Physical Kinetics. Course of Theoretical Physics [10] 9780750626354

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PHYSICAL KINETICS by

E. M. LIFSHITZ and L. P. PITAEVSKII Institute of Physical Problems, U.S.S.R. Academy of Sciences Volume 10 of Course of Theoretical Physics

Translated from the Russian by J. B. SYKES and R. N. FRANKLIN

ELSEVIER BUTTERWORTH HEINEMANN

AMSTERDAM · BOSTON · HEIDELBERG · LONDON · NEW YORK · OXFORD PARIS · SAN DIEGO · SAN FRANCISCO · SINGAPORE · SYDNEY · TOKYO

Related Titles Other titles in the COURSE OF THEORETICAL PHYSICS by L.D. Landau and E.M. Lifshitz

Vol. 1 Mechanics (3rd edition) Vol. 2 The Classical Theory of Fields (4th edition) Vol. 3 Quantum Mechanics - Non-relativistic Theory (3rd edition) Vol. 4 Quantum Electrodynamics (2nd edition) Vol. 5 Statistical Physics, Part 1 (3rd edition) Vol. 6 Fluid Mechanics (2nd edition) Vol. 7 Theory of Elasticity (3rd edition) Vol. 8 Electrodynamics of Continuous Media (2nd edition) Vol. 9 Statistical Physics, Part 2 Introduction to Superconductivity (2nd edition) by A.C. Rose-Innes and E.M. Rhoderick Concise Encyclopedia of Magnetic and Superconducting Materials by J. Evetts

Butterworth-Heinemann is an imprint of Elsevier The Boulevard, Langford Lane, Kidlington, Oxford, 0X5 1GB 30 Corporate Drive, Suite 400, Burlington, MA 01803, USA Translated from Fizicheskaya kinetika, 'Nauka', Moscow 1979 First Published in English by Pergamon Press pic 1981 Reprinted 1989, 1993, 1995, 1999, 2002, 2005, 2006 (twice), 2007, 2008 (twice) Copyright © 1981, Elsevier Ltd. All rights reserved No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the publisher Permissions may be sought directly from Elsevier's Science & Technology Rights Department in Oxford, UK: phone (+44) (0) 1865 843830; fax (+44) (0) 1865 853333; email: [email protected]. Alternatively you can submit your request online by visiting the Elsevier web site at http://elsevier.com/locate/permissions, and selecting Obtaining permission to use Elsevier material Notice No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made British Library Cataloguing in Publication Data Lifshitz, E. M. Physical Kinetics (course of theoretical physics; vol. 10) 1. Plasma (ionized gases) I. Title II. Pitaevskii, L. P. III. Series 530' .4'4 QC718 80-42162 Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress ISBN: 978-0-7506-2635-4 For information on all Butterworth-Heinemann publications visit our website at www.elsevierdirect.com Printed in the United States of America Transferred to Digital Printing, 2010

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NOTATION Particle distribution function / (Chapters I-VI); momentum distribution function always relative to d3p. Occupation numbers of quantum states n(p) for electrons and N(k) for phonons (Chapters VII and IX-XI); momentum distribution always relative to ά*ρΙ(2ιτΗγ. Collision integral C; linearized collision integral I. Thermodynamic quantities: temperature T, pressure P, chemical potential μ, particle number density N, total particle number Jf, total volume Ύ. Electric field E, magnetic induction B; unit electric charge e (electron charge —e). In estimates: characteristic lengths L; atomic dimensions and lattice constant d; mean free path / ; speed of sound u. Averaging is denoted by angle brackets f. The ratio τ ~ Ι/ϋ is called the mean free time. For a rough estimate of the collision integral, we can put C{f)

if - /ο)/τ ~ - (vll)if - /o).

(3.12)

By writing the difference / - f0 in the numerator we have taken account of the fact that the collision integral is zero for the equilibrium distribution function. The minus sign in (3.12) expresses the fact that collisions are the mechanism for reaching statistical equilibrium, i.e. they tend to reduce the deviation of the distribution function from its equilibrium form. In this sense, τ acts as a relaxation time for the establishment of equilibrium in each volume element of the gas. §4. The H theorem A gas left to itself, like any closed macroscopic system, will tend to reach a state of equilibrium. Accordingly, the time variation of a non-equilibrium distribution function in accordance with the transport equation must be accompanied by an increase in the entropy of the gas. We shall show that this is in fact so. The entropy of an ideal gas in a non-equilibrium macroscopic state described by a distribution function / is S = |/log(e//)dVdr;

(4.1)

see SP 1, §40. Differentiating this expression with respect to time, we have

= -jlogf^dVdr.

(4.2)

Since the establishment of statistical equilibrium in the gas is brought about by collisions of molecules, the increase in the entropy must arise from the collisional

12

Kinetic Theory of Gases

part of the change in the distribution function. The change in this function due to the free motion of the molecules, on the other hand, cannot alter the entropy of the gas, since this part of the change in the distribution function is given (for a gas in an external field U (r)) by the first two terms on the right-hand side of the equation dfldt = - v . V/ - F. d//dp + C(f). Their contribution to the derivative dSldt is - [ l o g / [ - v . a / / a r - F . a / / a p ] d V d r = J[v.a/ar + F.a/a P ](/log//e)dVdr. The integral over dV of the term involving the derivative dldr is transformed by Gauss's theorem into a surface integral; it gives zero on integration through the whole volume of the gas, since / = 0 outside the region occupied by the gas. Similarly, the term involving the derivative d/dp, on integration over d*p, becomes an integral over an infinitely distant surface in momentum space, and likewise gives zero. The change in the entropy is therefore expressed by dSldt = - f log / . C(f) dY dV.

(4.3)

This integral can be transformed by a device which, with a view to later applications, we shall formulate for the general integral / 2 ) = I(g),

(10.1)

§10

Approximate Solution of the Transport Equation

31

where ß = m/2T; the linear integral operator 1(g) is defined by Kg) -¡I

tWoi(g' + gi - g - gi) d3Pi da,

(10.2)

corresponding to the collision integral (3.9), and the equilibrium distribution func­ tion ist fo(v) = (Nßmlm3TTm)e-ßv\

(10.3)

An efficient method of approximately solving equation (10.1) is based on expanding the required functions in terms of a complete set of mutually orthogonal functions, which may with especial advantage be taken as the Sonine polynomials (D. Burnett 1935). These are defined by* Sr'ix) = j j exx-' -j£ (e"V +i ),

(10.4)

where r is any number and s is a positive integer or zero. In particular, S r °=l,

Sr\x) = r+\-x.

(10.5)

The orthogonality property of these polynomials for a given r and different s is

Jo

e~xxrSrs(x)Srs\x) dx = T(r + s + \)8js !.

(10.6)

We shall seek the solution of (10.1) as the expansion g(v) = (ß/N)v ¿ AsSHßv2). s=\

(10.7)

By omitting the term with 5 = 0, we automatically satisfy the condition (7.4), the integral being zero because the polynomials with 5 = 0 and s ^ 0 are orthogonal. The expression in parentheses on the left of (10.1) is the polynomial Sl^ißv2), and this equation therefore becomes - yS\l2(ßv2) = (ßlN) ¿ AJ(vSj/2).

(10.8)

5=1

Multiplying both sides scalarly by yfo(v)SlV2(ßv2) and integrating over d3p, we tThe distribution function is everywhere taken to be defined in momentum space. This, however, does not prevent it from being expressed for convenience in terms of the velocity v = p/m. tThey differ only in normalization and affix numbering from the generalized Laguerre polynomials:

32

Kinetic Theory of Gases

obtain a set of algebraic equations ¿α,,Α5=^δπ,

1 = 1,2,...,

(10.9)

with flu = - (ß2IN2) |/oV. SU(vS35/2)d3p = (ß2/4N2){vS^/2,vSy,

(10.10)

the notation being {F, G} = |/o(»)/o(»i)|v - vi|A(F)A(G) d3p d3Pi Ar, Δ(Ρ) = F(v') + F(vl) - F(v) - F(vi).

(10.11)

There is no equation with / = 0 in (10.9), since a0s = 0 because of the conservation of momentum: A(vS3/2) = Δ(ν) = 0. The thermal conductivity is calculated by sub­ stituting (10.7) in the integral (7.7). The condition (7.4) shows that this integral (with € = \mv2) can be put in the form K = -|J/oS] /2 (ßD 2 )v.gd 3 p and the result is κ = 5Αι/4.

(10.12)

The advantage of expanding in Sonine polynomials is shown by the simplicity of the right-hand side of equations (10.9) and the expression (10.12). The calculations are entirely similar for the viscosity. The solution of (8.6) is sought in the form gaß = - (β2ΙΝ2)(ναυβ - \ν%β) Σ BsSMßv2).

(10.13)

Substitution in (8.6), multiplication by U(v)Sl5^v2)(vavp-W8afi), and integration over d3p leads to the set of equations i > l s B s = 5o,o, 1=0,1,2

(10.14)

where bis = (ß3IN2){(vavß-\v28aß)Slsi2,

(vavß-W8aß)S55l2}.

(10.15)

§10

Approximate Solution of the Transport Equation

33

The viscosity is found from (8.9) as i)=kmB0.

(10.16)

The infinite set of equations (10.9) or (10.14) is approximately solved by retaining only the first few terms in the expansion (10.7) or (10.13), i.e. by artificially terminating the set. The approximation converges extremely rapidly as the number of terms increases: in general, retaining just one term gives the value of κ or η with an accuracy of 1-2%.t We shall show that the approximate solution of the linearized transport equation for monatomic gases by the above method gives values of the kinetic coefficients that are certainly less than would follow from the exact solution of the equation. The transport equation may be written in the symbolic form I(g) = L,

(10.17)

where the functions g and L are vectors in the thermal conduction problem, and tensors of rank two in the viscosity problem. The corresponding kinetic coefficient is determined from the function g as a quantity proportional to the integral -|/og!(g)d3p;

(10.18)

see §9. The approximate function g, however, satisfies not equation (10.17) itself but only the integral relation | /ogl(g) d'p = | foLg d'p,

(10.19)

as is evident from the way in which the coefficients in the expansions of g are determined. The statement made above follows immediately from the "variational principle" whereby the solution of (10.17) gives a maximum of the functional (10.18) within the class of functions that satisfy the condition (10.19). The validity of this principle is easily shown by considering the integral

- J fo(g-,etx)]do'

= C(f).

(11.1)

The right-hand side is zero for any function / that does not depend on the direction of p, and not only for the Maxwellian function / 0 as in the case of the Boltzmann equation. This is because of the assumption that the magnitude of the momentum is unchanged in the scattering of light particles by heavy ones: such collisions evidently leave steady any energy distribution of light particles. In reality, equation (11.1) corresponds only to the zero-order approximation with respect to the small quantity mi/m2, and energy relaxation occurs in the next approximation. If the concentration and temperature gradients are not too large (these quantities varying only slightly over distances of the order of the mean free path), / may be sought as the sum / = /ο(ρ,χ) + δ/(ρ,0,χ), where δ/ is a small correction to the local-equilibrium distribution function / 0 and is linear in the gradients of c and T. In turn, we seek 8f in the form 6/ = cos0.g(p,x),

(Π.2)

38

Kinetic Theory of Gases

where g is a function of p and x only. In substituting in (11.1), it is sufficient to retain the / 0 term on the left-hand side; in the collision integral, the / 0 term disappears: C(f) = gN2v j F(p, a)(cos 0' - cos 0) do'; the function g, which is independent of the angles, has been taken outside the integral. This integral may be simplified as follows. We take the direction of the momen­ tum p as the polar axis for the measurement of angles. Let φ and φ' be the azimuths of the x-axis and the momentum p' relative to this polar axis. Then cos 0' = cos 0 cos a + sin 0 sin a cos(

0. One further first-order effect is the presence in a moving gas of an additional surface heat flux (i.e. restricted to a layer at the wall with thickness ~ I) q^rf, proportional to the normal gradient of the tangential velocity: qLrf=

X2 = Y -T--.

A comparison of (14.15) with the expressions (9.3) shows that the corresponding quantities xa are the vectors ¿1 = qsurf,

¿2 =

ην,.

The "equations of motion" (9.1) are the relations (14.12) and (14.14); writing these as ¿1 = ΤφΧ2, ¿2 = ημΤ2Χι, we obtain the required relation φ = Τημ

(14.16)

(L. Waldmann 1967).

PROBLEMS PROBLEM 1. Two vessels containing a gas at different temperatures T\ and Ti are connected by a long tube. As a result of thermal slip, a pressure difference is established between the gases in the two vessels (the thermo-mechanical effect). Determine this difference. SOLUTION. The boundary condition at the surface of the tube for Poiseuille flow under the influence of the pressure and temperature gradients, with allowance for thermal slip, is v = μ dTldx at r = R (where R is the tube radius and the x-axis is along the length of the tube). Wefindin the usual way (see FM, § 17) the velocity distribution over the tube cross-section:

The mass of gasflowingthrough a cross-section of the tube per unit time is p7fjR4dP ,

2dT

where p is the gas density. In mechanical equilibrium Q = 0, whence dP = 8ημ άΊ^ dx IF dx Integration over the whole length of the tube gives the pressure difference: Pi-Pi =

(SwlR2)(T2-Ti)

(if Ί7 - T\ is fairly small, η and μ may be taken as constants). An estimate of the order of magnitude of the effect by means of (14.13) and (8.11) gives 8PIP ~ (12/R2)8T/T.

§14

Phenomena in Slightly Rarefied Gases

55

The velocity distribution over the tube cross-section when Q = 0 is

ÉL

»=μ(|τ-ΐ)

dx'

The gas flows along the walls in the direction of the temperature gradient (v > 0), and near the axis of the tube it flows in the opposite direction (v < 0). PROBLEM 2. Two tubes of length L and different radii (R\ Ti), the difference being small. As a result of thermal slip, a circulatory motion of gas is established in the tubes. Find the total gas flow through the tube cross-sections. SOLUTION. Dividing (1) in Problem 1 by R4 and integrating along a closed contour formed by the two tubes, we have Q = P|£(T

2

-T

1

)(R

2

2

-R

1

2

)^^.

The flow takes place in the direction shown in Fig. 2.

τ

-«—2/?,

2R0

PROBLEM 3. Determine the force F acting on a sphere of radius R immersed in a gas where a constant temperature gradient VT = A is maintained. SOLUTION. The temperature distribution within the sphere is given by T =

3Κ2 K\ + 2Κ2

Ar cos 0,

where κ\ and Κ2 are the thermal conductivities of the sphere and the gas; r and Θ are spherical polar coordinates with the origin at the centre of the sphere and the polar axis along A (see FM, §50, Problem 2). Hence we find for the temperature gradient along the surface of the sphere

111 R dO

3Κ2 K\ + 2Κ2

A sin 0.

The laminar flow of the gas resulting from the thermal slip is determined only by the one vector A. The corresponding solution of the Navier-Stokes equation may therefore be sought in the same form as in the problem of liquid flow past a sphere moving in it (see FM, §20): v= - a

A + n(A.n) , , 3 n ( A . n1 ) - A , +b— y r r

where n = r/r; the additive constant in v is omitted, since we must have t ) = 0 a s r->». The constants a and b are found from the conditions i>r = 0,

ve = (ßlR)dTlde

at

r^R;

56

Kinetic Theory of Gases

their values are a = bIR2 = - 3κ2Αμ/2(κ, + 2κ2). The force on the sphere is F = 8παηA = - \2τη\μΚκι^ΎΙ{κ\ + 2κ2). For the surface effects considered in these Problems to be in fact small compared with the volume effects, the temperature must vary only slightly over the radius of the tube in Problems 1 and 2, and over the radius of the sphere in Problem 3. PROBLEM 4. Two vessels joined by a long tube contain gas at the same temperature and at pressures Pi and P2. Determine the heat flux between the vessels which accompanies Poiseuille flow in the tube (the mechano-caloric effect). SOLUTION. According to (14.14) and (14.16), the heat flux along the walls of the tube is q' = IrrKq'smt = 2ττ1?Τημ dV/dr. From the condition of mechanical equilibrium of the liquid in a steady flow, we have 2ττΚη dVldr = TTR2 dPIdx = TTR2(P2 - Pi)/L. Hence, finally, q' = 7TjR^(P 2 -Pi)/L.

§15. Phenomena in highly rarefied gases The phenomena discussed in § 14 are no more than correction effects associated with higher powers of the ratio of the mean free path I to the characteristic dimensions L of the problem; this ratio was supposed still small. If the gas is so rarefied, or the dimensions L are so small, that I / L ä l , the equations of fluid dynamics become completely inapplicable, even with corrected boundary con­ ditions. In the general case of any Í/L, it is in principle necessary to solve the transport equation with specified boundary conditions on solid surfaces in contact with the gas. These conditions depend on the interaction between the gas molecules and the surface, and relate the distribution function for particles incident on the surface to that for particles leaving it. If this interaction amounts to scattering of molecules without chemical transformation, ionization, or absorption by the surface, it is described by the probability w(Tf9T)dTf that a molecule with given values of Γ strikes the surface and is reflected into a given range dT'; the function w is normalized by the condition iw(T,Ddr=l.

(15.1)

With this function, the boundary condition for the distribution function /(Γ) becomes f Jn.v(Γ\Γ)η.ν/(ΓΜΓ = - η . ν ' / ( Γ )

with

n.v'X).

(15.2)

§15

Phenomena in Highly Rarefied Gases

57

The integral on the left multiplied by dr' is the number of molecules incident on unit area of the surface per unit time and scattered into a given range dF; the integration is taken over the range of values of Γ that corresponds to molecules moving towards the surface (n being a unit vector along the outward normal to the surface of the body). The expression on the right of (15.2) is the number of molecules leaving unit area of the surface per unit time. The values of Γ on each side of the equation must correspond to molecules moving away from the surface. In equilibrium, when the temperature of the gas is the same as that of the body, the distribution function must have the Boltzmann form for both the incident and the reflected particles. Hence it follows that the function w must satisfy identically the equation ί

Jn. v