 dk dk3 σ 2 (k2,k3 ;k1 )δ(k1 − k2 − k3 ) dk s˙coll = kB 1 2 h¯ (2π)3   (1 + f (k1 )f (k2 )f (k3 ) × δ (E(k1 ) − E(k2 ) − E(k3 )) ln f (k1 )(1 + f (k2 )(! +f (k3 )  × f (r,k2,t)f (r,k3,t)(1 + f (r,k1,t))  (10.5.14) − f (r,k1,t)(1 + f (r,k2,t))(1 + f (r,k3,t)) ≥ 0. 10.5.3 The Collision Term at Moderately Low Temperatures There is a considerable body of work by several authors with the goal of obtaining a kinetic equation that describes the dissipative dynamics of the excitations and the condensate at temperatures below the transition point. In the temperature region where naλ2q 1, and nλ3q 1, the excitations are particle-like, rather than phonons which dominate processes at very low temperatures. The first work along this lines was that of U. Eckern [172] and of T. R. Kirkpatrick and J. R. Dorfman [382, 383, 385, 384]. Kirkpatrick and Dorfman derived a kinetic equation that consisted of two terms. One, C12 , involve interactions of two particles and the

422

Quantum Gases

condensate; the other, C22 , involving the mutual interactions of two particles, has the form of a Uehling–Uhlenbeck collision operator. Later work by A. Griffin, T. Nikuni, and E. Zaremba [697, 698, 277, 278] used different methods than those of Kirkpatrick and Dorfman, but they obtained the same kinetic equation. Including these terms, one has the kinetic equation   ∂ ∂ ∂ ∂ ∂ f (r,k,t) − + (E(k) + h¯ k · V s ) (E(k) + h¯ k · V s ) ∂t ∂(hkα ) ∂rα ∂rα ∂ h¯ kα = C12 (f ) + C22 (f ).

(10.5.15)

For this range of temperatures, C12 (f ) is given by Eq. (10.5.10) with σ 2 = 1. The particle–particle collision integral, C22 (f ), has the Uehling–Uhlenbeck form discussed earlier in this chapter. Taking the first terms in the scattering length expansions, one finds that C22 (f ) =

πU02   k1,k2,k3,k4 δ(k1 + k2 − k3 − k4 ) × δ(E1 + E2 − E3 − E4 ) h¯ 2 × [δKr (k − k3 ) + δ(k − k4 ) − δ(k − k1 ) − δ(k − k2 )] × {f (r,k1,t)f (r,k2,t)[1 + f (r,k3,t)][1 + f (r,k4,t)] − f (r,k3,t)f (r,k4,t)[1 + f (r,k1,t)][1 + f (r,k2,t)]}.

(10.5.16)

In the next section, we will consider the kinetic equation and transport equations in the low-temperature region where naλ2q 1. This involves those interactions of the excitations with the condensate that are included in the collision term, C12 . This should be a reasonable approximation as the temperature approaches absolute zero and leads some results already obtained by other methods. In particular, this collision term leads to an expression for the coefficient of shear viscosity that diverges as T −5 . We will discuss this in a later section. 10.6 The Two-Fluid Hydrodynamic Equations for the Very-Low-Temperature Region The most useful phenomenological theory of the transport properties of liquid helium below the lambda point for its superfluid phase transition was provided by L. Landau and I. M. Khalatnikov [411, 359]. These LK equations describe the dynamics of liquid helium as consisting of superfluid and normal fluid components, and have close similarities to the Navier–Stokes equations of ordinary fluids. The LK equations successfully describe the unusual properties of helium including second sound. The LK equations have the interesting property that they require a total six transport coefficients instead of the three that usually characterize a monatomic

10.6 Two-Fluid Hydrodynamic Equations at Very-Low-Temperatures

423

classical fluid. That is, the LK equations require coefficients of thermal conductivity, shear viscosity, and four coefficients of bulk viscosity, while for classical simple fluids, only three coefficients are needed. The question arises as to whether it is possible to derive the LK equations from a microscopic theory – if not for liquid helium itself, which presents a number of complicating features, then for a dilute Bose gas.25 Here we answer the question in the affirmative, and in the interest of brevity, we simply sketch the derivation from the kinetic equations derived earlier, by applying the Chapman–Enskog method to Eqs. (10.5.15). In order to simplify the discussion, retain only the collision term C12 , given by Eq. (10.5.10), on the right-hand side of this equation. This restriction to the simplest dynamical processes involving the condensate should be a reasonable approximation at very low temperatures, defined by the conditions nλ3q 1, and naλ2q 1. We begin by defining the macroscopic variables in terms of the distribution function, f (r,k,t). The local density of the fluid, n(r,t), including the superfluid component, is defined by Eq. (10.4.31)26 and is ˆ ˆ ψˆ † (r)ψ(r) n(r,t) = T r ρ(t) = nc (r,t) + T r ρˆ 1 ψˆ † (r)ψˆ  (r)    1 dk k + nc U0 −1 = nc (r,t) + 2 (2π)3 E(k)

 k + n c U 0 dk + f (r,k,t) + · · · . (2π)3 E(k)

(10.6.1)

The momentum density, J(r,t) = T r ρ(t) ˆ ˆj, where ˆj is given by Eq. (10.4.34), can be expressed in terms of the distribution function as  dk h¯ kf (r,k,t) + · · · . (10.6.2) J(r,t) = mn(r,t)V s + (2π)3 We have neglected terms of O(∇ 2, 2 ∇ 2, 4 ∇) and used Eq. (10.6.1) to replace the condensate density, nc (r,t), with the total density, n(r,t). The energy density, (r,t), is ˆ  (r), (r,t) = T r ρ(t)ˆ

(10.6.3)

ˆ U0 † h¯ 2 ∂ ψˆ † (r) ∂ ψ(r) ˆ ψ(r). ˆ ψˆ (r)ψˆ † (r)ψ(r) · + 2m ∂r ∂r 2

(10.6.4)

where ˆ (r) =

Then we find that (r,t) = K (r,t) + I (r,t),

(10.6.5)

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Quantum Gases

where the kinetic energy density is 1 K (r,t) = mn(r,t)V 2s (r,t) + V s (r,t) · [J(r,t)−mn(r,t)V s (r,t)], 2

(10.6.6)

and the potential energy density is n2 U0 1 + I (r,t) = 2 2



 dk dk {E(k) − k − nU0 } + E(k)f (r,k,t) + · · · . 3 (2π) (2π)3 (10.6.7)

In the preceding equations and in what follows, we use the notation E(k) = (k2 + 2nU0 k )1/2 , with the total density replacing the condensate density in Eq. (10.4.83).27 Finally, we can obtain an expression for the function, h(r,t) defined in Eq. (10.4.48) or Eq. (10.4.59), in terms of the macroscopic variables and the distribution function as

 dk ∂E(k) 1 nU0 + − U0 h(r,t) = m 2m (2π)3 ∂n  1 dk ∂E(k) f (r,k,t) + · · · . + m (2π)3 ∂n

(10.6.8)

10.6.1 The LK Hydrodynamic Equations Here we will outline a derivation of the two-fluid equations for the case of very low temperatures using the collision operator28 C12 . Thus, we consider the kinetic equation, Eq. (10.5.15), with expression Eq. (10.5.10) for the collision term, C12 (f ). The equation of motion of the superfluid velocity, Vs, is given by Eq. (10.4.49) with the function, h(r,t), given by Eq. (10.6.8). The procedure we will use is the familiar one of constructing a normal solution to the kinetic equation. We first obtain a set of conservation equations by making use of the collision invariants of C12 . These equations contain quantities such as a stress tensor and a heat flow vector that can only be evaluated if the distribution function is known. We assume that the system is close to a local equilibrium state and then construct the distribution function as an expansion in powers of the gradients of the macroscopic variables. This final step allows us to close the conservation equations – that is, to express the stress tensor and heat flow vector as expansions in powers of the gradients of the fluid variables. When applied to the kinetic equation Eq. (10.5.15), this process leads to the very-low-temperature form of the Landau–Khlatnikov two-fluid equations.

10.6 Two-Fluid Hydrodynamic Equations at Very-Low-Temperatures

425

It is easy to check that the collision term conserves momentum and energy but not the number of excitations. That is,  dk (10.6.9) h¯ kC12 (f (r,k,t)) = 0, (2π)3  dk E(k)C12 (f (r,k,t)) = 0. (10.6.10) (2π)3 By multiplying the kinetic equation by h¯ k, and using the conservation of momentum, we obtain the following lengthy equation for the time dependence of the momentum density, J(r,t),  ∂Jα ∂ Vs,α (Jβ − ρVs,β ) + Vs,β (Jα − ρVcsα ) + ∂t ∂rβ  dk n2 U0 ∂E(k) f (r,k,t)ρ δα,β + δα,β + ρVs,α Vs,β + 3 2 (2π) ∂ρ      dk ∂E(k) ∂E(k) δα,β dk ρ , + f (r,k,t)kα + k − E(k) + 2 (2π)3 ∂ρ (2π)3 ∂kβ (10.6.11) and, similarly for the energy,

   ∂ dk ∂ 1 Vs2 ∂ 2 =− . f (r,k,t) (E(k) + h¯ k · V s ) + Jα h + ∂t ∂rα 2 (2π)3 ∂ hk 2 ¯ α (10.6.12) Here ρ = mn(r,t), and all variables are to be evaluated at r,t. To these equations, we must add the conservation of mass equation ∂ρ(r,t) + ∇ · J(r,t) = 0. ∂t

(10.6.13)

The mass conservation equation does not follow from the kinetic equation, Eq. (10.5.9), with collision operator C12 given by Eq. (10.5.10), since this is a kinetic equation for the distribution functions for excitations, which does not necessarily conserve mass. However, when the condensate is taken into account, mass must be conserved. The Local Equilibrium Distribution Function In order to find a normal solution to the kinetic equation, we begin by determining the form of the distribution function for which the collision term, C12 (f ), vanishes. We denote this function by floc (r,k,t).29 We see that the collision integral will vanish if

426

Quantum Gases

floc (r,k2,t)floc (r,k3,t)[1 + floc (r,k1,t)] = floc (r,k1,t)[1 + floc (r,k2,t)][1 + floc (r,k3,t)].

(10.6.14)

Since we expect the distribution function to have the form of a local equilibrium distribution function, we look for solutions of the form floc (r,k,t) =

1 eA(r,k,t)

−1

.

Then we find that Eq. (10.6.14) can be written A(r,k1,t) − A(r,k2,t) − A(r,k3,t) = 0.

(10.6.15)

Therefore, the function A(r,k,t) must be a linear combination of the collision invariants, so that30 A(r,k,t) = β(r,t)[E(k) + h¯ k · C(r,t)]

(10.6.16)

and floc (r,t) = [exp(β(r,t)(E(k)) + h¯ k · C(r,t)) − 1]−1 .

(10.6.17)

Here β(r,t) and C(r,t) are to be determined by Galilean invariance and by requiring that the average values of the collision invariants obtained by using the normal solution, be determined entirely by the local equilibrium distribution function, as in the classical case. That is require that   dk dk (10.6.18) hkf hkf ¯ (r,k,t) = ¯ loc (r,k,t), 3 (2π) (2π)3   dk dk E(k)f (r,k,t) = E(k)floc (r,k,t), (10.6.19) 3 (2π) (2π)3 It is important that the thermodynamic relations obtained by replacing f (r,k,t) by floc (r,k,t) are those appropriate for a superfluid in local thermodynamic equilibrium. As usual, we can identify β(r,t) with the inverse temperature, β(r,t) = (kB T (r,t))−1 . The velocity vector C(r,t) is further specified by the requirement that the thermodynamic properties of the system be invariant under Galilean transformations, leading to the conclusion that the distribution function should depend on velocity differences [545]. From this, we obtain C(r,t) = V s − un, where un is the velocity of the normal fluid. There is no chemical potential term in the local equilibrium Bose distribution, Eq. (10.6.17), because the number of excitations is not conserved. In order to obtain the Euler ideal two-fluid equations, we will need the local equilibrium values of the thermodynamic functions. The local condensate density can be expressed in terms of the local density and the local equilibrium distribution

10.6 Two-Fluid Hydrodynamic Equations at Very-Low-Temperatures

427

function by replacing the distribution function in Eq. (10.6.1) with floc (r,k,t). The local equilibrium chemical potential can be obtained by making the same replacement of the distribution function in Eq. (10.6.8) by making the identification μ(r,k,t) = mh(r,k,t). Thus, the local equilibrium chemical potential μ can be expressed as

  dk dk 1 ∂E(k) ∂E(k) − U0 + . floc (r,k,t) μ = nU0 + 3 3 2 (2π) ∂n (2π) ∂n (10.6.20) The local equilibrium pressure can be identified in the conservation of momentum equation, Eq. (10.6.11), and is

 dk n2 U0 ∂E(k) 1 +  . (10.6.21) − E(k) + n p= k 2 (2π)3 ∂n 2 The local equilibrium internal energy density is   dk n2 U0 1 dk floc (r,k,t)E(k) + (E(k) − k − nU0 ). I = + 3 2 (2π) 2 (2π)3 (10.6.22) The local equilibrium entropy density can be identified from the H -theorem, Eq. (10.5.14), namely  dk [(1 + f (k)) ln(1 + f (k)) − f (k) ln f (k)]. (10.6.23) s = kB (2π)3 An equivalent form, useful for our purposes here is  dk 1 d sδα,β = floc (1 + fl ) × kα [E(k) + h¯ k · (V s − un )]2, 3 2kB T (2π) dkβ (10.6.24) If one carries out integrations by parts of the expression Eq. (10.6.24), one obtains the expression, Eq. (10.6.23), with f replaced by floc . The normal fluid density appears as  dk ρn (V s − un ) = floc (r,k,t)h¯ k, (10.6.25) (2π)3 and the superfluid density, ρs , is ρ s = ρ − ρn .

(10.6.26)

The Euler equations can then be obtained from the hydrodynamic equations to be derived in the following by using local equilibrium values for the thermodynamic functions and by setting all of the transport coefficients equal to zero.

428

Quantum Gases

To proceed further, we now look for solutions of the kinetic equation, Eq. (10.5.15) keeping only C12 . We expand the solution about the local equilibrium distribution in the form given by Eq. (10.3.10), that was used earlier to obtain the linearized U–U equation, Eq. (10.3.11). That is, we write f (r,k,t) = floc (r,k,t) + floc (r,k,t)(1 + floc (r,k,t)) (1) (r,k,t) + · · · , (10.6.27) where  (1) is to be of first order in the gradients, and the neglected terms are to be of higher order in the gradients. This expression is inserted into the kinetic equation in the same way as is done using the Chapman–Enskog method to solve the Boltzmann equation. That is, the left-hand side of Eq. (10.5.9) is evaluated using only the local equilibrium distribution function, floc (r,k,t), and the ideal two-fluid equations are used to express the time derivatives of the macroscopic quantities as linear combinations of their first-order gradients. The unknown function appears only in the collision operator. The algebra is very lengthy, and here we present only the linearized version of the final result. The linearization consists of replacing the local equilibrium distribution function by the total equilibrium distribution function, feq , after the first-order equation has been obtained. In this way, we obtain hydrodynamic equations that are linear in the hydrodynamic fields.31 The equation for the function  (1) (r,k,t) is then32 L12  (1) (r,k.t) feq (1 + feq ) =− kB T 

 1 ∂T ∂E(k) ∂(Jα − ρun,α ) × E(k) − T ∂ρ s ∂ρ ∂rα  



∂T 1 ∂T ∂E(k) k ∂E(k) ∂un.α ∂un.α + ρ+ s E(k) − ρ − T ∂ρ s ∂s ρ ∂ρ 3 ∂k ∂rα ∂rα



 k ∂E(k) ∂E(k) h¯ kα sT ∂E(k) ∂ ln T 1 δα,β Dα,β , + − E(k) + − kβ ρn ∂ h¯ kα ∂rα 2 3 ∂k ∂kα (10.6.28) where Dα,β is the symmetric, traceless stress tensor given by

∂un,α ∂un,β 2 ∂un,γ + − Dα,β ≡ , ∂rβ ∂rα 3 ∂rγ

(10.6.29)

and the linearized collision operator, L12, acting on the first-order gradient correction to the equilibrium distribution function, is given by

10.6 Two-Fluid Hydrodynamic Equations at Very-Low-Temperatures

L12  (1) =

nπU02 h¯ (2π)3



 dk1

429

 dk2

dk3 σ 2 (k1,k2 k3 )

× δ(k1 − k2 − k3 )δ (E(k1 ) − E(k2 ) − E(k3 )) × [δ(k − k1 ) − δ(k − k2 ) − δ(k − k3 )] × feq (k1 )feq (k2 )(1 + feq (k1 ))   ×  (1) (k2 ) +  (1) (k3 ) −  (1) (k1 ) .

(10.6.30)

In the preceding equations, the equilibrium distribution function, feq , is given by feq (k) = (exp βE(k) − 1)−1 . In addition, s is the equilibrium entropy density, and un is the average velocity of the normal fluid. For small wave numbers, the dependence of the function σ (k2,k3 ;k1 ) is given by Eq. (10.5.12). We know from the H -theorem that L12 is negative definite, except for the known zero eigenvalues, and it is isotropic. The latter property allows us to separate the solution of Eq. (10.6.28) as a sum of irreducible tensor parts, scalar, vector, and traceless symmetric tensor. It will be convenient to further separate the scalar part as a sum of two pieces, as given next. That is, we look for solutions of the four linear kinetic equations in the form of a products of irreducible tensors with scalar functions of momentum, A(k),B(k),C1 (k),C2 (k), as shown here:  



1 + feq ∂E(k) k2 k2 (10.6.31) L12 kα kβ − δα,β B(k) = feq kα kβ − δα,β 3 kB T ∂k 3  

1 + feq E(k) ∂E(k) h¯ 2 sT L12 (k)kα A(k) = feq kα − kB T k ∂k ρn (10.6.32)    1 + feq ∂E(k) E(k) ∂T − L12 C1 (k) = feq kB T ∂ρ T ∂ρ s (10.6.33)   1 + feq ∂E(k) k ∂E(k) ρ + L12 C2 (k) = feq kB T ∂ρ 3 ∂k



∂T ∂T 1 ρ+ s E(k) . (10.6.34) − T ∂ρ s ∂s ρ The requirement that these equations be soluble is that the right-hand sides are orthogonal to the conserved quantities. It is this property that allows us to realize that there are two scalar functions, C1 and C2 , that appear in the general solution to Eq. (10.6.28), since there are two combinations of scalar terms on the right-hand side of this equation, each of which is orthogonal to the conserved quantities. This leads us to Eqs. (10.6.33) and (10.6.34).

430

Quantum Gases

When we combine the solutions of Eq. (10.6.28) with the conservation equations, Eqs. (10.6.11) and (10.6.12), as well as with Eqs. (10.4.49) and (10.4.12) for the velocity of the superfluid, we obtain the following set of equations, which are of Navier–Stokes-type, dissipative hydrodynamic equations with a total of six transport coefficients: the coefficient of shear viscosity, η; the coefficient of thermal conductivity, κ; and four coefficients of bulk viscosity, ζ1,ζ2,ζ3,ζ4 . The conservation of mass for this fluid is expressed as ∂ρ + ∇ · (ρn un + ρs V s ) = 0. ∂t The conservation of momentum equation becomes   3 ∂ ρn un,α + ρs Vs,α ∂ 2 + pδα.β + ρn un,α un,β + ρs Vs,α Vs,β ∂t ∂rα !  "    ∂ ρs Vs,γ − un,γ ∂un,γ ∂ ηDα,be + δα,β ζ1 , + ζ2 = ∂rβ ∂rγ ∂rγ the superfluid velocity equation ! "    ∂ ρs Vs,γ − un,γ ∂un,γ ∂ ∂Vs,α μ Vs2 , + ζ4 =− + + ζ3 ∂t ∂rα m 2 ∂rγ ∂rγ

(10.6.35)

(10.6.36)

(10.6.37)

and the energy conservation equation

   μ Vs2 ∂  ∂ + + + sT un,α + ρn un,α un · (un − V s ) ρn un,α + ρs Vs,α ∂t ∂rα m 2 = ∇ · (κ∇T ) .

(10.6.38)

The transport coefficients are obtained in terms of the functions A(k),B(k),C1 (k), C2 (k), where the coefficient of thermal conductivity, κ, is  dk ∂E(k) κ=− feq (1 + feq ) × kE(k) A(k). (10.6.39) 3 (2π) ∂k The coefficient of shear viscosity, η, is  1 dk ∂E(k) feq (1 + feq ) × k 3 η=− B(k), 3 15 (2π) ∂k and the four coefficients of bulk viscosity are

 dk ∂E(k) k ∂E(k) + C1 (k), feq (1 + feq ) × n ζ1 = − (2π)3 ∂n 3 ∂k

 dk ∂E(k) k ∂E(k) ζ2 = − feq (1 + feq ) × n + C2 (k), (2π)3 ∂n 3 ∂k

(10.6.40)

(10.6.41) (10.6.42)

10.6 Two-Fluid Hydrodynamic Equations at Very-Low-Temperatures



ζ3 = −

dk ∂E(k) feq (1 + feq ) × C1 (k), (2π)3 ∂ρ

ζ4 = ζ1.

431

(10.6.43) (10.6.44)

The identity of the two bulk viscosities, ζ1 and ζ4 , follows, in general, from the Onsager reciprocal relations as applied to this system [545]. The LK equations given in Eqs. (10.6.35)–(10.6.38) have the important property that the vanishing of the dissipative terms requires the system to be in a local equilibrium state with the correct relations between the local thermodynamic functions [545, 431]. The First Enskog Approximation In order to obtain explicit expressions for the transport coefficients, we must determine the functions, A,B,C1 , and C2 by solving the kinetic equations Eqs. (10.6.31)– (10.6.34). This can be done the same way as the analogous equations are solved in the classical case. That is, the unknown functions are expanded in a convenient set of orthogonal polynomials, and the coefficients are determined by carrying out integrations involving the kinetic operator, L12 , acting on functions proportional to one of the polynomials in the set. Here we give only a simple example. Consider Eq. (10.6.32) for the function A(k). If we expand this function as a sum of orthogonal polynomials, the first term in the expansion will be

E(k) h¯ 2 sT − , (10.6.45) A(k) ≈ A k ρn where A is a constant. This constant can be determined by inserting this approximate solution in Eq. (10.6.32), multiplying this equation by

E(k) h¯ 2 sT − kβ k ρn and integrating over the variable k. This is the quantum analog of first Enskog approximation in classical kinetic theory. In this way, we obtain 

2   dk E(k) h¯ 2 sT 1 feq (1 + feq )kα kβ × − A= kB T h¯ (2π)3 k ρn



−1  E(k) h¯ 2 sT E(k) h¯ 2 sT dk − − kβ . × L12 kα × (2π)3 k ρn k ρn (10.6.46) The relevant integrations, while not overly difficult, are lengthy, and here we quote only the results [384]33 for the first Enskog approximations for the six transport coefficients for very low temperatures, where naλ2q 1. They are

432

Quantum Gases

κ ≈ 0.09kB c,

(10.6.47)

η  4.56 × 10−4 m(naλ2q )7 c,

(10.6.48)

ζ1 = ζ4  4.94 × 10−3 (naλ2q )−1 c,

(10.6.49)

ζ2 ≈ 1.67 × 10−3 mn(naλ2q )−1 c,  −1 ζ3 ≈ 1.52 × 10−2 mn2 aλ2q .

(10.6.50) (10.6.51)

 −1 c = a 2 π 3/2 (3βm)1/2 nλ3q .

(10.6.52)

Here

Notice that the shear viscosity grows very dramatically with T as T −5 at very low temperatures, compared to the dependencies of the other transport coefficients. This indicates that the transport of momentum in the plane perpendicular to that of the fluid is very slow. The restrictions on the energies and momenta on the two excitations indicate that, on the average, an excitation decays into two nearly parallel excitations, with little transfer of momentum in the orthogonal plane. The growth of the shear viscosity as the temperature is lowered is an indication that the mean free path or mean free time for shear viscosity is large compared to those for the other transport properties. This is due to the fact the the number of excitations decreases as the temperature is lowered, bringing into question the assumption that it is possible to maintain a local equilibrium state at very low temperatures. In addition, the variation of the temperature dependences of the transport coefficients, from T −5 for η to T 2 for κ, indicates that there is strong dependence of the transport coefficients on the dynamical processes described by the collision operator C12, that was used here.34 The T −5 dependence of the shear viscosity was predicted by Khalatnikov, based on very general arguments that should apply apply to all superfluids. Using many-body perturbation theory, the same result was also obtained by S. K. Ma and V. N. Popov [359, 447, 538]. Experimental results for liquid helium at low temperatures show a dramatic increase in the viscosity as the temperature is lowered [688]. The results for the bulk viscosities given earlier have been corrected by Griffin, Nikuni, and Zaremba [278], who took into account a term neglected by the earlier work of Kirkpatrick and Dorfman.35 10.6.2 Two-Fluid Hydrodynamics at Moderately Low Temperatures The description of hydrodynamic flows for the moderately low-temperature case requires Chapman–Enskog like solutions of the complete form of Eq. (10.5.15), which describes the decays of two excitations into one due to interactions of the

10.6 Two-Fluid Hydrodynamic Equations at Very-Low-Temperatures

433

excitations with the condensate, as given by C12 , as well as interactions between two excitations where the final state also has two excitations, as described in the collision term C22 in Eq. (10.5.16). In this temperature region, the excitations are nearly particle-like, so these latter interactions may be thought of as being binary collisions and are described by the Uehling–Uhlenbeck equation.36 One might argue that as one approaches the transition point from below, collisions between two excitations as described by C22 should become the dominant dynamical processes in the gas. The analyses of Kirkpatrick and Dorfman as well that of Griffin, Nikuni, and Zaremba are based on this supposition. The latter authors considered the solution of the kinetic equation with collision operator C12 + C22 and used, as a zeroth-order approximation for the distribution function, the local equilibrium solution for C22 , which is denoted as Fl (r,k,t) and has the form Fl (r,k,t) = [exp (β(r,t) (E(k)) + h¯ k · C(r,t) − μ(r,t)) − 1]−1 .

(10.6.53)

One expands the distribution function in the kinetic equation, Eq. (10.5.15), about this local equilibrium distribution function in the same way as done in Eq. (10.6.27); that is, f (r,k,t) = Fl + Fl (1 + Fl ) 1 + · · · .

(10.6.54)

It is immediately clear that the collision integral C12 does not vanish when the local equilibrium distribution function is Fl in the Chapman-Enskog expansion of the total distribution function. This results in a new term in the kinetic equation, C12 (Fl ), which acts as a source term in the equation for 1 . The previously mentioned authors include this source term when obtaining the Chapman–Enskog solution and then derive the Landau–Khalatnikov equations valid for this temperature region with associated transport coefficients. This differs from the treatment of this region by Kirkpatrick and Dorfman, who used as the zeroth-order, local distribution function given by Eq. (10.6.17). Their local equilibrium distribution function has the property that both collision operators C12 and C22 vanish for this function, and it is more consistent with the Chapman–Enskog procedure whereby the collision terms vanish for the zeroth-order distribution function. Griffin et al. argue that their zeroth approximation allows for the condensate and normal fluid to have different chemical potentials initially. They show that the chemical potential equilibrate on an exponential time scale, which is reasonable. In the interest of not making the present chapter unduly lengthy, we recommend that the reader follow the analysis in the book and papers by A. Griffin, T. Nikuni, and E. Zaremba [697, 698, 278]. We have shown in this chapter that the Landau–Khalatnikov two-fluid equations can indeed be derived from a microscopic model of a dilute Bose gas below its transition temperature, and that it is possible to obtain numerical values for all of

434

Quantum Gases

the associated transport coefficients for a range of temperatures below the condensation temperature. We conclude this chapter by calling attention to extensive work presented in recent papers and a book by L. E. Reichl, E. D. Gust, M.-B. Tran, and Y. Pomeau, not discussed here due to lack of space, that argues for the presence of an additional term, C13 , in the kinetic equation that describes processes involving the interactions of three excitations with each other and with the condensate [550, 284, 286, 285, 551, 537, 626]. It is clear that the last word on this topic has not been said. Notes 1 This chapter was written by TRK and JRD. 2 An example of this would be the spectrum of particles of a three-dimensional, ideal Bose gas. What does it mean when we say that particles are in their ground state when, for large systems, the energy difference between the ground and first excited state is very small? This question is resolved by examining the size dependence of the chemical potential, which is proportional to this energy difference. 3 The Weyl representation of operators as well as the derivation of Eq. (10.2.11) can be found in J. H. Irving and R. W. Zwanzig [320], V. I. Tatarski [618], and R. G. Littlejohn [436], as well as the cited references and in many quantum mechanics texts, e.g. the book by L. E. Ballentine [20]. 4 The special case of hard-sphere interactions requires a careful treatment and involves the construction of the quantum mechanical binary collision operators in order to obtain the version of Eq. (10.2.12) that holds for such potentials. We will not deal with this issue here. 5 The description of the superfluid behavior in liquid helium requires a treatment of a dense quantum fluid and is not immediately accessible through appropriate generalizations of the Uehling–Uhlenbeck equation, as described in this chapter. For discussions of liquid helium, see the recent books by A. Griffin [277] and T. Leggett [431], among others. Some of the important earlier papers by L. Landau, L. Tisza, F. London, C. N. Yang, T. D. Lee, K. Huang, P. C. Hohenberg, and P. C. Martin are [623, 622, 411, 620, 439, 425, 314, 423, 428, 427, 310]. 6 The measurement of the superfluid component of the gas below the BEC point is discussed in a number of papers [78, 111, 327]. 7 Our discussion of transport in a condensed Bose gas is based on the work of U. Eckern [172] and of T. R. Kirkpatrick and J. R. Dorfman [385, 384, 382] with very similar but more recent studies, particularly those of A. Griffin, T. Nikuni, and E. Zaremba [697, 698, 278]. There are several excellent, related studies in the literature [115, 430, 4, 444, 431, 524, 612]. We mention some very recent work by L. E. Reichl, E. D. Gust, M.-B. Tran, and Y. Pomeau at the end of this chapter [550, 284, 286, 285, 551, 626, 537]. 8 Our treatment that follows will be along the lines of the work by S. V. Peletminskii, A. I. Sokolovskii, and V. S. Shchelokov [522]; by T. R. Kirkpatrick and J. R. Dorfman [385, 382, 384, 383]; and by U. Eckern R [172], E. Zaremba, T. Nikuni, and A. Griffin [697, 698, 278]. 9 We have changed the notation for the potential energy function from φ to V in order to prevent confusion with a phase angle, φ, to be introduced later. 10 This limit ignores the interesting and very important problem of describing the behavior of the Bose gas in a trap. One way of describing the trap is by means of a harmonic external potential that acts on all of the particles. Although a trapping potential should be included if one is to apply the theory to realistic situations, we will not do so here, in order to keep the presentation to a reasonable length. 11 Expansions in small parameters with the first few terms of the form given by Eq. (10.4.10) appear quite often for transport coefficients. Chapters 12 and 13 with have examples for transport coefficients where the small parameter is the gas density. There may be a generic reason for the ubiquity of expansions of this form, but it is not known to us.

Notes

435

12 O. Penrose and L. Onsager have estimated that superfluid helium at T = 0 is composed of about 8% particles in the zero momentum state and about 92% nonthermal excitations [523]. 13 In some descriptions, the condensate density in Eq. (10.4.11) is replaced by the superfluid density. Because the two densities are closely related, the two possibilities are physically equivalent. 14 We note that if the phase angle were constant and allowed to take any value between 0 and 2π , the expectation value of the ground state wave function would be zero. It will be important in what follows that this not be the case, and a special ensemble, the η-ensemble, will be used to restrict the initial value of the phase angle. 15 We have assumed that the pseudo-potential is repulsive. An attractive potential leads to very different behavior in which the long-wavelength, or small-wave-number, modes have imaginary frequencies, whenever k < 2nc |U0 |. This leads to the possibility of a collapse, whereby the attractive forces overcome the zero point energy of the bosons. For details, we refer to the literature [524]. 16 A vanishingly small field to obtain a non-zero-order parameter is often used in phase transitions that involve a broken symmetry. For example, a vanishing magnetic field is used for magnetic phase transitions to fix the direction of the magnetization vector. 17 Although we do not use a special notation, all of the averages defined in what follows are taken in the η-ensemble. 18 These identities can most easily be proved by replacing φ with λφ and taking the derivative with respect to λ. 19 This is consistent with a corresponding assumption in the theoretical treatment of the fluid at equilibrium. 20 We could express ρˆ1 in terms of the unitary operators defined earlier, but this representation will not be used here. 21 These equations can most easily be derived by taking the first and second derivative of the left-hand side with respect to γk and solving the resulting differential equation. 22 We now change the notation slightly to use r as the spatial variable instead of R. We also use the commutator identity [ab,cd] = a[b,c]d + [a,c]bd + ca[b,d] + c[a,d]b, with ab = aˆ †k−q/2 aˆ k+q/2,and cd = aˆ †k aˆ k1 , in what follows. 1 23 Most, if not all, of the books on condensed Bose gases discuss the Gross–Pitaevsky (GP) equation, which is a mean-field description of the condensate at T = 0. The GP equation is a nonlinear Schrödinger equation for the condensate wave function, and, as it is not a kinetic equation and not needed for our further discussions, it will not be discussed here. 24 See, for example, [216], chapter 12. 25 A general derivation of the LK equations for a condensed Bose gas, based on the use of projection operators to express the transport coefficients as integrals of time correlation functions, similar to Green–Kubo formulae for a classical fluid, can be found in the paper by Kirkpatrick and Dorfman [383]. There the general expressions are evaluated for a dilute, condensed Bose gas. 26 In what follows, we replace sums over wave numbers with integrals, keeping track of factors of the volume, , and of (2π )3 . In addition, we suppose that the various divergent integrals appearing in what follows can be regularized following the procedure of Lee, Yang, and Huang for similar integrals appearing in the equilibrium theory for the dilute Bose gas [314, 425, 423, 426, 427, 428, 429]. 27 In the interest of simplifying the equations somewhat, we abuse the notation and use the same symbol for E(k) as we did earlier in Eq. (10.4.83) for the energy expressed in terms of the condensate density. 28 As mentioned earlier, we simplify the discussion and limit ourselves to the simplest dynamical processes, appropriate for the very-low-temperature case. The more general case, where two or all three collision terms are included, can be treated by similar methods, but the calculations are lengthier [382, 385, 384, 697, 503, 698, 278, 550, 284, 286, 285]. 29 We have used the symbol floc (r,v,t) to describe a local Maxwell–Boltzmann distribution function. The fact that we use floc (r,k,t) for the local equilibrium distribution function for a condensed Bose gas should not cause confusion.

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Quantum Gases

30 It is worth pointing out that both of the collision terms, C12,C22, vanish for the local equilibrium form given by Eq. (10.6.17). If only the collision term, C22, is used, the conservation of the number of excitations leads to the inclusion of a chemical potential term in the contribution local equilibrium distribution function. We will remark on this point further on in this discussion. 31 It is possible to go further and obtain nonlinear equations, but we will not be concerned with them here. 32 Due to the large number of terms in the linearized equation, we present the right- and left-hand sides of the equation separately. The right-hand side of the equation is Eq. (10.6.28), and the left-hand side is Eq. (10.6.30). Thus, the full equation is obtained by equating the right-hand sides of these two expressions. 33 More details for these calculations can be found in chapter 18 of the book by A. Griffin, T. Nikuni, and E. Zaremba [278], as well as in their original papers. 34 This is in striking contrast with the temperature dependence of the classical transport coefficients as obtained from the Boltzmann equation, all of which have similar temperature dependences. 35 See Section 17.4 of [278]. The results for the transport coefficients given there are equivalent to the results of this chapter. 36 The form of the kinetic equation with two collision terms, C12 + C22, was carefully analyzed by A. Griffin, T. Nikuni, and E. Zaremba [697, 698, 278].

11 Cluster Expansions

11.1 Introduction Our work in this and the following chapters is principally devoted to finding solutions to the following basic questions: (1) Can one derive the Boltzmann equation, starting from the Liouville equation, as the first term in a systematic expansion of the collision operator for the oneparticle distribution function, where correlated collisions among three, four, and greater numbers of particles are taken into account, successively or otherwise? (2) What would replace the Stosszahlansatz in such a systematic theory? (3) What is the dependence of the transport coefficients on the density, as obtained from the generalized Boltzmann equation? (4) What physical properties, in addition to transport coefficients, would enable us to test the theory? In our previous discussions, we considered nonequilibrium processes in dilute gases and concentrated largely on the case where the mean free path length is large compared to the characteristic size of a particle and small compared to the characteristic size of the container. We generalized the treatment to include higherdensity effects when we modified the Boltzmann transport equation for hard spheres according to the Enskog prescription, as extended to the revised Enskog theory (RET) by later authors. The Enskog theory, including the RET, is not a systematic generalization of the Boltzmann equation to higher densities because it includes only excluded volume and other finite size effects, such as the difference in location of the centers of colliding particles, and only extends to higher densities the probability of uncorrelated binary collisions taking place in the gas. That is, the Enskog theory does not take into account any effects due to velocity correlations between two particles that may exist before their binary collision, correlations that require some time on the order of the mean free time to become established. In this and the two following chapters, we will see how such correlations can

437

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be included in the generalization of the Boltzmann equation to higher densities, and we will consider some of the important effects of these correlations. We will see that a systematic expansion of the corrections to the Boltzmann equation in powers of the density cannot be carried out, due to divergence problems, but that the first few terms in what one hopes is a well-behaved extension of low-density results to higher densities can be obtained. This will lead us to conclude that (1) for three-dimensional systems, there are logarithmic terms in the density expansion of transport coefficients; (2) dynamical correlations between particles in the fluid decay algebraically with time; and, as a result, (3) the transport coefficients in the Navier–Stokes equations for one- and two-dimensional fluids are not well defined. As a consequence of (2) and (3), these equations must be modified with corrections that are nonanalytic in the gradients of the hydrodynamic fields and/or depend on the size of the system. These effects are in stark contrast with the results of the low-density Boltzmann equation treatment of hydrodynamic flows and are due to the existence of long-time and long-range dynamical correlations in the fluid. Similar effects occur in three-dimensional systems but beyond the Navier–Stokes order; that is, complications do arise in the Burnett and higher-order hydrodynamic equations, which take into account terms of cubic and higher orders in the gradients. As we shall discuss further on, for linearized hydrodynamic equations in three dimensions, there are an infinite number of terms with fractional powers between second and third order in the gradients. The prediction of the existence of longrange spatial correlations was confirmed by light scattering experiments on fluids with stationary temperature gradients, and the long-time, algebraic decays of the time correlation functions were first discovered in computer studies of the velocity autocorrelation function for gases of hard disks or of hard spheres. In this chapter, we first present a selective review of earlier attempts to obtain a generalized Boltzmann equation beyond the dilute gas limit. Then we show how cluster expansion methods can be used to generalize the Boltzmann equation beyond the dilute gas limit as a systematic expansion of a generalized collision operator in powers of the density. This expansion is very similar to virial expansions of the equilibrium, thermodynamic quantities for a moderately dense gas. In Chapter 12, the attempt at generalizing virial expansion methods to nonequilibrium quantities will be shown to fail, due to dynamical correlations produced by collisions in the gas.1 There we present a way to sum the most divergent terms in the virial expansion, which leads to better-behaved, but still problematic, expressions for the higher-density corrections to the kinetic equation for the singleparticle distribution function and for the transport coefficients. In neither case is it possible to represent the results as a power series expansion in the density where each successive term depends on the dynamics of successively larger numbers of particles. The resummation of the divergences can be carried out by direct

11.1 Introduction

439

summation of the divergent terms once they are identified and expressed in a form suitable for resummation. Another way of accomplishing this resummation is done by combining the BBGKY hierarchy equations with an appropriate cluster expansion of the higher-order distribution functions. From this, one may obtain a series of closure approximations of increasing order of accuracy and complexity. In the n-th order closure approximation, an n + 1–particle cluster function is set equal to zero for all pre-collisional configurations that enter the nth-order hierarchy equation. This results in an equation for the distribution function that is free of the most serious divergence problems of the virial expansion. We will show, although it is not immediately obvious, that the method based upon the hierarchy equations is equivalent to summing the most divergent terms in the virial expansion of the collision operator or of the transport coefficients, themselves. The Green–Kubo formulae discussed earlier, in Chapter 6, allow us to apply these methods to obtain expressions for transport coefficients without having to use the Chapman–Enskog method to obtain expressions for them. We show that terms proportional to the logarithm of the density appear in a natural way from this analysis. The presence of these terms is confirmed by means of computer simulated molecular dynamics of various model systems. The renormalization of the nonequilibrium virial expansion will be shown to lead to a number of important consequences, many of which have been verified by computer simulated molecular dynamics, as well as by laboratory experiments. One of these results, the presence of long-time tails in time correlation functions,2 was first discovered in computer simulations by B. J. Alder and T. E. Wainwright in 1967 [9, 10, 673] and has played a central role in the further development of kinetic theory by stimulating theorists to provide explanations for these “tails,” based on kinetic theory. This, in turn, provided a microscopic understanding of the closely related mode-coupling theory [218, 333, 346, 228, 229] as well as of other methods to describe these long-time tails [157, 191, 706, 192, 193, 156, 707, 155, 195, 196, 555].3 The long-time tails and their consequences will be the subjects of Chapter 13. The presence of longtime tails in the Green–Kubo correlation functions have a profound effect on the transport properties of the gas. In Chapter 14, we consider the effects of long-time tails on the derivation of hydrodynamic equations using kinetic theory, and other methods.4 We will consider the derivation of hydrodynamic equations for two- and three-dimensional gases and show some of the unexpected pathologies that arise on deriving these equations.5 Some History The starting point for generalizing the Boltzmann equation to higher densities is the Liouville equation, or, equivalently, the BBGKY hierarchy equations that express the time development of an n-particle distribution function in, among other terms,

440

Cluster Expansions

a set of collision operators that act on the distribution function for n + 1 particles. Since this is an open set of equations, one must find some way to close the hierarchy. For many decades, researchers devised several methods to close the hierarchy equations, none of them satisfactory, except at low densities. The first goal of these workers was to derive the Boltzmann equation from the hierarchy, something that they managed to do. Apart from the fact that in these efforts some questionable assumptions were made, none of them succeeded in going beyond the Boltzmann equation in a systematic way. The first person who devised a systematic scheme was N. N. Bogoliubov in 1947 [55, 106]. He introduced a functional assumption that has a reasonable physical foundation and is, in many respects, a generalization of the Chapman–Enskog method for constructing solutions to the Boltzmann equation. He based his argument on a picture of three stages in the dynamics of a gas: In the first stage, called the initial stage, the gas starts from some initial state, and the particles undergo a few subsequent collisions with each other. During this interval, the time development of the distribution functions is determined by the initial condition, free streaming, and the collision dynamics, but the gas has hardly any collective properties. In Bogoliubov’s picture, this stage lasts just a few mean free times and is followed by the kinetic stage. In this stage, the two-particle and higher distribution functions are functionally dependent6 on the single-particle distribution function.7 Then, in Bogoliubov’s picture, the gas reaches the third state, after a sufficiently large number of mean free times. This stage is called the hydrodynamic stage, where the single-particle distribution function is a functional of the hydrodynamic densities and therefore depends on time only through the time dependence of the hydrodynamic fields. The gas is characterized by three length scales: a, the characteristic size of a particle; , the mean free path of a particle in the gas; and L, the characteristic size of the vessel containing the gas, or the length scale that describes the spatial variation of the hydrodynamic fields. Of course, this picture does not apply to all gases under all conditions, but it is supposed to describe a gas under the conditions8 that a  L. The kinetic stage develops on times of the order of /v, and the hydrodynamic stage develops over times on order L/v, or, for diffusive processes, of order L2 /(v), where v is a typical velocity of a particle in the gas (e.g. the speed of sound). Bogoliubov assumed an initial ensemble whereby particles that are separated by distances much greater than the range of the forces at t = 0 may become correlated only through collisions. This assumption, together with the functional hypothesis, allowed Bogoliubov to derive the Boltzmann equation from the BBGKY hierarchy and to indicate how the corrections to it can be obtained. Later, S. T. Choh and G. E. Uhlenbeck used Bogoliubov’s method to obtain, for the first time, the three-particle correction to the Boltzmann equation [95, 179]. Then M. S. Green

11.2 Generalizing the Boltzmann Equation

441

[270, 271, 274] and E. G. D. Cohen [101, 99, 100, 98, 102] independently showed that Bogoliubov’s functional method could be replaced by a much simpler method based on the use of cluster expansions similar to those used for obtaining virial expansions for thermodynamic properties of a gas in equilibrium. M. S. Green, J. V. Sengers, and co-workers were able to show that the three-body Enskog correction to the Boltzmann collision operator was, in fact, part of the Choh–Uhlenbeck term for hard-sphere particles, and that most of the remaining contributions could be identified as the results of sequences of three or more binary collisions taking place among the three hard spheres [273, 582, 308, 583, 585]. Not too long after that, J. R. Dorfman and E. G. D. Cohen [153, 154, 581, 71],9 among others, discovered that the cluster expansion is plagued by divergence difficulties after the first one or two terms, and new methods were required to renormalize the theory. This was accomplished by K. Kawasaki and I. Oppenheim [350], who were able to show that the most divergent terms in the virial expansion of the transport coefficients could be expressed as “ring diagrams” and could be summed to obtain a more well-behaved expression.10 In this chapter, we present the cluster expansion method used to obtain a generalized kinetic equation in the form of a density expansion of the collision operator. This work will set the stage for the discussion of its divergence problems to be given in Chapter 12. 11.2 Generalizing the Boltzmann Equation The Boltzmann transport equation of 1872 occupies a position in nonequilibrium statistical mechanics not unlike that of the perfect gas law for equilibrium systems [57]. Both the perfect gas law and the Boltzmann equation have proven to be extremely useful, but it is clear that they can be applied only to very dilute gases. The perfect gas equation of state was extended to higher densities by means of cluster expansion methods based upon the factorization properties of n-particle equilibrium distribution functions for particles interacting with short-range forces. Similarly, the efforts of M. S. Green and E. G. D. Cohen to extend the Boltzmann equation to higher densities were based on cluster expansion methods, assuming the factorization of nonequilibrium n-particle distribution functions and evolution operators [270, 271, 99, 100, 274]. We briefly illustrate this procedure since it served as the basis for generalizing the Boltzmann equation as a virial expansion – that is, by expressing the collision operator as a power series in the density, with terms that depend on the dynamics of a fixed number of particles. The first, Boltzmann, term depends only on the dynamics of two particles, the next term depends on the dynamics of three particles, and so on.

442

Cluster Expansions

General Cluster Expansions In Chapter 7, we outlined the use of cluster expansions to obtain a density expansion of the two-particle distribution function needed for use in the Enskog and revised Enskog equations. The cluster expansions introduced in Section 7.8 can be applied to any set of functions, WN (x1,x2, . . . ,xN ), of the position and momentum variables, xi = (r 1,p i ) for N = 1,2, . . . particles, possibly multiplied by a product of N identical functions, φ(xi ), typically of momentum, #φ(xi ). We assume that WN is symmetric in these variables – that is, it does not change under permutations of the particle indices. For the cluster expansion to be useful, we assume that the function, WN , has the factorization property, namely that if all members of a group of particles – say particles 1, . . . ,j – are sufficiently far from the remaining particles, j + 1, . . . ,N, then W(x1, · · · ,xN ) = W(x1, · · · ,xj )W(xj +1, · · · ,xN )(1 + (O(dj,N−j ))−α ). (11.2.1) The quantity ds,N−s is defined as the smallest distance between two particles, each one in a different subset, and α > d, where d is the number of spatial dimensions of the system. Then one writes cluster expansions of the function WN in different forms depending on which few-particle function is of interest.11 The one-particle cluster functions, U (x1 |x2 . . . ,xj ), called one-particle Ursell functions, corresponding to this set of W functions are defined recursively for N = 1,2, . . . by the cluster expansion WN (x1,x2,x3, . . . ,xN ) = U1 (x1 )WN −1 (x2,x3, . . . ,xN ) +

N 

U2 (x1 |xj )WN −2 (x2, . . . ,xj −1,xj +1, . . . ,xN )

j =2



+

U3 (x1 |xj ,xk )WN −3

2≤j , the time it takes a particle, on the average, to travel over the range, a, of the interaction potential. The source of the difficulty, as we show in what follows, can be expressed simply. The n-particle collision operators for n = 3,4, . . . , particles have contributions from sequences of collisions, such as those illustrated in Fig. 11.2.3 of the previous chapter. In these sequences, the particles can move arbitrarily long distances between their previous and their next

467

468

Divergences, Resummations, and Logarithms

collision in the sequence. There is no way to cut off the range of integration. As a result, the virial expansion treats incorrectly an important physical property of the actual dynamics in a gas – namely that particles travel, on the average, a mean free path or two between collisions and the probability of a particle avoiding a collision for a time t since its previous one decays exponentially as exp[−t/t ]. This exponential decay is a collective effect not properly described by the individual terms in the virial expansion. As we will see in this chapter and the next, resolving the divergence difficulties in the nonequilibrium virial expansions leads to, among other things, the appearance of logarithmic terms in the density expansions of transport coefficients and the appearance of algebraic, long-time decays of the Green–Kubo time correlation functions. The appearance of nonanalytic terms – in this case, logarithms – in the density expansion of the transport coefficients for particles with short-ranged forces was an unexpected result [520]. Until the mid-1960s, it had been supposed that there was a kind of “super statistical mechanics”1 for systems of particles interacting with short-ranged, pairwise, central forces. This notion was based on the idea that cluster expansion methods pioneered by J. E. Mayer, with the Mayer f -function expansions for the development of virial expansions of the thermodynamic properties of such systems and further developed with the introduction of Ursell and Husimi cluster expansions, could be directly extended to nonequilibrium systems using the time-dependent cluster expansions described in the previous chapter [271, 101, 99, 100, 274, 102]. The discovery of the divergence problems in the nonequilibrium virial expansions and the appearance of ln na d terms showed that nonequilibrium systems are fundamentally different from equilibrium ones for classical systems with short-ranged forces. We discuss this fundamental difference further in Section 14.5. On the other hand, it was known already in the 1950s that nonanalytic terms exist in the density expansions of properties of quantum systems with short-ranged interactions. For dilute Bose gases in three dimensions with an s-wave scattering length a, the crossover from free-particle dynamics to phonon dynamics takes place at a wave number k ∝ (na)1/2 . This, in turn, leads to an expansion in powers of (na 3 )1/2 in both the ground state energy per particle and in the ground state pressure. Similarly, in dilute Fermi gases, the existence of a Fermi wave number kF ∝ n1/3 leads to an expansion in powers of (na 3 )1/3 for the same quantities. Physically, in both cases, the nonanalyticity arises from collective effects: phonons in the Bose case and the existence of a sharp Fermi surface in the Fermi case [426–428]. The divergence problem to be discussed in what follows also has some similarities to one that occurs in the equilibrium virial expansions for a system of classical particles interacting with long-ranged forces. For example, a calculation

12.2 Divergences in the Nonequilibrium Virial Expansion

469

of the equilibrium properties of ionic solutions requires resummations of terms that can be represented by ring diagrams, among others, and leads to terms of order n3/2 , where n is the number density of charged particles [467, 18, 656, 12, 468]. 12.2 Divergences in the Nonequilibrium Virial Expansion To illustrate the singular behavior of the collision operators beyond the Boltzmann collision operator, we consider the first term in the virial expansions that shows this behavior. That is, we consider gases in two dimensions, and we examine the time dependence of the three-body collision integral, J3 (f1 |t), which is expressed as an integral involving the operator, Ct (1,2,3), given in the previous chapter as Eq. (11.2.70). We will be concerned with the contributions to this operator from three successive binary collisions, namely those illustrated in Fig. 11.2.3, which we will refer to as correlated sequences of three binary collisions. When the Enskog overlapping configurations are removed from the products of three successive binary collision operators, the relevant integral is  t  t1  (hs) ¯ − (1,2)(1 + P12 ) dt1 dt2 T J3 (f1 |t) = dx2 dx3 0 0 4 (0) (0) × S−(t−t1 ) (1,2,3)T− (1,3)S−(t (1,2,3) 1 −t2 ) 5 (0) (1,2,3) St(0) (1,2,3)#3i=1 fi (xi ,t). × (T− (1,2) + T− (2,3)) S−t 2 (12.2.1) 12.2.1 Digression about Three-Body Events for Short-Range Potentials Since we are considering here sequences of time-separated binary collisions, it is possible to generalize this discussion to include the case where the particles interact with short-range forces and where we can use the binary collision operator expansions of the n- particle streaming operators, as described by Eq. (6.21), which is appropriate for time-separated binary collisions. We can further simplify this binary collision expansion by replacing the binary collision operators for a colliding pair, i,j , say, T(i,j,N.z) in this equation by using adjusted binary collision operators that treat the binary collisions as instantaneous and occurring at the point where the incoming and outgoing trajectories in the relative coordinate frame intersect with the apse line of the collision. We denote this point by a(κ,g) ˆ κ, ˆ where g is the relative velocity of the colliding particles before the collision, a(κ,g) ˆ is the spatial separation between the two particles at collision along the apse line, κ. ˆ We use the notation κˆ for the unit vector along the apse line in order to avoid confusion with a Fourier transform vector, k, later on in this chapter. Thus, we will replace the binary collision operator T(i,j,N,z) by Ta (i,j ), given by

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Divergences, Resummations, and Logarithms



Ta (i,j ) =

d κB(g ˆ ˆ ˆ ij )κ)(b ˆ κˆ − 1), ij − a(κ,g ij , κ)δ(r

(12.2.2)

where B(g, κ) ˆ is the collision function defined in Chapter 2, which depends upon the differential collision cross section and the relative velocity of the colliding particles. The operator bκˆ replaces all velocities to its right by the restituting velocities for the (i,j ) collision. For hard-sphere particles, this construction is simply the hard sphere T(i,j ) operator. Having defined the binary collision operator Ta , we note that there is a corresponding barred operator, T¯ a , defined by    ¯ a = d κB(g ˆ ˆ δ(r ij − a(κ,g ˆ ij )κ)b ˆ κˆ − δ(r ij + a(κ,g ˆ ij )κ) ˆ . (12.2.3) T ij , κ) In what follows, we will treat these adjusted binary collision operators as if they were hard-sphere binary collision operators, since they become hard-sphere binary collision operators in the hard-sphere limit.2 In this approximation, we replace the streaming operators with a binary collision form, given by ⎡ ⎤  (bc) T¯ a (i,j )⎦ . S¯−t (1, . . . ,n) = exp −t ⎣L0 (1, . . . ,n) − (12.2.4) 1≤i 0. In other words, if v12 (t − t2 ) a, there is a very limited range of impact parameters for the (1,3) collision that will lead the scattering of particle 1 into the appropriate solid angle. The time integrations over t1 and t2 can be done simply, because of the delta functions in the operators Tσ (i,j ). It only remains to perform the integration over z3 . Thus, the volume in coordinate space for particle 3 is roughly given as

a d−1 d−1 v12 dτ, dx3 ∝ a v12 τ where τ is the time between the first and last (1,2) collision. We also assume that the v3 integration is well behaved. Therefore, the contribution to the three-particle collision integral from recollision events taking place over a time interval td to t is

 t a d−1 C + O(td /t), d = 3 dτ ≈

x3 ∝ . (12.2.8) D ln(t/td ) d = 2 v τ 12 td Here C,D are constants, and td is a time after which the small angle approximation is valid. We see that this integral converges for d ≥ 3, but it diverges logarithmically in two dimensions.6 Here we will provide a detailed analysis of the mean free path damping, leading to the appearance of terms logarithmic in the density expansions for transport properties.7 The long-time divergence for the two-dimensional case is the first indication of the failure in generalizing the Boltzmann equation to higher densities by expanding the collision operator simply in powers of the density, in analogy to equilibrium virial expansions. In a moment, we will show that the divergence difficulty is not just a pathology of two dimensional systems. It applies to three-dimensional systems as well, but appears one order higher in the density. The geometric analysis of the three-body terms given previously can easily be extended to collision operators involving higher numbers of particles [154]. For three-dimensional systems, the three-body terms are finite, but a divergence appears in the next order in the density. An examination of the dynamical events such as that

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Divergences, Resummations, and Logarithms

Figure 12.2.3 One of the many four-body ring events. These are correlated sequences of four collisions among the four particles.

illustrated in Fig. 12.2.3 shows that the phase space of only one of the field particles is restricted by the condition that a (1,2) collision take place at time t. Each of the other field particles only has to lie in a collision cylinder with length proportional to the duration of the completed sequence, τ . Therefore, the four-body corrections will be of the order ln t in three dimensions and of order t in two dimensions. In general, the n-body collision operator will grow with time as (t/td )n−(d+1) , for n > 3 when d = 2, and for n > 4 when d = 3. These estimates are valid for times long compared to a/ < v > , where < v > is the average speed of the particles. The dynamical events that produce the most divergent terms in each order of the density are called ring events. They are correlated sequences of l binary collisions taking place among l particles, for l ≥ 3, where the final binary collision produces the correlation by linking the first and last binary collisions. All of the other collision sequences among l particles place more geometrical restrictions on the velocities of the particles than the ring events and are therefore less divergent.8 The ring term contribution of the collision operator may be generated schematically9 in terms of sequences of binary collisions with collision operators that have the general form 

¯ a (12)S (0) ∗ (1 + P12 ) dx2 T



l−2 dx3 Ta (1,3)S 0 ∗ (1 + P13 ) Ta (12) (12.2.9)

12.2 Divergences in the Nonequilibrium Virial Expansion

475

for l ≥ 3. As a result of expressing the ring collision operators in a form suitable for summation as a geometric series, the most divergent terms in each order of the density up to logarithmic corrections can be summed together leading to a ring kinetic equation that may be written in a reasonably compact form, to be be discussed in Section 12.3. This resummation is often referred to as the Kawasaki– Oppenheim resummation, since the identification of the most divergent terms in the virial expansion of transport coefficients as ring events and their resummation was first done by K. Kawasaki and I. Oppenheim [350], in the context of evaluating the Green–Kubo expressions for transport coefficients. Divergences and Long-Range Correlations The preceding discussion served to clarify the source of the difficulty in making a virial expansion of the collision operator in order to construct an extension of the Boltzmann equation to higher densities:10 the terms in the virial expansion of the collision operator depend upon the dynamics of fixed numbers of particles moving in infinite space. This gives rise to contributions growing with time, leading us to the conclusion that an important physical phenomenon has not been properly treated. Particles in a real gas are unlikely to travel arbitrarily large distances without colliding with other particles. Typically, the free trajectories between collisions are on the order of a mean free path, because free motion is always interrupted by collisions with other particles in the gas. This mean free path damping is a collective effect not accounted for in the individual terms of the preceding virial expansion. Instead, to obtain physically meaningful results for long times, we must resum the terms in the virial expansion in order to incorporate the collective effects of particle collisions in the collision operator. We note that the fact that almost all of the terms in the virial expansion of the generalized Boltzmann collision operator diverge does not mean that there is no way to generalize the Boltzmann equation. It just implies that the insistence on a power series expansion does not lead to results that are useful for long times, meaning times on the order of or longer than the mean free time between collisions. We can, even now, anticipate some results to be expected from a more well-behaved and physically relevant representation of the collision integral. The effect of collisional damping of free paths ought to lead to an exponential damping factor in the time dependence of three-body collision integrals. This would lead eventually to integrals roughly of the form     ∞ exp[−αnt] (na 3 )2  b + b ln(na 3 ) d = 3 d dt . (12.2.10) ∝ na (na 2 ) c + c ln(na 2 ) d = 2 td − 1 T On the basis of this, we should expect the presence of logarithmic terms in the density expansions of typical nonequilibrium properties such as transport

476

Divergences, Resummations, and Logarithms

coefficients.11 In addition, the renormalization of the divergences in all of the higher-order density terms leads to contributions that to leading order are of the same order in the density, independent of the number of particles. Each additional particle adds a factor n and each additional factor of t adds a factor 1/n, as time must be scaled by the mean free time. This is an indication of the difficulties one encounters when trying to determine the coefficients b or c in Eq. (12.2.10). In a few relatively simple cases – particularly Lorentz gases, discussed later in this chapter – these difficulties have been overcome successfully.12 When the gas is not in equilibrium, the correlated collision sequences produce equal time correlations that can range over distances much larger than a mean free path, even when the same gas in equilibrium has only short-range correlations [371, 372, 373, 374]. The collisional damping suggests that these correlations extend over distances of a few mean free path lengths. However, we shall see in the following chapters that conservation laws give rise to hydrodynamic processes that, in a nonequilibrium gas, produce equal time correlations that extend over distances large compared to a mean free path and, in many cases, over distances on the order of the size of the container.

12.3 Ring Kinetic Equations Here we will derive a kinetic equation that includes the sum of the most divergent terms in each order of the density. In this approximation, the kinetic equation will include the Boltzmann collision operator plus the ring collision operator and, if required, additional non-ring Choh–Uhlenbeck terms. As mentioned earlier, a careful consideration of the various terms in the n-particle contribution to the collision operator reveals that for each n the ring terms are the most divergent ones.13 Kawasaki and Oppenheim showed that these most divergent terms can be summed, leading to a renormalized collision operator that, as we will see, incorporates the effects of the mean free path damping [350]. Here we will follow another, rather heuristic, method leading to the same results, which is based upon the BBGKY hierarchy equations combined with cluster expansion methods and which leads to the same results [187]. We will use this BBGKY method in later chapters too. Before we proceed, it is important to mention one issue of concern. A great asset of the virial expansion method, applied to equilibrium functions, is that in some cases, it allows one to make approximations in a systematic way and find bounds on the errors made by truncating the series at a certain order [279, 568, 475]. The renormalization methods for the kinetic equation, whether by summing series or by using hierarchy methods, make it

12.3 Ring Kinetic Equations

477

very difficult to obtain systematic approximations for nonequilibrium properties, or at least to establish sharp error bounds.14 Almost any density expansion of nonequilibrium quantities obtained by these methods is based on uncontrolled – or, at best, intuitively motivated – approximations. Nevertheless, one may obtain results, such as estimates for the Choh–Uhlenbeck plus leading ring contributions to the transport coefficients for three dimensional systems, of orders n,n2 ln n, and n2 relative to their Boltzmann values. These can be compared with the results of computer-simulated molecular dynamics and those of laboratory experiments. This will be discussed later in this chapter. We have identified in Eq. (12.2.9) the general, but so far schematic, form of the most divergent terms in the collision operator for the n-particle contribution to the kinetic equation for f1 (1,t). The l-particle ring terms have a definite structural form, which is represented by the binary collision sequences contributing to the l-particle collision operator, Eq. (12.2.9). Examples of correlated collision sequences among three particles are illustrated in Fig. 11.2.3, b, c, and d. To effect the resummation of these terms, we generalize the Kawasaki–Oppenheim resummation [350] for time correlation functions to apply to the kinetic equation for the single-particle distribution function. To do this we must first resolve one feature of the virial expansion of the collision operator that now appears as an unphysical property of this expansion. That is, the single-particle distribution functions in the n-particle collision operators, Jn(d) (f1 |t), are all evaluated at the same time t and appear as products, #n1 f1 (xi ,t). Physically, it would be more reasonable to expect that the distribution function for each particle be evaluated at the time of its first appearance in the collision sequence. While we will not give the detailed derivation of the expression for the ring terms here, since we obtain the same result with somewhat less effort later, we can motivate the results by considering the ring term contribution to the kinetic operator, J3(d) (f1 |t), given by Eq. (12.2.5). Now we pay close attention to the single-particle free streaming operators and remove the restriction to spatially homogeneous systems, to write this expression as  J3(d) (f1 |t) =





t

dx2 dx3

dt1 0

t1

¯ a (1,2)(1 + P12 ) dt2 T

0

4 (0) (0) × S−(t−t (1,2)Ta (1,3)(1 + P13 )S−(t (1,2) 1) 1 −t2 ) 5 (0) (0) (1,2)S (3) × Ta (1,2)St−t t−t 2 1 ×

3 # i=1

f1 (xi ,t).

(12.3.1)

478

Divergences, Resummations, and Logarithms

If we now examine and slightly generalize Eq. (11.2.30) of the previous chapter, we can see that (0) (3)f1 (x3,t) = f1 (x3,t1 ) + · · · , St−t 1 (0) St−t (1,2)f1 (x1,t)f1 (x2,t) 2

= f1 (x1,t2 )f1 (x2,t2 ) + · · · ,

(12.3.2) (12.3.3)

where we have given only the first terms in the cluster series relating the distribution functions at one time, t, to the distribution functions at an earlier time, t1 or t2 . With these changes, J3(d) (f1 |t) becomes, to leading order in the density,  t  t1  (d) dt1 dt2 T¯ a (1,2)(1 + P12 ) J3 (f1 |t) = dx2 dx3 0 0 4 (0) × S−(t−t (1,2)Ta (1,3)(1 + P13 )f1 (x3,t1 )S(t(0)1 −t2 ) (1,2) 1) × Ta (1,2)f1 (x1,t2 )f1 (x2,t2 ) + · · · .

(12.3.4)

The most divergent terms in the power series expansion for the kinetic equation have the schematic form given by Eq. (12.2.9). Keeping track of the time variables and realizing that the sum of all the ring terms must be written as a time-ordered exponential, we obtain a ring kinetic equation for the single-particle distribution with a Boltzmann collision operator in the exponential. Since we are about to derive this equation in another way, we will give the explicit expression for the resummed collision operator as Eq. (12.3.17) [632, 646]. Hierarchy Method for the Ring Resummation In this method, we begin with a restatement of the first two BBGKY hierarchy equations, but, for the time being, we will neglect here many effects due to the finite duration of collisions, and to the presence of boundaries. The equations, under these assumptions, become  ∂f1 (x1,t) (12.3.5) + L0 (x1 )f1 (x1,t) = dx2 T¯ a (1,2)f2 (x1,x2,t) ∂t  ∂f2 (x1,x2,t)  + L0 (x1 ) + L0 (x2 ) − T¯ a (1,2) f2 (x1,x2,t) ∂t    ¯ a (1,3) + T ¯ a (2,3) f3 (x1,x2,x3,t). = dx3 T (12.3.6) Here the binary collision operator, Ta (i,j ), is given by Eq. (12.2.2). Our goal is to use the hierarchy equations, together with cluster expansions of the distribution function, to obtain a generalization of the Boltzmann equation (GBE) that is free of the divergence problems encountered in the virial expansions of the collision

12.3 Ring Kinetic Equations

479

operator. To that end, we apply a cluster expansion for the distribution functions of the following form:15 f2 (x1,x2,t) = f1 (x1,t)f1 (x2,t) + g2 (x1,x2,t),

(12.3.7)

f3 (x1,x2,x3,t) = f1 (x1,t)f1 (x2,t)f1 (x3,t) + f1 (x1,t)g2 (x2,x3,t) + f1 (x2,t)g2 (x1,x3,t) + f1 (x3,t)g2 (x1,x2,t) + g3 (x1,x2,x3,t).

(12.3.8)

This cluster expansion is similar to an Ursell expansion and is used here to formulate, in a compact way, a closure approximations for the hierarchy equations. Using this cluster expansion, one may construct approximate solutions to the hierarchy equations by setting correlation functions, gs (x1, . . . ,xs ,t), of successively higher orders equal to zero. If we set g2 = 0 for pre-collisional phases and replace Ta with T0 , where T0 (i,j ) = δ(r i − r j )t0 (vi ,vj ), with

(12.3.9)

 t0 (vi ,vj ) =

d κB(g, ˆ κ)(b ˆ κ− ˆ 1),

(12.3.10)

then Eq. (12.3.5) becomes the usual Boltzmann equation.16 The next approximation, and the one we use here, is to keep g2 but to set g3 = 0 for pre-collisional configurations. It is clear one can proceed in this way in order to obtain a closed set of equations for a general distribution function, fn . These equations become considerably more complicated with increasing levels of closure. Here we will explore the consequences of setting g3 = 0 in Eq. (12.3.8), using Eq. (12.3.7) for f2 , and inserting the resulting expression for f3 in Eq. (12.3.6). In doing so, we obtain a coupled set of equations for f1 and g2 . We find that the equation for g2 (x1,x2,t) takes the form ∂g2 (x1,x2,t) + (L0 (x1 ) + L0 (x2 ) − Ta (1,2)) g2 (x1,x2,t) ∂t = Ta (1,2)f1 (x1,t)f (x2,t) + a (1,2|t)g2 (x1,x2,t),

(12.3.11)

where a (1,2|t) is a two-particle kinetic operator given by17  a (1,2|t) = dx3 [Ta (1,3)(1 + P13 ) + Ta (2,3)(1 + P23 )] f1 (x3,t).

(12.3.12)

We have explicitly noted the time dependence of the operator a (1,2|t) caused by the time dependence of the single-particle distribution functions. This time

480

Divergences, Resummations, and Logarithms

Figure 12.3.1 A repeated ring event with five collisions among four particles. Such events have phase volumes that are less divergent than the ring events.

dependence requires us to write the solution of Eq. (12.3.11) with a time-ordered exponential operator and is given by g2 (x1,x2,t)

   τ = T exp − dτ1 [L0 (x1 ) + L0 (x2 ) − Ta (1,2) − a (1,2|τ1 )] g2 (x1,x2,0) 0    τ  t dτ T exp − dτ1 [L0 (x1 ) + L0 (x2 ) − Ta (1,2) − a (1,2|τ1 )] + 0

0

× Ta (1,2)f1 (x1,τ1 )f1 (x2,τ1 ).

(12.3.13)

Here the symbol T indicates a time-ordered exponential. The presence of the operator Ta (1,2) in the exponential operators appearing on the left-hand side of Eq. (12.3.13) indicates that this expression for g2 will include not only the resummation of ring terms in the kinetic equation for f1 but also the resummation

12.3 Ring Kinetic Equations

481

of repeated ring terms of all orders. This can easily be seen by expanding the exponential operators in powers of the operator Ta (1,2). The zeroth-order term in Ta (1,2) provides the resummation of the ring terms in the kinetic equation, and the first-order term represents the resummation of two repeated ring events, and so on. The repeated ring events are less divergent than the ring events involving the same number of particles. Such an event, for four particles, is illustrated in Fig. 12.3.1. The four-body repeated rings would lead to contributions to the GBE, of the form  t1 −td  t−td 1 ∼ (ln t)2 . dt1 dτ (12.3.14) (t − t1 )(t1 − τ ) td td The time, td , on the order of a/ < v >, marks the onset of the 1/t behavior and is needed for the convergence of this estimate. This contribution, while divergent at large times, t, is less divergent than the four-body ring term, which grows as the first power of the time. This being the case, we should neglect the term Ta in the exponential operators, so that the equation for g2 becomes ∂g2 (x1,x2,t) + (L0 (x1 ) + L0 (x2 ) − a (1,2|t)) g2 (x1,x2,t) ∂t (12.3.15) = T¯ a (1,2)f1 (x1,t)f1 (x2,t), with solution

  t  dτ1 [L0 (x1 ) + L0 (x2 ) − a (1,2|τ1 )] g2 (x1,x2,0) g2 (x1,x2,t) = T exp − 0   τ   t dτ T exp − dτ1 [L0 (x1 ) + L0 (x2 ) − a (1,2|τ1 )] + 0

0

× Ta (1,2)f1 (x1,τ1 )f1 (x2,τ1 ).

(12.3.16)

The first term includes the effect of the initial correlation terms contained in g2 (0). We recall that g2 appears in the first hierarchy equation in the combination T¯ a (1,2)g2 (x1,x2,t), and it is this combination we have in mind in the discussion here. We assume that the initial correlations between two particles are non-zero only over a distance of a few particle diameters. Then one sees that the initial condition term on the right-hand side of Eq. (12.3.16) consists of events similar to ring events but restricted by the condition that the first interaction of the two particles, as described in g2 (x1,x2,0), is taking place precisely at a time, t, earlier than the final interaction of the two particles in the collision sequence, while in the ring events responsible for the second term, the time interval between the first and last collision in the sequence, τ, has the range 0 ≤ τ ≤ t. The initial condition term is, at most, the sum of contributions from less divergent events than the ring terms described by the second term [154]. It is important to point out that the initial condition term does have a hydrodynamic mode component that will eventually

482

Divergences, Resummations, and Logarithms

result from the hydrodynamic modes of the kinetic operator in the exponential acting on g2 (0). Therefore, for reasons discussed in the next chapter, this term will decay algebraically with time. We see that any assumption that the initial condition terms decay exponentially is incorrect. Anticipating the long-time tail discussion in the following chapter, we can argue that the initial conditions terms decay one power of time faster than the time decays resulting from the second term in Eq. (12.3.16).18 In the following, the initial condition term will be neglected compared to the full ring term, and in any case, its contribution should be studied together with the resummation of similar, less divergent terms, for a consistent treatment of the kinetic equation beyond the ring terms. Combining the results for the two-particle correlation function, Eq. (12.3.16), with the first hierarchy equation, we obtain the ring kinetic equation for the singleparticle distribution function as  ∂f1 (x1,t) ¯ a (1,2)f1 (x1,t)f1 (x2,t) + L0 (x1 )f1 (x1,t) = dx2 T ∂t   τ    t + dx2 T¯ a (1,2) dτ T exp − dτ1 [L0 (x1 ) + L0 (x2 ) − a (1,2|τ )] 0

0

× Ta (1,2)f1 (x1,τ1 )f1 (x2,τ1 ).

(12.3.17)

By combining Eq. (12.3.16) with Eqs. (12.3.5) and (12.3.7), we have obtained the summation of the most divergent terms in each order of the density for the GBE. However, this solution so far is very formal, it contains a time-ordered exponential and is more than we will actually need for our further work. Instead of pursuing the consequences of the nonlinear GBE, we will turn our attention to the linearized version, which can be used to obtain linear hydrodynamic equations and expressions for transport coefficients in low-density systems, which are obtained after the resummation of the ring events is carried out. The Linearized Ring Kinetic Equation In this section, we discuss the linearized version of Eq. (12.3.17) with the eventual goal of deriving linearized hydrodynamic equations using the linearized GBE. Of course, we must carry out the linearization of the distribution functions in a way that is consistent with the approximations we have made so far. We linearize the distribution functions about a Maxwell–Boltzmann distribution, nφ0, at temperature T , uniform number density n, and in a frame where the equilibrium mean velocity is zero. Then we can write the linearized forms of the distribution functions as f1 (xi ,t) = nφ0 (vi ) (1 + χ1 (xi ,t)) ,

(12.3.18)

12.3 Ring Kinetic Equations

483

f2 (xi ,xj ,t) = n2 φ0 (vi )φ0 (vj )   × 1 + χ1 (xi ,t) + χ1 (xj ,t) + χ2 (xi ,xj ,t) ,

(12.3.19)

f3 (xi ,xj ,xk,t) = n φ0 (vi )φ0 (vj )φ0 (vk )  × 1 + χ1 (xi ,t) + χ1 (xj ,t) + χ1 (xk,t) 3

+ χ2 (xi ,xj ,t) + χ2 (xi ,xk,t)

 + χ2 (xj ,xk,t) + χ3 (xi ,xj ,xk,t) ,

(12.3.20)

and so on. To be consistent with the previous approximation, we set χ3 = 0. As may be checked easily, one has Ta (i,j )φ0 (vi )φ0 (vj ) = 0. Then the first two hierarchy equations, on neglect of repeated ring contributions, are of the form ∂χ1 (x1,t) + L0 (1)χ1 (x1,t) ∂t  =n

dx2 T¯ a (1,2) (χ1 (x1,t) + χ1 (x2,t) + χ2 (x1,x2,t)) ,

∂χ2 (x1,x2,t) + (L0 (1) + L0 (2)) χ2 (x1,x2,t) ∂t = T¯ a (1,2) (χ1 (x1,t) + χ1 (x2,t)) + (1 + P12 )n × (χ2 (x1,x2,t) + χ2 (x2,x3,t)) .

(12.3.21)

 dx3 φ0 (v3 )Ta (1,3) (12.3.22)

Without the final term containing χ2, the first of these two equations is the linearized Boltzmann equation. The final term contains a correction to this due to correlations between the two colliding particles. These correlations are described by the second equation, also linear, but with an inhomogeneous term, Ta (1,2) (χ1 (1) + χ1 (2)). One can read the the first term on the right-hand side of the second equation as describing a first collision between particles 1 and 2. The second term describes changes of the pair distribution function due to collisions of the initially colliding pair, or again exchange partners introduced at preceding collisions, each with new particles in the gas, as described by the last terms on the right-hand side of Eq. (12.3.22). The ring events contained here are completed with the final (1,2) collision that is described by the third term on the right-hand side of Eq. (12.3.21) and lead to the linear form of the GBE.19 Since Eqs. (12.3.21) and (12.3.22) are linear and translation invariant, their analysis can be simplified by taking Laplace and Fourier transforms of the functions χ1 (xi ,t) and χ2 (xi ,xj ,t). Therefore, we define the transforms of these two functions by  ∞  dt dr i exp[−zt − ik · r i ]χ1 (xi ,t), (12.3.23) χ1,z (k,vi ) = 0

484

Divergences, Resummations, and Logarithms

and





∞

χ2,z (k1,vi ,k2,vj ) =

dt

dr i dr j exp[−zt − ik1 · r i − ik2 · r j ]χ2 (xi ,xj ,t).

0

(12.3.24) We also use the Fourier representation of the binary collision operators so that    Ta,k (1,2) = d κˆ exp −ik · a(κ,g) ˆ κˆ B(κ,g)(b ˆ (12.3.25) κˆ − 1), ¯ a,k=0 and Ta,k=0 (1,2) = t0 (1,2), ¯ a,k . Note that both T with a similar definition of T defined by Eq. (12.3.10). We can now express Eqs. (12.3.21) and (12.3.22) in terms of the transformed functions. We obtain [z + ik · v1 ] χ1,k,z (v1 ) = χ1,k (v1,t = 0) + k (v1 )χ1,k,z (v1 )   dq ¯ Ta,l (v1,v2 )φ0 (v2 )χ2,q,l,z (v1,v2 ), + n dv2 (12.3.26) (2π)d   z + iq · v1 +il · v2 − q (v1 ) − l (v2 ) χ2,q,l,z = χ2,q,l (v1,v2,t = 0) + Ta,−l (v1,v2 )χ1,k,z (v1 ) + Ta,q (v1,v2 )χ1,k,z (v2 ).

(12.3.27)

Here the wave vector l = k − q, and the operator k (vi ) is a linearized, wavenumber-dependent, collision operator given by    (12.3.28) k (vi ) = dvj feq (vj ) Ta,0 (i,j ) + Ta,k (i,j )Pij . Notice that, unlike the collision operators, a (i,j |τ ), appearing in Eq. (12.3.15), this collision operator is time independent. We neglect the initial correlations between the two particles in Eq. (12.3.27) and then combine Eqs. (12.3.26) and (12.3.27) to obtain a closed expression for the single-particle function, χ1,k,z , as   (12.3.29) z + ik · v1 − k (v1 ) − Rk,z (v1 ) χ1,k,z (v1 ) = χ1,k (v1,t = 0), where the ring collision operator, Rk,z (v1 ), is given by   dq ¯ a,l (v1,v2 ) Rk,z (v1 ) = dv2 feq (v2 )T (2π)d   × Gq,l,z (v1,v2 ) Ta,−l (v1,v2 ) + Ta,q (v1,v2 )P12 ,

(12.3.30)

with  −1 Gq,l,z (v1,v2 ) = z + iq · v1 +il · v2 − q (v1 ) − l (v2 ) . As before, l = k − q.

(12.3.31)

12.3 Ring Kinetic Equations

485

For sufficiently low densities – that is, for a  – the wave number dependence of the operator, k,q , and l in Eqs. (12.3.29) and (12.3.31) can be neglected since the wave number dependence leads to terms that are higher order in the density. We can also neglect, for ka 1, the wave number k dependence in the binary collision operators T¯ a,(k−q) . Then the equation for χ1,k,z (v1 ) becomes   (12.3.32) z + ik · v1 − (v1 ) − Rz (v1 ) χ1,k,z (v1 ) = χ1,k (v1,t = 0), where the linearized Boltzmann collision operator, (v1 ), is the limit as k → 0 of k, given by Eq. (8.2.9) and repeated here, as  (12.3.33) (v1 ) = dv2 feq (v2 )t0 (1,2)(1 + P12 ), and the ring operator, Rz (v1 ), is   dq feq (v2 )T¯ a,−q (v1,v2 ) Rz (v1 ) = dv2 (2π)d   ˜ q,l,z (v1,v2 ) Ta,q (v1,v2 )(1 + P12 ) , ×G and

  ˜ q,l,z (v1,v2 ) = z + i(q · v1 + l · v2 ) − (v1 ) − (v2 ) −1 . G

(12.3.34)

(12.3.35)

Equations (12.3.32), (12.3.34), and (12.3.35) are the main starting points for our analysis to follow. This ring kinetic equation is a generalization of the linear Boltzmann equation that includes, in addition to the binary collision term, the contributions from the ring events;20 that is, the most divergent terms in each order in the density, are contained in the operator Rz (v1 ), defined earlier.21 One can easily follow the dynamics in the ring operator as consisting of a first and and a last binary collision between the same two particles, or particles connected to the final pair through the action of the particle exchange operators, Pij , sandwiched between ˜ k.l,z . sequences of uncorrelated binary collisions, as described by the operator G These are precisely the ring events, appearing here in a particular form that results from the application of the Laplace and Fourier transforms. This picture is also clear from an inspection of the time-dependent version of Eq. (12.3.29) obtained by inverting the Laplace transform. It is   ∂ + ik · v1 χ1,k (v1,t) = k (v1 )χ1.k (v1,t) ∂t    t dq dt1 dv2 feq (v2 )T¯ a,l (v1,v2 ) + (2π)d 0 3  2 × exp −t1 iq · v1 +il · v2 − (v1 ) − (v2 )   × Ta,q (v1,v2 )(1 + P12 ) χ1,k (v1,t − t1 ). (12.3.36)

486

Divergences, Resummations, and Logarithms

Here one sees the time development of the ring collision term – namely, there is a binary collision at time t − t1 – then over the time interval t1 , there are sequences of an arbitrary number of uncorrelated binary collisions and free motion between them as described by the product of two Boltzmann propagators, exp{−t1 (iq · v1 +il · v2 −(v1 )−(v2 ))}, followed by a binary collision at time t. The Ring Kinetic Equation for Tagged Particle Motion Much the same kind of arguments as were used for the development of the ring kinetic equation for the single-particle distribution function can be applied to study the distribution function for the motion of a tagged particle, or a very dilute concentration of tagged particles, in a gas of otherwise identical particles. If one realizes that the tagged particle is always followed so that permutation operators exchanging particles never apply to it, one can write down the ring kinetic equation for the Laplace transform of the deviation from equilibrium of the distribution of the (D) (v1 ), by making a slight change to the ring kinetic equation tagged particles, χ1,k,z of a pure gas. This equation is  (D)  (D) (v1 ) = χ1,k (v1,t = 0), (12.3.37) z + ik · v1 − a,D (v1 ) − RD,k,z (v1 ) χ1,k,z where D identifies these equations as appropriate for describing the motion of a tagged particle, often referred to as self-diffusion:22  a,D (v1 ) = dv2 feq (v2 )Ta,0 (v1,v2 ), (12.3.38) and





RD,k,z (v1 ) =

dv2

dq ˜ D,q,l,z Ta,q (v1,v2 ), (12.3.39) feq (v2 )T¯ a,−q (v1,v2 )G (2π)d

where

  ˜ D,q,l,z (v1,v2 ) = z + iq · v1 +il · v2 − a,D (v1 ) − (v2 ) −1 . G

(12.3.40)

This equation is needed for the evaluation of the ring contributions to the Green– Kubo formula for the coefficient of self-diffusion. 12.4 Applications to Green–Kubo Correlation Functions Almost everything discussed so far in this chapter has a direct application to the calculation of the time correlation functions that appear as integrands in the Green– Kubo formulae for transport coefficients, discussed23 in Chapter 6. As we will see more explicitly in this chapter the following chapter, the results for transport coefficients for dilute gases24 in dimensions greater than 2, as obtained from the

12.4 Green-Kubo Correlation Functions

487

Green–Kubo formulae, are identical, to leading order in the density, to the results obtained for transport coefficients as obtained from the linearized Boltzmann equation. For dilute gases – that is, gases where the ideal gas equation of state is a good approximation – the evaluation of the Green–Kubo time correlation functions of the kinetic parts of the currents for the coefficients of shear viscosity and thermal conductivity leads directly to the ring kinetic equation, Eq. (12.3.29), and in a similar manner, to the ring equation, Eq. (12.3.37), needed for the coefficient of tagged particle diffusion. We begin by expressing the Green–Kubo time correlation functions in terms of one- and two-particle distribution functions that obey linearized BBGKY hierarchy equations. Here we consider only the time correlation functions for the kinetic parts of the microscopic currents in the Green–Kubo formulae. The kinetic parts of the microscopic currents, Jξ,K , for the various transport coefficients are25 JD,K = v1x , Jξ,K (N ) =

N 

(12.4.1) jξ,K (vi ),

(12.4.2)

i=1

where D denotes the coefficient of self-diffusion, and ξ = η,λ, denote the coefficients of shear viscosity and thermal conductivity, respectively, with jη,K (vi ) =  2 mvix viy , and jλ,K (vi ) = mvi − (d + 2)kB T vix /2. The kinetic–kinetic parts of the time correlation functions, ρξ,K (t), can be expressed in terms of a one-particle distribution function, (d) K , [268, 269, 305, 189, 193], as   Jξ,K (0)Jξ,K (−t) 0 ρξ,K (t) ≡ 2 1 Jξ,K   −1  2 dv0 φ0 (v0 ) dx1 jξ,K (v1 )jξ,K (v0 ) × (d) = jξ,K (v1 ) K (x1,t|x0 ), (12.4.3) where the single-particle distribution function, (d) K (x1,t|v0 ), is the one resulting from the initial ensemble  1 δ(vi − v0 ), ρv0 (,t = 0) = ρeq () Nφ0 (v0 ) i=1 N

(12.4.4)

and is used for either transport coefficient, ξ = η,λ. For tagged particle diffusion, the formulation is slightly different. Here

488

Divergences, Resummations, and Logarithms



 v1,x (0)v1,x (−t)  2  ρD (t) ≡ v1,x    2 −1 dv0 φ0 (v0 ) dx1 v1,x v0,x (d) = v1,x D (x1,t|x0 ),

(12.4.5)

where the single-particle distribution function, (d) D (x1,t|x0 ), results from a similar initial ensemble that identifies particle 1 as the tagged particle, given by ρD,v0 (,t = 0) =

1 ρeq ()δ(v1 − v0 ). φ0 (v0 )

(12.4.6)

Notice that (d) (x1,t|v0 ) is to be read as a function of x1 , but does not depend on the spatial coordinates because of the translation invariance of the initial distribution. At this point, we are in a position to use the methods just developed for obtaining kinetic equations – ring kinetic equations in particular – for the distribution func(D) (v1,t), but we may set k = 0, since the functions now are tions χ1,k (v1,t) and χ1,k spatially homogeneous. The initial values of these distribution functions are δ(v1 − v0 ) + nφ0 (v1 ), V δ(v1 − v0 ) (d) . D (v1,v0,t = 0) = V (d) K (v1,v0,t = 0) =

(12.4.7) (12.4.8)

Using the arguments of the previous section, we see that distribution functions (d) (d) ξ (v1,t|v0 ),D (v1,t|v0 ) satisfy the linear kinetic equations, Eqs. (12.3.29) or (12.3.37), respectively, with k = 0. That is,   (d) z − (v1 ) − Rz (v1 ) (d) (12.4.9) K,z (v1 |v0 ) = K (v1,t = 0|v0 ),  (d)  (12.4.10) z − D (v1 ) − RD,z (v1 ) D,z (v1 ) = (d) D (v1,t = 0|v0 ), Notice that because the single-particle distribution functions depend on velocities but not on position, due to translational invariance, the kinetic equations they satisfy are spatially homogeneous versions of Eqs. (12.3.29) and (12.3.37). The Enskog Modification of the Ring Kinetic Equation for Self Diffusion For hard-sphere systems, it is possible to extend much of the analysis of this chapter to higher densities if one makes approximations similar to those made in the derivation of the revised Enskog equation discussed in a previous chapter. That is, one takes into account excluded volume and collisional transfer effects in each binary collision occurring in the ring kinetic equation but ignores all collision sequences other than single binary collisions and the most divergent events making up the ring

12.4 Green-Kubo Correlation Functions

489

approximation. If one makes a “topological ordering” of the diagrammatic expansion of the full collision operator [632], one obtains the revised Enskog operator as the zero-loop approximation. With some additional efforts, one may show the oneloop approximation, in addition, contains the ring diagrams with mostly Enskogdecorated vertices, and so on. In addition, as shown in Chapter 7, the revised Enskog operator has the equilibrium distribution as stationary solution (also in nonuniform equilibrium states) and the transport coefficients resulting from it satisfy the Onsager symmetries. Notice that each additional loop with a given number of particles adds an additional geometric constraint. Essentially, this approximation consists of “decorating” each binary collision, both in the “Boltzmann term” and in the ring sum with excluded volume effects, replacing all binary collision operators ¯ and T forms, and expressing the ring collision operators by their appropriate T in an irreducible form, with no hidden overlapping contributions, as discussed in the arguments leading to Eq. (11.2.70), for example. The analysis leading to the proper form of this Enskog modified ring kinetic equation was carried out, for the case of self-diffusion, by J. R. Dorfman and E. G. D. Cohen on the basis of cluster expansion methods [155], and by using diagrammatic techniques by H. van Beijeren and M. H. Ernst [632, 646]. It is subtle and lengthy. Here we consider the ring contributions to the distribution functions for self-diffusion, (d) D (v1,v0,t), or its Laplace transform, needed for the velocity autocorrelation function appearing in the Green–Kubo formula. We provide only the final results and refer to the literature for the details. One of the essential features of these derivations is that the Boltzmann propagators appearing in the ring collision operator are, in this approximation, replaced by the propagators obtained by linearizing the revised Enskog equation, and the binary collision operators now are those appropriate for hard spheres of radius a. The equations for the self-diffusion function are somewhat different due to the tagging of one particle. These equations are 

(d) (d) z − D,E (v1 ) − R(E) D,z (v1 ) D,z (v1,v0 ) = D (v1,v0,t = 0),

(12.4.11)

where26 

RD,z (v1 )(d) D,z (v1,v0 )

=



 dq ¯ −q (1,2) dv2 feq (v2 ) χ (n)T d (2π)     ¯ −q (1,3)H (1,3,q) 1 + ng2 (q) dx4 P24 + dx3 T    (E) × GD,q,z (1,2) χ (n)Tq (1,2) + dx5 Tq (1,5)H (1,5,q) × (v1,v0 ),

(12.4.12)

490

and

Divergences, Resummations, and Logarithms

 −1 . G(E) D,q,z (1,2) = z + iq · v12 − D,E (v1 ) − nA−q (v2 ) − χ (n)−q (v2 ) (12.4.13)

The operator D,E was defined in Eq. (7.5.80). Here the action of the operator Aq on a function of the form φ0 (v2 )g(v2 ) is given by   Aq φ0 (v2 )g(v2 ) = iq · v C(q) − χ (n)f (q)  × dv4 φ0 (v4 )v4 g(v4 ), (12.4.14) where C(q) is the Fourier transform of the direct correlation function and f (q) is the Fourier transform of the Mayer f -function, both defined in Chapter 6. The function H (1,3,q) is the Fourier transform with respect to r 12 of the function H (r 1,r 3 |r 2 ), defined in Eq. (7.3.7), and g2 (q) is the Fourier transform of the pair correlation function. In the following chapter, we will use the Enskog modification of the linearized ring equation for self-diffusion in order to provide an approximate form for long-time tails in the velocity autocorrelation function for hard spheres at elevated densities. The Enskog modification to the ring equation for the collective coefficients – shear viscosity and thermal conductivity – requires further complicated modifications and will not be discussed here. 12.5 Logarithms in the Density Expansions of Transport Coefficients In what follows here, we return to the case of moderately dilute gases and show how the mean free path damping is incorporated in the ring equations and how it leads to log n terms in the density expansions for the transport coefficients. We note that the ring collision operator has the form of an integral over wave vectors, q. A convenient separation of this integral into different regions is provided by the natural length scales characteristic of the gas [335, 337, 555, 374]. The length scales of interest in what follows are the mean free path length, , and a characteristic macroscopic length, usually taken to be the size of the container, L. For purposes of this discussion, we shall assume that the gas is dilute enough so that the ordering of these lengths is given by a  L, where a is the range of the inter-particle forces. The ordering of space scales leads naturally to an ordering of the different regions in wave number space that are important for the analysis of the wave vector integral. For q ≥ a −1, where q = |q|, processes are described that take place over distances comparable to the range of interactions. In the intermediate region, −1 ≤ q ≤ a −1, one must expect that ring events taking place over distances of a few mean free paths should be important. That is,

12.5 Logarithms in the Density Expansions of Transport Coefficients

491

one has to consider the contributions of 3,4,. . .-particle ring events with damped trajectories. The contributions of these few particle ring events will be determined by a combination of the geometrical conditions required for the event to take place, and the collisional damping due to collisions with other particles in the gas that restricts the range of free motion of the particles participating in the ring events to be on the order of a mean free path. We will show that logarithmic terms in the density expansions, described in Eq. (12.2.10), arise from this intermediate region of wave numbers. The small q region provides us with a new phenomenon; namely, the contribution from microscopic hydrodynamic modes of the kinetic equation become important on long length scales or, equivalently, small wave numbers. In the following chapter, we will focus our attention on this long-wavelength or small-q region where q ≤ l −1 , introduce the long-time tails in the time correlation functions for transport coefficients, and discuss their striking contributions to the transport coefficients and, ultimately, to the forms of the hydrodynamic equations. 12.5.1 Fluid Dynamics from the Ring Equation We return to the ring kinetic equation for a low-density fluid, Eq. (12.3.32), and use by now standard projection operator methods to derive hydrodynamic equations together with expressions for the transport coefficients that appear in them. We may use the derivation presented in Section 3.8, replacing the operator Lk with 2 L(R) k (z) ≡ ik · v − (v) − Rz (v). The collision invariants, 1,v, and mv are zero eigenfunctions of both 0 and Rz , so that we can go through all the steps take in Section 3.8 with resulting form of the generalized hydrodynamic equations27 z ψi | χk  +

d+2 

  ij ψj  χk 

j =1

= ψi | χk (t = 0 − k 2

d+2 

   Uij (k,z) ψj  χk ,

(12.5.1)

j =1

where the matrix elements, using the projected operator, Lˆ (R) k (z) are given by   (12.5.2) ij (k) = ψi |iv · k|ψj , 

−1 (12.5.3) (kˆ · v)|ψj . Uij (k,z) = ψi |(kˆ · v) z + Lˆ (R) k (z) The k- and z-dependent hydrodynamic equations remain the same as in Section 3.8 but with Uij (k,z) given by Eq. (12.5.3). In particular, if we take kˆ = xˆ , the expressions for the k- and z-dependent kinetic contributions to the coefficients of shear viscosity and thermal conductivity are

492

Divergences, Resummations, and Logarithms



−1

(12.5.4) η(k,z) = βm vx vy z + Lˆ (R) vx vy , k (z)

 −1 βmv 2 − (d + 2)

βmv 2 − (d + 2) λ(k,z) = kB vx z + Lˆ (R) . (z) vx k 2 2 (12.5.5) We consider the general form for Uij (k,z) Eq. (12.5.3), and expand the inverse operator in powers of the ring operator nRz , keeping only the zeroth-order and firstorder terms in Rz since higher powers of the ring operator are of the same order as terms we have neglected already. Thus, a consistent expression for Uij (k,z) is Uij (k,z) = Uij(B) (k,z) + Uij(R) (k,z), where

0 1 Uij(B) (k,z) = ψi (kˆ · v) [z − ]−1 ψj (kˆ · v) , 0 Uij(R) (k,z) = ψi (kˆ · v) [z − ]−1 Rk,z (v) 1 × [z − ]−1 ψj (kˆ · v) .

(12.5.6)

(12.5.7)

(12.5.8)

The second term contains the ring operator, Eq. (12.3.34), and is    dq (R) feq (v1 )φ0 (v2 )ψi (kˆ · v) [z − ]−1 Uij (k,z) = dv1 dv2 (2π)d  −1 × Ta,−q (v1,v2 ) z + i(q · v1 + l · v2 )) − (v1 ) − (v2 )   (12.5.9) × Ta,q (v1,v2 )(1 + P12 ) [z − ]−1 (kˆ · v1 )ψj (v1 ). The first terms, Uij(B) , are independent of the wave number and are the Boltzmann equation contributions to the matrix elements. In the limit z → 0, the elements Uij(B) (z → 0) are the Boltzmann equation values of the Navier–Stokes transport coefficients discussed in Chapter 3. The second terms, Uij(R) , are, for small z and k, the contribution to the generalized Navier–Stokes transport coefficients from the ring kinetic operator. The standard Navier–Stokes transport coefficients are expressed by the elements Uij (k = 0,z → 0), if these limits exist, since the elements appear in Eqs. (3.8.24) as k 2 Uij . By considering non-zero values of |k| in Uij (k,z), one can, in principle, continue the hydrodynamic equations beyond the Navier–Stokes equations. In this way, we may regard these quantities as generalized transport coefficients. In the next chapter, we will show that by keeping the k dependence in the inverse operator appearing in Eq. (12.5.9), we obtain correction terms to the Navier–Stokes equations for gases in three dimensions of order k 5/2 and higher. For the moment, we will focus solely on the Navier–Stokes coefficients alone and neglect this additional k dependence.

12.5 Logarithms in the Density Expansions of Transport Coefficients

493

Unlike the Boltzmann equation contributions discussed in Section 3.8, the next term, the ring contribution, has the form of an integral over wave vectors, q. As described earlier, a convenient separation of this integral into different regions is provided by the natural length scales, a, the range of the potential the mean free path length, , and a characteristic macroscopic length, L, usually taken to be the size of the container, and we are considering the case when a  L. The ordering of space scales leads naturally to an ordering of the different regions in wave number space that are important for the analysis of the wave vector integral. Therefore, we separate the wave-number integration into two pieces, 0 ≤ q ≤ q0 and q0 ≤ q, with q0 ∼ −1 . For q ≥ a −1, processes are described that take place over distances comparable to the range of interactions. In the intermediate region,  ≤ q ≤ a −1, one must expect that ring events taking place over distances of a few mean free paths should be important. That is, one has to consider the contributions of 3,4, . . .-particle ring events with damped trajectories. The contributions of these few particle ring events will be determined by a combination of the geometrical conditions required for the event to take place, as well as the collisional damping that restricts the free motion of particles to be on the order of a mean free path. We will show that logarithmic terms in the density expansions, described in Eq. (12.2.10), arise from this intermediate region of wave numbers. For large values of the wave number q, on the order of an inverse mean free path or greater, wave number is no longer a small parameter useful for perturbation expansions of eigenfunctions and eigenvalues of the kinetic operators. For our purposes, therefore, it is useful to express all of the matrix elements Uij(R) (0,z) as Uij(R) (0,z) = Uij(R,),

(12.5.10)

where the superscripts, < or >, refer to the small-wave-number region of integration and the large-wave-number region, respectively. Here we will focus on the large-wave-number contributions, Uij(R,>) , and in Chapter 13, we will focus our attention on the long wavelength or small q region where q −1 . Tagged Particle Diffusion Before completing this discussion, we will briefly consider the corresponding analysis for the ring kinetic equation for tagged particle diffusion given by Eq. (12.3.37). For tagged particle diffusion, there is only one conserved quantity, namely the number of tagged particles in the gas. This leads to a simplified projection operator on the unit function, PD (k) = |ψD (v) ψD (v)|, and its complement, PD,⊥ = 1 − P D , which, when applied to the ring kinetic equation for tagged

494

Divergences, Resummations, and Logarithms

particle diffusion leads to a diffusion type equation for the local density deviation, n(k,z), as   (12.5.11) z + k 2 D(k,z) n(k,z) = n(k,t = 0), where

 D(k,z) =

dv1 feq (v1 )(kˆ · v1 )HD,k,z (v1 )(kˆ · v1 ),

(12.5.12)

where  HD,k,z (v1 ) = PD,⊥ z + ik · PD,⊥ v1 PD,⊥   −1 − PD,⊥ D (v1 ) + RD,k,z (v1 ) PD,⊥ PD,⊥ .

(12.5.13)

The ring operator RD,k,z (v1 ) is given by Eq. (12.3.39). Viscosity as an Example To illustrate the application of these  results, we use  the equation for the transverse (η) momentum density, ui (k,z) ≡ ψη,i (k,v)|χ1,k,z , from Eq. (12.5.1), as  (η)  (η) (12.5.14) z + k 2 Uηi ,ηi (k,z) ui (k,z) = ui (k,t = 0). We discuss this equation in some detail, as an example of the analyses of the generalized hydrodynamic equations. The generalized coefficient of kinematic viscosity in the Navier–Stokes equation for this velocity field is then Uηi ,ηi (k = 0,z) ≡ ν(0,z). The ring contribution to the kinematic viscosity is given, for small z, by    dq (R) ν (0,z) = βmn dv1 dv2 φ0 (v1 )φ0 (v2 )v1x v1y []−1 (2π)d   ¯ a,−q (v1,v2 ) z + iq · (v1 − v2 ) − (v1 ) − (v2 ) −1 ×T × Ta,q (v1,v2 )(1 + P12 ) []−1 v1x v1y .

(12.5.15)

We focus now on the wave number integration. Density Logarithms in the Fluid Transport Coefficients The region q > q0, where  is small but finite, contains the contributions from dynamical events that take place over distances of a few mean free paths or less. It is in this region that the mean free path damping of free motion of particles between collisions is important. If we were to expand the denominator obtained

12.5 Logarithms in the Density Expansions of Transport Coefficients

495

from the ring sum in powers of the density – that is, in powers of [(v1 ) + (v2 )] – we would recover the original density expansion which is term by term divergent. However, the mean free path damping is obtained expressing the Boltzmann collision operators, (v1 ) and (v2 ), as a sum of terms, one of which provides the necessary damping, and the other terms can be used to generate an expansion of the contribution of ν (R,>) to the viscosity for moderately dilute gases, which, as we expect, will contain powers of the density and terms logarithmic in the density. We write (vi ) = I (vi ) − ν(vi ),

(12.5.16)

where ν(vi ) is the low-density equilibrium collision frequency   ν(vi ) = n dv3 d κB( ˆ κ,g ˆ i3 )φ0 (v3 ), and the “interacting and exchange” part of the collision operator, I (vi ), is given by   I ˆ κ,g ˆ i3 )  (vi ) = n dv3 φ0 (v3 ) d κB( × {bκˆ (i,3) + (bκˆ (i,3) − 1) Pi3 } .

(12.5.17)

This term is the contribution to the collision operators due to real collisions of particle i with a third particle, and to real and virtual collisions with a third particle but with exchange of particle i with particle 3. The separation of the Boltzmann operators given by Eq. (12.5.16) is useful in the ring operators since the collision frequencies, ν(vi ), provide the needed collision damping. Thus, there is a correction to the Boltzmann viscosity coming from the large q region, given by    dq ν (R,>) (0,zN S ) = mβn dv1 dv2 φ (v )φ (v )v v [z − ]−1 d 0 1 0 2 1x 1y (2π) q>q0 ¯ × Ta,−q (v1,v2 )(v1,v2 )Ta,q (v1,v2 )(1 + P12 ) × [z − ]−1 v1x v1y , with

(12.5.18)

 −1 (v1,v2 ) = z + ν(v1 ) + ν(v2 ) + iq · (v1 − v2 ) − I (v1 ) − I (v2 ) . (12.5.19)

We proceed by expanding (v1,v2 ) in powers of the interacting parts of the collision operators as  −1 (v1,v2 ) = z + ν(v1 ) + ν(v2 ) + iq · (v1 − v2 )  −1  I  + z + ν(v1 ) + ν(v2 ) + iq · (v1 − v2 )  (v1 ) + I (v2 ) −1  + ··· . (12.5.20) × z + ν(v1 ) + ν(v2 ) + iq · (v1 − v2 )

496

Divergences, Resummations, and Logarithms

This expression can be converted into a time convolution integral by writing  ∞ dt exp [−zt] exp [−tA] (v1,v2 ) = 0  ∞  t + dt dt1 exp [−zt] exp [−(t − t1 )A] B exp [−t1 A] + · · · , 0

0

(12.5.21) where A = (ν(v1 ) + ν(v2 ) + iq · (v1 − v2 ))   B = I (v1 ) + I (v2 ) . The presence of the collision frequency in the exponentials provides the mean free path damping not taken into account in the virial expansions. By expanding the exponential in powers of B, we generate ring terms with the mean free path damping properly incorporated in the expressions. If we insert the time convolution expansion of (v1,v2 ) in the expression Eq. (12.5.18) for ν (R,>) (0,z), we obtain28    dq (R,>) ν (0,0) = mβn dv1 dv2 φ (v )φ (v )v v d 0 1 0 2 1x 1y q>q0 (2π)  ∞   ¯ a,−q (v1,v2 ) × []−1 T dt exp −t(ν(v1 )+ν(v2 )+iq ·(v1 −v2 )) 0  ∞  t   + dt dt1 exp −(t − t1 ) (ν(v1 ) + ν(v2 ) + iq · (v1 − v2 )) 0 0 I    ×  (v1 ) + I (v2 ) exp −t1 (ν(v1 ) + ν(v2 ) + iq · (v1 − v2 ))  + · · · Ta,q (v1,v2 )(1 + P12 ) []−1 v1x v1y . (12.5.22) The first term on the right-hand side vanishes since it represents a contribution from two successive collisions between particles 1 and 2 without an intervening collision with another particle in the gas. The next term describes three-body correlated collision events with exponentially damped free particle motions. To see this, we extend the range of the q integration to the full space. In doing so, we add a term of order nd . This shows that, indeed, the logarithmic terms and other ring terms involving a restricted number of collisions result from q-values larger than q0 . The wave number integration leads to a delta function corresponding to the dynamics of the collision sequence. To illustrate this procedure, we consider the contribution from the recollision sequence (1,2)(1,3)(1,2). This is described by using the term in I (v1 ) involving the (1,3) collision without particle exchange. The wave number, q, integration leads to a delta function   δ a(κˆ 1,g 12 ) − a(κˆ 2,g 12 ) + (t − t1 )v1 + t1 v1 − tv2 ) . (12.5.23)

12.5 Logarithms in the Density Expansions of Transport Coefficients

497

The primed velocity, v1, results from the intermediate (1,3) collision. We have encountered this delta function earlier in this chapter, as Eq. (12.2.7), apart from some minor changes. It is important to note that the exponential damping of the time intervals between collisions now appears in the expressions for ν (R,>) . The delta functions lead to geometrical factors of t −d+1, and the exponential damping leads to a finite result,29 but with a logarithm of the density coming from the integral 

∞

dt exp [−νt] t −d+1 ∼ ν d−2 ln ν.

(12.5.24)

td

Here td is a lower limit on time of order a/ < v >, imposed to avoid unphysical divergences for t → 0 . We have limited our discussion here to the renormalized recollision contribution to the coefficient of kinematic viscosity, but a similar analysis can be made for other events involving particle exchanges, and similar analyses can be given for the coefficients of thermal conductivity and for the diffusion coefficient for tagged particle diffusion. The analysis of the generalized ring diffusion coefficient of a tagged particle is based upon Eq. (12.5.12). Otherwise, the analysis is much the same as that for the other transport coefficients. By including the Boltzmann term, the Choh–Uhlenbeck three-body contributions, and some four-body contributions, such as Enskog ones, that are not included in the ring terms but are required for the evaluation of bμ(3), we find that the first few terms in the density expansion of the transport coefficients become μ(2) > = 1 + aμ(2) na 2 + cμ(2) (na 2 ) ln(na 2 ) + · · · , μ0

(12.5.25)

for two-dimensional gases, and μ(3) = 1 + aμ(3) na 3 + bμ(3) (na 3 )2 + cμ(3) (na 3 )2 ln(na 3 ) + · · · , μ0

(12.5.26)

where μ is the coefficient of shear viscosity, thermal conductivity, or tagged particle diffusion, and μ0 is the low-density Boltzmann equation value for the corresponding transport coefficient. It is important to note that the density expansions for the transport coefficients of a two-dimensional gas, Eq. (12.5.25), are incomplete descriptions of the transport in these gases since, as we will show in the next chapter, the long-wavelength contributions from the ring kinetic operator, denoted by U (R,) (k,z). There are no singularities in each of the terms as functions of wave number or z for small values of these quantities.

12.6 The Diffusion Coefficients for the Classical and Quantum Lorentz Gases 499

Figure 12.5.1 The coefficients of self-diffusion, D/D0 ; viscosity, η/η0 ; and thermal conductivity, λ/λ0 , for hard spheres, reduced by their Boltzmann values as obtained from molecular dynamics, are plotted as functions of density nσ 3, where σ is the diameter of the spheres. The data points are the results of simulations by W. W. Wood and J. E. Erpenbeck [206] for the coefficient of self-diffusion and B. J. Alder, D. M. Gass, and T. E. Wainwright for all three transport coefficients [8]. The solid curves represent the expansion given by Eq. (12.5.26) for each transport coefficient using the values for the coefficients given in Table 12.1, including the (3) Enskog theory approximation for bμ,E . The dashed lines represent the first two terms in the expansion, 1 + aμ(3) na 3 . This figure is courtesy of J. V. Sengers.

This follows from the fact that singularities are shielded by the mean free path damping. An expansion of Uij(>) (k,z) in powers of k may be possible. 12.6 The Diffusion Coefficients for the Classical and Quantum Lorentz Gases As in our previous discussions of the Lorentz gas given in Chapter 5, we consider a point particle moving in an array of non-overlapping hard-sphere scatterers, placed at random in space, with number density n. Here we consider the generalization of the Lorentz–Boltzmann equation to a somewhat higher density. Exactly the same kinds of density expansion methods, modified somewhat to a system of one moving particle and fixed scatterers, can be applied to the Lorentz gas. The virial expansion of the generalized Lorentz–Boltzmann equation has the same divergence difficulties as found for gases of moving particles. Thus, the contribution to the diffusion coefficient from the three-body collision operator in the generalized Lorentz–Boltzmann equation, involving correlated collisions of the moving particle with two scatterers, is finite in three dimensions and has a logarithmic divergence in two dimensions. All the higher-order terms in the virial expansion of the collision operator are divergent for both two- and three-dimensional Lorentz gases. The source of the divergence is the same for this model as in the gas of moving

500

Divergences, Resummations, and Logarithms

Figure 12.6.1 The figure on the left is a recollision event of the ring type with two scatterers for a particle in a random Lorentz gas. On the right are two events involving the moving particle and three scatterers that are equally divergent. Both diagrams labeled (a) are ring recollision events, while diagram (b) is a non-ring recollision event in which the moving particle traverses the middle scatterer twice without interacting with it. Both ring and non-ring events must be taken into account when the most divergent terms in the density expansion are summed.

particles. The terms in the virial expansion depend only on the collisions of the moving particle with isolated groups of scatterers, so the free flights of the moving particle between collisions can be arbitrarily long and are undamped. There are some complications due to the fact that the scatterers are fixed in space, so that both the ring and similarly divergent but non-ring terms must be resummed. An example of a ring diagram and a similarly divergent non-ring event involving three scatterers and the moving particle are illustrated in Fig. 12.6.1a and b. The additional most divergent contributions can also be resummed, leading to a generalized ring kinetic equation for the Lorentz gas. Due to the simplicity of the Lorentz gas, it is possible to calculate more terms in the density expansion of the diffusion coefficient, especially for two-dimensional models, than has been done for a gas of moving particles. The theoretical work on this model has been carried out by J. M. J. van Leeuwen, A. Weijland [657, 676], and C. Bruin [69, 70]. For two- and three-dimensional hard-sphere Lorentz gases, the density expansion of the (inverse of) the diffusion coefficients31 are, up to and including the known coefficients,

12.6 The Diffusion Coefficients for the Classical and Quantum Lorentz Gases 501

va 16 64 = na 2 − (na 2 )2 ln(na 2 ) D (2) 3 9 2 2 − 4.68(na ) + 24.10(na 2 )3 ln(na 2 ) + · · ·

(12.6.1)

and va (3) DLG

= 3π(na 3 ) + 19.05(na 3 )2 + 0.645(na 3 )3 ln(na 3 ) + · · · .

(12.6.2)

Here a is the radius of the scatterers, v is the speed of the moving particle, and (d) is the diffusion coefficient in d dimensions. Bruin carried out a systematic DLG (2) comparison of the theoretical results for va/DLG with the results of computer simulations of the Lorentz gas as a function of density. He compared (a) the computed results for the inverse diffusion coefficient with the sum of the first four terms in the density expansion given by Eq. (12.6.1) as a function of density, and (b) the computed results with the low-density value subtracted as a function of density. From Eq. (12.6.1), we see that this combination, when divided by (na 2 )2, should be 

 64 va/D (2) − (16/3)na 2 /(na 2 )2 = − ln(na 2 ) − 4.68 + · · · . 9

(12.6.3)

These two comparisons are presented in Fig. 12.6.2. One can see that the computed values follow the theoretical predictions at low densities with deviations due to higher-order terms. Although this work was done in 1974, it remains one of the important demonstrations of the presence of logarithms in the density expansion of transport coefficients. A much more recent confirmation of the presence of logarithms of the density in the expansion of the diffusion coefficient was discussed in Chapter 5 in connection with the chaotic behavior of the moving particle in a thermostatted Lorentz gas with random placement of non-overlapping scatterers. In this case, too, the computed values for the diffusion coefficient, obtained as a side result of the computation of the Lyapunov exponents for the chaotic motion of the particle, agree with the theoretical predictions only when the logarithmic terms are included in the density expansion of the diffusion coefficient [642]. While the results just discussed represent the most extensive calculations of terms in the density expansion of diffusion coefficients for a classical Lorentz gas, it has been possible to make some progress along these lines for a three-dimensional quantum mechanical Lorentz gas with hard-sphere scatterers. A physical realization of this system would be the motion of an electron in helium gas at low temperatures – a system for which experimental results for the mobility are available [380, 381, 689]. K. I. Wysokinski, W. Park, D. Belitz, and T. R. Kirkpatrick [689, 690] calculated the first few terms in the expansion of the mobility, μq , in terms of the density for helium, expressed as the ratio, χ, of the

502

Divergences, Resummations, and Logarithms

Figure 12.6.2 These figures show the inverse of the coefficient of diffusion for a moving particle in a hard-disk Lorentz gas, as obtained from computer simulations. The upper figure shows the density dependence of vσ/Dn∗ compared with the sum of the theoretical values for the first four terms in Eq. (12.6.1). Here σ is the radius of a scatterer, v is the speed of the moving particle, and n∗ = nσ 2 is the reduced density. The solid line is the theoretical result for this quantity given by the first four terms in Eq. (12.6.1). The lower figure shows the same quantity with the low-density value subtracted from it as expressed in Eq. (12.6.3). The solid line shows the value of the first logarithmic term appearing in Eq. (12.6.3). The agreement of the theory with the simulation results at low densities is evident. These figures are taken from the papers of C. Bruin [69, 70].

thermal de Broglie wavelength, λq , to the mean free path, , of the electrons in the gas.32 The relevant expansion is μq = 1 + μ1 χ + μ2 χ 2 + μ2 χ 2 ln χ + O(χ as / l) + o(χ 3 ). (12.6.4) μB Here χ = λq /(π), with  = 1/(4πnas2 ), where as is the s-wave scattering length for the electron–helium collisions, n is the number density of the helium atoms, and the thermal de Broglie wavelength is λq = (2π 2 h¯ 2 β/m)1/2 , with m, the mass of the electrons. The Boltzmann expression for the mobility, μq,B , is μq,B =

eβ . 3(2πβm)1/2 nπas2

(12.6.5)

The known coefficients are μ1 = −π 3/2 /6, μ2 = (π 2 − 4)/32, μ2 = 0.236 . . . .

(12.6.6)

12.7 Final Remarks

503

√ Figure 12.6.3 The functions f (χ ) = μ2 ln χ + μ2 ± 2χ π are plotted as functions of χ . Here χ = λq /(π ), proportional the ratio of the de Broglie wavelength of the electrons to the classical mean free path length. The six measured values of these quantities are indicated, and the two solid curves correspond to the two possible bounds on including the next term, μ3 χ . The two dashed curves represent these bounds without the logarithmic term. This figure is taken from the paper of K. I. Wysokinski, W. Park, D. Belitz, and T. R. Kirkpatrick [689, 690].

The value for μ2 required a rather difficult and lengthy calculation. Wysokinski et al. also succeeded in placing upper and lower bounds for the next correction term. √ This is of the form μ3 χ 3, where μ3 is bounded above and below by −2 π μ2 ≤ √ μ3 ≤ 2 π μ2 . In Fig. 12.6.3, the functions f± (χ ) =

√ μq /μq,B − 1 − μ1 χ = μ2 ln χ + μ2 ± 2χ π 2 χ

are plotted as a function of χ . The experimental values of these quantities for six values of χ are indicated, and the two solid curves correspond to the two possible bounds for μ3 χ . The two dashed curves represent the theoretical values without the logarithmic term. Due to the lack of experimental data at sufficiently low densities, the logarithmic term does not show up in recognizable form. However, the good agreement between the theoretical expression, without adjustable parameters, and the experimental data (except perhaps for the lowest density one, which has large error bars) lend considerable credence to the actual presence of the logarithmic term. 12.7 Final Remarks We conclude this chapter with two observations. (1) We have been able to obtain the first few terms in the density expansions of the transport coefficients for

504

Divergences, Resummations, and Logarithms

three-dimensional hard-sphere gases. The coefficients of the first logarithmic terms in the density are known, but to go beyond these results one must either use approximate theories, such as the Enskog theory or to do better, one must analyze many terms that were neglected in the ring summation. For simplified gases such as the classical Lorentz gas with randomly placed scatterers, it has been possible to determine the coefficients of a few of the terms beyond the first density logarithm, but for other gases this has not been done. (2) The discovery of the long-time tails, to be discussed in the next chapter has resulted in a change in direction of modern kinetic theory from the goal of developing a systematic approach to calculating the values of transport coefficients to the goal of understanding the structures of the equations of fluid dynamics which, as a consequence of the existence of long-time tails, are more complex and more subtle than previously suspected [180]. Notes 1 We mention here that perhaps a more sophisticated and deeper form of “super statistical mechanics” may underly parallel developments in the theories for nonequilibrium behavior of very different physical systems, as described in Section 15.4. 2 This is purely formal since in the low-density limits taken in the calculations below, the difference between the Ta and the T¯ a operators becomes unimportant 3 Collisional transfers of momentum and energy for completed collisions are accounted for in this construction. 4 This is the case of particular interest when we study the applications of the kinetic equations to the evaluation of Green–Kubo time correlation functions. 5 We refer the reader to the references given earlier for the discussions of the divergences in these density expansions. For a useful review, see [164]. 6 More detailed discussions of the geometrical factors that determine the time dependence of the three-body ring terms can be found in [153, 581, 154, 335, 337, 336]. 7 The reasons for the divergence are the same as those appearing in the inverse Knudsen number expansion for the drag force on a large object moving in a dilute gas. Similarly, the drag force will have terms proportional to the logarithm of the inverse Knudsen number [458, 517, 291, 586]. 8 As we will see later in this chapter, the Lorentz gas with fixed hard-disk or hard-sphere scatterers has non-ring events in the l-scatterer terms that are as divergent as the ring events for the same number of scatterers [657, 676, 69, 70]. In wind-tree models, which are two-dimensional Lorentz models with parallel diamond shaped scatterers, non-ring terms play an important, and for overlapping trees, even a dominant role in diffusion of the moving particles [300, 301]. 9 It is worth pointing out that for any finite time, t, the ring terms have a finite value, albeit a value proportional to logarithms or powers of t. However, to evaluate the Green–Kubo integrals for transport coefficients, one extends the time integrals to infinity. 10 It is reasonable to ask then why virial expansions are useful for determining the equilibrium properties of a gas with short-range forces. The answer lies in the fact that for equilibrium properties, the terms in the virial expansion decay to zero rapidly when the distance between any two subsets of particles increases beyond a few interaction lengths. The situation is quite different for the equilibrium properties of systems of particles that interact with long-range potentials, such as Coulomb potentials. There, as mentioned previously, the calculation of equilibrium properties requires resummation of divergent terms in the equilibrium virial expansions [18]. An interesting exception to the absence of long-range correlations in an equilibrium fluid occurs for particles with ellipsoidal shapes, say, that interact with short-ranged, non-central, repulsive forces. They can form liquid crystals where the nematic or smectic phases have correlations that can extend over large distances [120].

Notes

505

11 This was first suggested by M. S. Green (private communication, 1963) based upon his intuition that the t −2 decay appearing in the three-body collision terms for three dimensions would lead, in some way, to logarithmic terms in the density. 12 It is also interesting that such density logarithms must be taken into account when comparing the theoretical and computer simulation results for the Lyapunov exponents for random Lorentz gases [642], as was mentioned previously in Chapter 5. 13 This analysis consists of noting that any additional collision among the same number of particles brings in additional constraints on the integrations involved, making them less divergent. An example is the four-particle repeated ring event illustrated in Fig. 12.3.1. 14 An exception is the case of tagged particle diffusion for hard spheres at low density and finite times [647]. 15 We are deliberately avoiding using a cluster expansion based on successive numbers of particles, since that led to the divergence problems we are trying to overcome here. 16 The restriction to pre-collision situations is very important since setting g2 = 0 for all phases of the two particles, including post-collision phases, would ignore velocity correlations produced by a collision. Setting g2 = 0 for post-collisional configurations but not pre-collisional ones, leads to an anti-Boltzmann equation [107]. Setting g2 = 0 for pre-collisional configurations is another version of the Stosszahlansatz, while setting g3 = 0 for pre-collisional configurations, as we shall do here, may be considered as a relaxation of the Stosszahlansatz, since it takes into account some kinds of pre-existing velocity correlations between colliding particles. 17 Based on our discussion in Chapter 11 of the overlapping, Enskog terms in the generalization of the Boltzmann equation to higher densities, we have changed all but the leftmost, Ta (2,3), barred binary collision operators to unbarred forms to remove from the ring sum some of the finite or less divergent contributions, involving overlapping configurations. 18 This can be seen most simply by noticing that the initial condition term in Eq. (12.3.16) is effectively the time derivative of the second term. Thus, if the analysis of the second term leads to an algebraic time decay of some quantity, the initial condition term will decay one power of time faster. Equivalently, one can rephrase this argument in terms of Laplace transforms. The relative power of the decays of the initial condition term and the full ring term is consistent with the analysis of Dorfman and Cohen of initial condition terms related to the divergences in the density expansion of the kinetic equation [154]. 19 The time ordering of the collision sequence reads from right to left, so that the time of collisions increases from right to left. 20 There are higher order in density contributions to each of the divergent terms in the virial expansion of the kinetic equation for the single-particle distribution function. For hard spheres, part of these corrections come from Enskog type overlap contributions for each of the binary collisions. This is discussed in detail in a paper by J. R. Dorfman and E. G. D. Cohen [155] as well as in the PhD dissertation of H. van Beijeren [632]. 21 This statement is somewhat questionable for two-dimensional systems. As we will see in the following chapter, long-time tail contributions to the ring operator diverge and are no longer just density corrections to the Boltzmann operator. Consequently, we must then add ring ˜ q,l,z (v1,v2 ), leading to “rings contributions to the Boltzmann operators in the ring propagator, G within rings,” and so on. In the next chapter we will discuss a self-consistent treatment, suggested by this argument, and show that it leads to a more correct description of the asymptotic long-time behavior of time correlations functions, for example. 22 Notice that D is independent of the wave number, k, as it reduces to the first term in Eq. (12.3.28). 23 There is an extension of Green–Kubo theory appropriate for transport properties when nonlinear effects must be included in the description of hydrodynamic flows in a fluid. The theory, due to K. Kawasaki and J. Gunton and to T. Yamada and K. Kawasaki [349, 692]. Extended kinetic equations, including nonlinear effects, will be presented in Chapter 14. 24 As we will see in the next chapter, the evaluation of the Green–Kubo formulae for gases in two dimensions has divergence difficulties stemming from long-time tail effects, differing from the divergences in virial expansions discussed earlier in this chapter,

506

Divergences, Resummations, and Logarithms

25 In addition to the kinetic currents discussed here, the microscopic currents associated with the coefficients of shear viscosity and thermal conductivity have terms that depend on the interaction potentials. We will not discuss these terms here. The current associated with the coefficient of self-diffusion is purely kinetic without additional potential energy terms. 26 In [155], the terms containing H (1,3,q) or G2 (q) were left out, since they do not contribute to the long-time tail in the velocity autocorrelation function. However, they do contribute to the short-time behavior and to the diffusion coefficient [632]. In addition, they are needed to maintain time-reversal symmetry of the ring operator, and the H (1,3,q) term shows up as Aq in the zero-loop approximation to the full collision operator. 27 Note that the inner products are defined with the equilibrium velocity distribution functions, φ0 (v), as weight functions. The binary collision operators, 0 , contain in their integrands the full equilibrium distribution functions, feq (v) = nφ0 (v). 28 We are interested in the asymptotic, long-time evaluation of these functions, which corresponds to taking the z → 0 limit. 29 We have implicitly excluded d = 1, and we have consistently done so. 30 See also the papers of Gervois, et al. [248, 249]. 31 J. M. J. van Leeuwen and A . Weijland use the fact that if one fixes the initial value of the direction of motion of the moving particle, vˆ , say, its average velocity direction will be the same for all later times [657]. That is to say, the velocity unit vector is an eigenfunction of the collision operator describing the average motion of the moving particle. This observation makes it convenient to calculate the inverse of the diffusion coefficient. 32 The mobility, vE /E, is defined as the ratio of the average velocity of particles in a field to the field strength. For small field strengths, the mobility can be expressed in terms of quantum Green–Kubo formula. It is directly proportional to the diffusion coefficient, with a factor that depends on the ratio of the particle’s charge to its mass.

13 Long-Time Tails

In the previous chapter, we derived a linear kinetic equation that takes into account uncorrelated binary collisions as described by the linearized Boltzmann equation as well as the sum of the most divergent terms in each order of the density expansion of the collision operator. We also outlined how the same procedure allows us to obtain the Boltzmann and ring contributions to the transport coefficients using the Green–Kubo expressions for them in terms of time correlation functions. We then explored one of the important consequences of the ring kinetic equations, namely the incorporation of the mean free path collisional damping of particle trajectories and the appearance of density logarithms in the density expansions of transport coefficients. In this chapter, we turn to the analysis of the ring kinetic equation for a description of long-wavelength, hydrodynamic processes in the gas. In particular, we will see that the ring kinetic equation contains contributions due to the coupling of hydrodynamic modes at a microscopic level, which provides the basis for mode-coupling theory at low density. These results will provide a deepening of our understanding of the microscopic foundations and limitations of the equations for fluid dynamics and for the associated transport coefficients. We will describe the long-time behavior of the Green–Kubo correlation functions whose time integrals are the transport coefficients. Of particular importance is the appearance of the so-called long-time tails of the Green–Kubo correlation functions that decay with time, t, in d dimensions, as (t/t )−d/2, where t is the mean free time between collisions. Our study will reveal some fundamental problems with the attempts to provide a microscopic foundation for the equations of fluid dynamics. We will see that the Navier–Stokes equations describe fluid flows in three-dimensional systems, but a systematic extension of these equations in integer powers and higher-order gradients of the fields is not possible.1 For two-dimensional systems the situation is even more problematic. In this case, it is impossible to derive the Navier–Stokes equations. For both two- and three-dimensional systems, we will see in this and the following chapter that the corrected forms of the hydrodynamic equations contain 507

508

Long-Time Tails

nonanalytic terms, and in many situations, the equations are only well behaved if the finite size of the system is taken into consideration. 13.1 Introduction In the previous chapter, we derived generalized equations of hydrodynamics, using projection operator methods developed in Section 3.8. These methods project the deviation from equilibrium of the single-particle distribution function onto the space spanned by the conserved quantities – or, equivalently, the zerothorder hydrodynamic modes – which to lowest order in a wave-number expansion are the Fourier transforms of the densities of the conserved quantities – particle number, momentum and (kinetic) energy. We obtained expressions for the transport coefficients, which in the Navier–Stokes limit are given by Green–Kubo formulae. In the previous chapter, we showed how collision sequences taking place on spatial scales of the order of a mean free path length, as described by the ring kinetic equation, lead to logarithms in the density expansion of transport coefficients. In this chapter, we will analyze contributions from dynamical events taking place on spatial scales of many mean free path lengths leading to long-time tails in the Green–Kubo time correlation functions. Preliminary Remarks on Long-Time Tails As we show in what follows, central to our discussions will be small-wave-number contributions to the quantities Uij(R) (k,z), defined by Eq. (12.5.9) in the previous chapter. They lead to wave number integrals of the form2 

−1  , dq z + aq 2 + bk · q + ck 2

q