Statistical mechanics and the physics of fluids [2 ed.] 8876421440

Table of contents :
Cover
Title
Contents
Preface
1 Statistical ensembles and Boltzmann's entropy
2 The first principle of thermodynamics and the thermodynamic functions
3 Gases and plasmas at equilibrium
4 The states of matter
5 Thermodynamic fluctuations and liquid structure
6 Critical phenomena
7 Diffusion and brownian motion
8 Viscosity, sound waves, and inelastic scattering from liquids
9 Heat transport
10 Supercooling and the glassy state
11 Non-newtonian fluids: especially liquid crystals and polymers
12 Turbulence
References

Citation preview

1

APPUNTI

Mario Tosi Scuola Normale SUperiore · Piazza dei Cavalieri, 7 56100 Pisa, Italy [email protected] Patrizia Vignolo Scuola Normale Superiore Piazza dei Cavalieri, 7 56100 Pisa, Italy [email protected]

Statistical mechanics and the physics offluids

Mario Tosi and Patrizia Vignolo

Statistica I mechanics and the physics of fluids

I

IE. DIZIONI DELLA

NORMALE

© 2005 Scuola Nonnale Superiore Pisa Seconda edizione Prima edizione: 1997 ISBN: 88-7642-144-0

Contents

Preface

VII

1 Statistical ensembles and Boltzmann's entropy

1

Statistical ensembles and Boltzmann's entropy I. I. The key principle of statistical mechanics . . . . . . . .

1

1.2. Density of microscopic states . . . . . . . . . . . . . . . 1.3. The state of thermal equilibrium and the canonical ensemble. . . . . . . . . . . 1.4. Fluctuations . . . . . . . . 1.5. Grand canonical ensemble 1.6. Boltzmann's entropy . . . I. 7. Second principle of thermodynamics . 1.8. Third principle of thermodynamics .

2 The first principle of thermodynamics and the thermodynamic functions 2.1. Thermodynamic variables . . . . . . . . . . . . . . . . . 2.2. Thermodynamic identity and the first principle of thermodynamics . . . . . . . . . . . . 2.3. Helmholtz free energy and variational principle . . . . 2.4. Gibbs free energy and grand thermodynamic potential . 2.5. Specific heats and compressibilities 2.6. Fluctuation phenomena . . . . . . . 2. 7. Free energy from partition function . 2.8. The classical limit . . . . . . . . . . 2.9. Some general results of equilibrium statistical mechanics

3 Gases and plasmas at equilibrium 3. I. Ideal gases and their entropy .

I 2 3 4 5 5

6 7

9 9 11 12 13 14 15 17 18 19

21 21

VI

3.2. 3.3. 3.4. 3.5. 3.6. 3.7. 3.8. 3.9.

Mario Tosi and Patrizia Vignola

Detailed balance for fermions at thermal equilibrium Fermi and Bose gases . . . . Free electrons in metals . . . Bose-Einstein condensation . Hannonic oscillators Black-body radiation . . . . Phonons in crystals . . . . . Classical real gases and weakly coupled classical plasmas

23 24 25 27 29 31 32 33

4 The states of matter 4.1. Equation of state of pure materials 4.2. Phase diagram . . . . . . . . . 4.3. A liquid open to the atmosphere 4.4. Clausius-Clapeyron equation and melting 4.5. Van der Waals condensation . . . . 4.6. Melting and Lindemann's law .. . 4.7. Brownian motion in the liquid phase 4.8. Flow properties of dense liquids

37 38 40 41 42 44 45 46 49

5 Thermodynamic flnctuations and liqnid structure 5.1. Thermodynamic fluctuations and their correlations 5.2. Some examples . . . . . . . . . . .. 5.3. Spatial correlations of density fluctuations and liquid structure . . . . . . . . 5.4. Connections between liquid structure and thermodynamics . . 5.5. Inhomogeneous fluid . . . . . . . . . 5.6. Direct correlation function . . . . . . 5.7. Liquid-structure theory for classical systems

51 51 53

57 58 60 62

6 Critical phenomena 6.1. Paramagnetism oflocalized spins . 6.2. Ising model and order-disorder transitions 6.3. Mean-field theory . . . . . . . . . . . 6.4. Ising magnetism in dimensionalities d = 1 and d = 2 6.5. Critical exponents . . . . . . . . . . . 6.6. Homogeneity hypothesis and scaling . 6.7. Liquid-vapour critical point . . . . .

65 65 67 67 69 71 72 74

55

VII Statistical mechanics and the physics of fluids

7

Diffusion and Brownian motion 7 .1. Correlations of fluctuations in time . 7.2. Relaxation time . . . . . . . . . . . 7.3. Brownian motion . . . . . . . . . . 7.4. Diffusion, mobility, and Nernst-Einstein relation. 7.5. Transport coefficients and Onsager relations 7.6. An example: thermoelectric effects . . . . . 7.7. Microscopic diffusion and memory function 7.8. Nyquist's theorem . . . . . . . . . . . . . . 7 .9. Mass diffusion and incoherent inelastic scattering

8

Viscosity, sound waves, and inelastic scattering from liquids 89 8.1. Hydrodynamic variables . . . . 89 8.2. Stresses in a Newtonian fluid and the Navier-Stokes equation . 90 8.3. Creeping flow past an obstacle . 93 8.3.1. Vorticity. . . . . . . . . 94 8.4. Transverse currents and isothermal sound propagation . 96 8.5. Brillouin doublet . . . . . . . . . . . . . . . . . . . . 98 8.6. Microscopic density fluctuations and inelastic neutron scattering . . . . . . . . . . . . . 99 8.7. Inelastic photon scattering from liquids . . . . . . . . 102 8.8. Two limiting examples of density fluctuation spectrum 103 8.9. Rigidity of liquids . . . . . . . . . . . . . . . . . . . . 104

9

Heat transport 9.1. Fourier's law . . . . . . . . . . . . 9.2. Thermodynamics with mass motion and entropy production . . . . . . . . . . . . . . 9.3. The effect of heat flow on sound wave propagation . . . 9.4. Light scattering and sound propagation: Rayleigh and Brillouin peaks . . . . . . . . . . . . . . . . 9.5. Heat conduction by electrons in liquid metals 9.6. Superfluid 4 Helium . . . . . . . . . . 9.7. Hydrodynamics of superftuid Helium in the two-fluid model . . . . . . . . . 9.8. Inelastic neutron scattering from superfluid 4 He

10 Supercooling and the glassy state 10.1. Macroscopic features of a glass and the glass transition 10.2. Kinetics of nucleation and phase changes 10.3. The free energy landscape . . . . . . . . . . . . . . .

77 77 79 79 81 82 83 84 86 87

107 107 108 111 112 113 115 118 120

125 125 128 130

VIII

Mario Tosi and Patrizia Vignolo

10.4. Atomic motions in the glassy state 10.5. Strong and fragile liquids . . . . . 10.6. Supercooled and glassy materials . 10.6.1. Phase diagram and amorphous branch of the hardsphere system . . . 10.6.2. Supercooled water 10.6.3. Metallic glasses . . 10.6.4. Superionic glasses 10.6.5. Glassy polymers .

132 133 135 135 136 137 138 138

11 Non-Newtonian fluids: especially liquid crystals and polymers141 11.1. Introduction to non-Newtonian flow behaviour . . . 141 11.2. Liquid crystal phases . . . . . . . . . . . . . . . . 143 11.3. Nematic liquid crystals and their phase transitions . 146 11.4. Viscosity in a uniaxial liquid . . . . . . . . 148 11.5. Flow birefringence and curvature elasticity . 150 11.6. Disclinations . . . . . . . . . . . . . . 152 11.7. Polymers . . . . . . . . . . . . . . . . 154 11.7 .1. The isolated polymer molecule . 154 11.7 .2. Polymer solutions . . . . . . . 155 11.7.3. Polymeric materials . . . . . . 157 11.8. Non-Newtonian behaviour in polymeric liquids 158 11.8.1. Reptation in concentrated polymer systems 159 11.8.2. Macroscopic flow phenomena in polymeric liquids 160 11.9. Colloidal dispersions and suspensions 161 11.lOSurfactant systems . . . . . . . . . . . . . . . . . . . . 163 12 Turbulence 12.1. Introduction . 12.2. Instabilities in fluids . . . . . . . . . . . 12.2.1. The Rayleigh-Taylor instability 12.2.2. Thermal convection and the Rayleigh-Benard instability . . . . . . . . . . . . . . . . . 12.2.3. The Kelvin-Helmholtz instability . . . . 12.3. Evolution of Benard convection with increasing Rayleigh number . . . . . . . . . . . . . . . . . 12.4. Energy cascade in homogeneous turbulence . . . 12.4.1. Energy cascade and Kolmogorov microscales 12.4.2. Kinetic energy spectrum . . . . . . . . . . 12.5. Diffusion in homogeneous turbulence . . . . . . . 12.5.1. Stochastic modelling of turbulent diffusion 12.6. Turbulent shear flows . . . . . . . . . . . . . . . .

167 167 169 169 170 171 173 174 175 177 178 178 179

IX

Statistical mechanics and the physics of fluids·

12.6.1. Reynolds stresses . . . . . . . 12.6.2. Lattice Boltzmann computing 12.7. Turbulence in compressible fluids .. 12.8. Turbulent behaviour of non-Newtonian fluids References

180 181 182 183

185

Preface

This volume collects the lecture notes of an introductory course on statistical mechanics, held at Scuola Normale Superiore di Pisa and mainly addressed to third-to-fifth year students in physics and chemistry. Starting from a set of notes first prepared in 1994, these lecture notes are now being reprinted with some major changes and additions in order to give an introduction to a number of recent developments in the study of fluids and of soft condensed matter. Three main themes are covered in the course. The first part gives a compact presentation of the foundations of statistical mechanics and their deep-lying connections with thermodynamics in Chapters I and 2. This is followed in Chapter 3 by applications to ideal gases of material particles ,and of excitation quanta, and by a brief introduction to a real classical gas and to a weakly coupled classical plasma. Chapter 4 then gives a broad oYerview on the three states of ordinary matter, with focus on the equation of state, on phase diagrams and phase transitions, and on dynamical phenomena in fluids. The second part of the course is devoted to fluctuations around the equilibrium state and their instantaneous and time-dependent correlations, the main emphasis being on classical fluids. Liquid structure and m introduction to critical phenomena are covered in Chapters 5 and 6, :md these are followed in Chapters 7 - 9 by a discussion of irreversible processes as exemplified by diffusive single-particle motions and by the d~-namics of density and heat fluctuations in liquids, including an intro,a""tion to superfluid 4 He. Finally, the third part of the course presents at an introductory level '!IOllile advanced themes of great topical interest: supercooling and the ~y state in Chapter 10, non-Newtonian fluids with special attention to ~mers and liquid crystals in Chapter 11, and dynamic instabilities and llmlbulence in Chapter 12. These Chapters, which are largely taken over liimI !he book of N. H. March and M. P. Tosi on "Introduction to Liquid ,~''

'j

'

XII

Mario Tosi and Patrizia Vignola

State Physics", are mainly meant to stimulate and address the reader to further study of the relevant literature. The presentation of the macroscopic properties of thermodynamic systems and of ideal gases takes after the book of L. D. Landau and E. M. Lifshitz, "Statistical Physics" (Pergamon, Oxford 1959) which is in our opinion unequalled in its formative value. The remaining parts of the course have greatly profited from the books published by N. H. March and M. P. Tosi on "Atomic Dynamics in Liquids" (Dover, New York 1991) and on ''Introduction to Liquid State Physics" (World Scientific, London 2002). It is a measure of the progress made in statistical mechanics over the last few decades that a rigorous presentation can now be pursued into the microscopic structural and dynamical properties of liquids, that a full understanding of critical phenomena has been achieved, and that topics so far reserved for the specialistic researcher, such as those covered in the last three Chapters, can now be included in a volume of lecture notes addressed to advanced students. Among the many other textbooks on the foundations and applications of statistical mechanics we shall cite here just a few for further reading, namely the books by K. Huang on "Statistical Mechanics" (Wiley, New York 1987) by S.-K. Ma on "Statistical Mechanics" (World Scientific, Singapore 1985) and by R. P. Feynman on "Statistical Mechanics - A Set of Lectures" (Benjamin, Reading 1972). A number of specific references to other books, review articles, and original research papers are given as the presentation develops, with the aim of providing for each topic covered in these lecture notes an introduction to the relevant scientific literature. Mario Tosi and Patrizia Vignola Pisa, October 2004

Chapter 1 Statistical ensembles and Boltzmann's entropy

The development of statistical mechanics in the second half of the nineteenth century was motivated by the wish to bridge the gap between the macroscopic description of many-particle systems obeying the laws of thermodynamics and the microscopic description which is in principle afforded by the laws of mechanics. This has not proved possible withont invoking a new key principle, which is embodied in the Gibbs statistical ensembles and in the statistical definition of the entropy given by Boltzmann. The crucial role in the statistical description of a macroscopic system in contact with a thermal bath is attribnted to the energy and to the number of particles, which are additive constants of the motion in the absence of couplings with the surroundings. Many aspects and consequences of the key principle of statistical mechanics have since been tested on laboratory systems as well as by the numerical solution of the equations of motion for model systems made of hundreds to thousands of particles.

1.1. The key principle of statistical mechanics The key principle of statistical mechanics is as follows: the probability that a system containing a given number N of particles in thermal equilibrium is in a microscopic state of energy En is Q- 1 exp(-En/knT), where Q = Ln exp(-En/ knT) is the canonical partitionfanction. Here kn = Boltzmann's constant (kn = 1.380658 x 10-23 J/K) T = temperature, and the sum runs over all accessible microscopic states. The notion of thennal equilibrium implies that the system is viewed as being very weakly coupled to a heat bath at a given temperature, and the coupling is not known precisely but has been on for a sufficiently long time. Notice that the probability that the system has energy En contains an additional degeneracy factor, counting the number of microscopic states at energy En.

2

Mario Tosi and Patrizia Vignolo

As a consequence, if In) is a microscopic state with energy En and A a quantum mechanical operator for a physical observable, then the expected value of the observable is (A)= Q- 1 l)nlAln)exp(-En/kBT)

(1.1)

n

where (n I A In) is the value of the observable in the state In). In particular the expected value of the energy of the system, that is the thermodynamic internal energy U, is obtained by taking A =Hamiltonian: i.e. U = (E) = Q- 1 LEnexp(-En/kBT).

(1.2)

n

Fluctuations in the "instantaneous" value of the energy (and of other properties) occur as the system is driven by the coupling with the heat bath to explore its various microscopic states. In the words of Richard Feynman, "this fundamental law is the summit of statistical mechanics, and the entire subject is either the slide-down from this summit as the principle is applied to various cases, or the climbup to where the fundamental law is derived and the concepts of thermal equilibrium and temperature clarified".

1.2. Density of microscopic states The laws of mechanics in principle allow a full description of the microscopic states of a closed system - that is, a system which does not interact with its surroundings or is at most subject to constant external fields, so that its energy is constant in time. It is preferable to use the description provided by quantum mechanics: (i) the microscopic states are labelled by quantum numbers, symbolically indicated by the index n; (ii) the stationary state In) can be taken to be an eigenfunction of the Hamiltonian of the system corresponding to the eigenvalue En of the energy; and (iii) each state can be taken as a function of the (space and spin) coordinates of the particles, whose square modulus describes their probability distribution. The whole set of stationary states has a spectrum of energy levels En. For a macroscopic system the energy spectrum usually is a continuum of allowed energy levels, and we describe it by the density of states g (E), defined by g(E)d E= number of microscopic states having energy between E and E+dE.

3 Statistical mechanics and the physics of fluids

We can also write g(E) = dG(E)/dE with G(E) =number of microscopic states having energy up to the value E. The density of states provides the "degeneracy factor" that is needed for a statistical description of a macroscopic system at thermal equilibrium.

1.3. The state of thermal equilibrium and the canonical ensemble To pass from a mechanical to a thermodynamic and statistical description we need to specify what is meant by thermal equilibrium state. The system under consideration is macroscopic and subject to external conditions that are described by variables such as the temperature T, the pressure p and the magnetic field H. The notion of temperature implies a weak and imprecisely known contact with the surroundings (the "thermal bath"): the system is not a strictly closed one. In the state of thermal equilibrium, at fixed external conditions the observed values of macroscopic physical quantities (such as the particle number density N / V or the magnetization M) do not change during the time of the experiment, provided that the time resolution of their observation is low compared to the time scale r of molecular motions. The time r may be visualized as an average time interval between collisions of a particle with another in a fluid, or in a solid as an average period of vibration of the atoms around their lattice sites. Since a thermodynamic system is not a strictly closed one, on the time scale r of molecular motions it fluctuates over its microscopic states. A low-resolution measurement yields the average value of a physical property, but the "instantaneous" value of each property (and in particular that of the energy) fluctuates around the average. We describe the macroscopic state through the probability w(E) that the system is in a microscopic state of energy E. Then the function W(E) = g(E)w(E) gives the probability W(E)dE that the energy lies in the range from E to E + d E. The underlying hypothesis is that the probability of occupation of a microscopic state is determined only by its energy and that other microscopic details are irrelevant. Notice that the energy is the only additive constant of motion which remains available to describe a closed system which neither translates nor rotates and has a fixed number N of particles. Specifically, at thermal equilibrium we take w(E)

=

Q- 1 exp(-E/ksT)

(1.3)

where Q is to be determined from the condition that W (E) is normalized to unity. The adoption of the above expression for w(E) is equivalent

4

Mario Tosi and Patrizia Vignolo

to the assertion that macroscopic systems in mutual contact at thermal equilibrium are statistically independent - a reasonable assumption if we bear in mind that their interactions occur in the contact areas and can have only negligible effects in the bulk of each system over times that are not too long. Indeed, let (1 +2) be the system formed by systems l and 2 at equilibrium with a thermal bath: the statistical independence of the component systems is expressed by w(l+Zl(E(l+Z)) = w(l)(E< 1l) x w 0, H(e) is its integral up to energy e, and an integration by parts has already been made to take advantage of the fact that the function 3(n(e)) ;ae has nonzero values only in a narrow range of energy around the chemical potentialµ,. By expansion of H(e) - H(µ,) into a power series in (e - µ,)we get at lowest order I= H(µ,)

+ (:ir 2 /6)(k 8 T)

2

[dh(e)/de],~µ.

(3.13)

The chemical potential µ,(T) is obtained from N /V, using h(e) = A.{e: (3.14)

27 Statistical mechanics and the physics of fluids

ere cntemal energy U(T), using h(e) = AVe 312 , is U(T) = (3/5)Nsp[l

+ (5n: 2 /12)(k 8 T/sp) 2 ]

(3.15)

!mi ~.ence the electronic contribution to the heat capacity of the metal is

(3.16) !cc.:erimentally, this contribution can be separated in low-temperature ni=-.;.rnrements from the contribution due to ionic motions, which is pro:r>:r~0nal to T 3 (see § 3.8) . .'c constant magnetic field H induces a preferential orientation of the ~:ec:.-onic spins. Neglecting the effect of temperature, the difference beC·• """ the two spin populations is

1

8F+/.lsH

::. . ''I= Nt-N+ =

0

18F-f,lsH 1

1 -g(e)de2

0

-g(e)de '.". µ, 8 Hg(ep) 2

.,., ~=·ear terms in the field, with /LB =Bohr magneton =lie/(2mk 8 ) = i c-~ Kelvin/Tesla. The induced magnetization is M = b.Nµ, 8 /V and cioe ;aramagnetic "Pauli" susceptibility is

(3.17)

l!:e many-body theory of electrons in metals confirms the above result i::r :=-'le electronic heat capacity, provided that g(ep) is taken as the real

:11:o:s;1y of states at the Fermi level. It also shows that the paramagnetic \jl'I: susceptibility, besides involving the true density of states, contains m. enhancement factor over the Pauli free-electron value. The interac·1:oc; may in fact drive a transition to a ferromagnetic state in which the .:e:e-c:ron spins are spontaneously aligned (see Chap. 6).

31.5. Bose-Einstein condensation ::";e properties of a Bose gas at low temperature are very different from :m>.e of a Fermi gas, owing to the fact that at T = 0 all the bosons can go :~r.c the single-particle state of zero kinetic energy. The Fermi sphere colcu:-;es into the point k = 0 and the spectrum of single-particle excitation ~".lergies ek+q - ek collapses into the "dispersion curve" (liq ) 2/2m. :..et us start with the gas of Bose particles at high temperature and de.:::e.:c;e T at constant N / V. The chemical potential rises from negative ,.J!Ccies and vanishes at temperature To such that N/V

= A(ksTo) 312 lo°" dz(z 112 /[exp(z)

-1]),

28

Mario Tosi and Patrizia Vignola

yielding k 8 T0 "" 3.3l(N/V) 213 tt 2 /m. This tells us that at To the de Broglie wavelength has become comparable to (V / N) I/ 3 and a macroscopic fraction of the particles in the gas begins to "condense" in the state at e = 0. In other words, T0 is the temperature of a phase transition in momentum space leading the particles of the gas to progressively condense from states with finite momentum into the state of zero momentum. For T < To the chemical potential remains zero and the numbers of particles with finite momentum and of particles in the condensate are Ne>O =

N,~o

1"°

3 2

de{g(e)/[exp(,Be) - 1]} = N(T /T0 ) 1

= N[l -

,

(3.18) (T / To)

312

].

The internal energy is determined only by the particles with e > 0 and is given by U

I

=

1"°

de{eg(e)/[exp(,Be)-1])

= 0.770Nk 8 T(T/T0 ) 312 ]

(3.19)

whence Cv = 5U /2T, S = 5U /3T, F = -2U /3 and p = 2U /3V ex T 512 • The pressure vanishes for T ---+ 0 (the condensate does not contribute to jt). All the thermodynamic functions reported above are continuous at T = T0 , but it can be shown that aCv /aT is discontinuous there. A long-debated question is whether a Bose-Einstein condensate is present in liquid 4 He at low temperatures and what may be its role in the superfluidity of this liquid. In general the interatomic forces in a quantal fluid induce correlations in the particle motions, increasing the kinetic energy with a gain of potential energy and of total energy. In fact, the momentum distribution in liquid 4 He is strongly distorted from that of the ideal Bose gas by the atom-atom interactions and the evidence is for a strong depletion of the condensate. Neutron inelastic scattering experiments at large momentum and energy transfers can probe the presence of a Bose-Einstein condensate [4] and show that a fraction of about 8% of the atoms in superfluid 4 He at very low temperature (T ::o 0.3 K) are in the condensate (see § 9.8). Bose-Einstein condensation in trapped vapours of alkali atoms has been realized in 1995 [5], [6], [7]. In these experiments a dilute gas of bosonic Rb or Na atoms is cooled to a final temperature of "" 50 nK iu~ide a magnetic trap providing an essentially harmonic potential well for the atoms. Observations of the density profile of the condensate cloud are

'.·

29 Statistical mechanics and the physics of fluids

performed by absorption of laser light in resonance with an atomic transition, after turning off the trap and allowing some time for free expansion of the cloud to occur. In the ideal gas limit the thermodynamic properties of the condensate cloud in a harmonic trap may be evaluated by a "semi-classical" approximation [8] involving a density of states g(e) ex [e - V(r)] 112 per unit volume, V (r) being the harmonic confining potential. This yields t 8 Tc ""liwN 113 for the condensation temperature (w being the frequency of the trap) and

No/N = 1- (T/Tc) 3

(3.20)

fur the condensate fraction. We see that the confinement changes the ,, !hermodynamic properties of the gas in a substantial way. In particubr.. a discontinuity in the heat capacity (rather than in its temperature derivative) is found to signal the condensation temperature. It should be moticed, however, that interaction effects in the condensed gas, though dilute, are important [9], [10], [11].

3.6. Harmonic oscillators \1ie leave at this point the quantum gases of material particles and open ·lliie discussion of ideal gases of excitation quanta by discussing the prop..mes of a thermodynamic assembly of harmonic oscillators having fre~ncy w. This is realized in a gas of diatomic molecules at moderate iemperatures, where the internal vibration of each molecule may be de..:ribed as a harmonic oscillator if one approximates with a parabola the &pe of the potential energy of the molecule as a function of the interllllJIC!ear distance iu the neighbourhood of its minimum.The vibrational ~levels are en= (n + 1/2)/iw (with n = 0, 1, ... ) and hence Qvib

= exp(-/iw/2)

L exp(-n/iw,B) = n

= exp(-/iw,B/2)/[1- exp(-/iw,B)], Fvib

= Nliw/2 + Nkn T ln[l -

Uvib

= Nliw/2 + N/iw/[exp(/iw,B) -

~: 'll'llo heat capacity [Cv ""

exp(-/iw,B)], l].

Nk8 (liw,6)2 exp(-/iw,B)] is essentially expolllmlial at low T and increases monotonically with T up to the classical d>eNks. from the above expression for Uv;b/ N, we notice that the contribuaimo hw/2 is the quanta! "zero-point energy" (without a corresponding

30

Mario Tosi and Patrizia Vignola

entropy term) and that the second tenn is due to thermal excitations and is the product of a quantum of energy nw times a Planck statistical factor (n(w)) = [exp(nw,8) - 1]- 1. We may therefore interpret the thermal tenn in the mean vibrational energy of an oscillator in the equilibrium assembly as being associated with the presence of (n(w)) quanta, each of them with energy nw equal to the separation between any energy level of the oscillator and the next higher level. This viewpoint can be made more precise. The relevant property of the wave functions ;(q) = (2hw) 112cf>n(q)acf>n(q) = (2hw) 112(n) 112 cf>n(q)cf>n-1 (q)

(n 2: 1).

Hence,

L(n + 1) 112 wn+1cf>n+1 (q)cf>n(q) = exp(-hwf3)(2hw) 112 L(n + 1) 1l w,,cf>,,+1 (q)cf>n (q) = (2hw) 1/ 2

n:;:O

2

n2::_0

=exp(-hwf3)(wq - ip/2)w(q). Thus w(q) obeys the differential equation aw(q)/3q = -2aqw(q), with "' = (w/h)tgh(hwf3/2). Integrating it and normalizing the probability distribution we get w(q) = (a/n) 1i 2 exp(-aq 2 ).

(3.21)

The mean square displacement is (q 2 ) = l/(2a), reducing to (q 2 ) = ;, 8 T / w 2 in the classical limit. The distribution of momenta for the oscil:aror is immediately obtained from the above result, bearing in mind the :orm of the Hamiltonian.

3.7. Black-body radiation We have already introduced the "black body" as an ideal gas of photons at oquilibrium with the walls of its container. The photons have energy hw l!ld momentum hk related by the dispersion relation w = ck (c = speed .Jf light) the values of k being determined by boundary conditions at the •.rnlls of the container. In evaluating the appropriate density of states l(w)dw =number of photon states with frequency in the range between "' and w + dw, we proceed as in our earlier evaluation of the density of ;tales g(e) for material particles. With a cubic container of side L and bearing in mind that there are two possible directions of polarization, we iind (3.22)

32

Mario Tosi and Patrizia Vignolo

The energy of radiation with frequency between (J) and (J) dU.,

+ d(J) is

= ll(J)(n((J))) f((J))d(J) = (II V /rr 2 c3 )(J)3 [exp(ll(J),B) -

1]- 1d(J).

(3.23) This expression reduces to the classical Rayleigh-Jeans formula at low frequencies (dU.,jd(J) ex (J)2 for ll(J) « k 8 T) and to the Wien formula at high frequencies (dU,,,/d(J) ex (J)3 exp(-l!(J),B) for /j(J) » k 8 T). The Planck function interpolates between these two limits. The thermodynamic functions of the gas take the following values (with er= Stefan-Boltzmann constant= rr 2 kV(601! 3c2 )): U =4crVT 4 /c, Cv = 3S = 16crVT 3 /c, F = -pV = -U/3.

3.8. Phonons in crystals As a consequence of crystalline order (periodicity in space) the smallamplitude vibrations of the atoms around their lattice positions in a crystal can be analyzed in terms of a set of independent harmonic oscillators, which represent vibrational waves propagating through the crystal. We consider for simplicity a monatomic crystal with one atom per unit lattice cell. The vibrational waves have frequency (J) and wave vector k, with a dispersion relation (J) j(k) where the index j labels three possible polarizations. The thermal properties of the harmonic crystal may therefore be described in terms of excitation quanta ("phonons") in analogy with black-body radiation. However, the dispersion relation for phonons is determined by the interatomic forces and is in general quite complex, leading to a structured spectrum f ((J)). However, in the limit of small lkl and (J) the propagating waves of lattice vibration see the crystal as an elastic continuum and reduce to sound waves, with a dispersion relation (J) = c,k (c, is the speed of sound propagation, that for simplicity we take as isotropic and independent of polarization). Therefore, f ((J)) = 3 V (J)2 / (2n 2 c;) at low frequencies. The vibrational heat capacity of the crystal becomes (3.24) at low temperatures, where thermal excitations are limited to low-frequency vibrations (typically, on a temperature range of a few degrees above the absolute zero). On the other hand, the frequency spectrum has an upper cut-off frequency ((J)m, say) and the classical limit C v = 3 N k 8 is reached for k 8 T » ll(J)m (in many solids this corresponds to T "" 300 K).

33 Statistical mechanics and the physics of fluids

J:t summary, we have seen in the examples of black-body radiation and ::E lattice vibrations how the introduction of the quanta of a field al,i:·• s us to map a complex problem into a perfect gas. We shall give 1no,lher illustration of this in Chap. 9, where we shall discuss the ex,:::'2lions of liquid 4 He at low temperatures. In such a system one may ,,,,,:!her apply the Bose-Einstein statistics directly to the atoms, which are ,;:-;:mgly interacting in the dense liquid phase, nor use the symmetry as.xiated with crystalline periodicity to describe the atomic dynamics in ;,c::ms of waves with a known dispersion relation. Nevertheless, how"'er complex the ground state of the system may be, we can describe 11:0 weakly excited states in terms of elementary excitations with a dis::,,,-;ion relation w(k) which is well defined but is a priori known only ,111: che sonnd-wave regime. The restriction to weakly excited states (i.e. l ;ow temperatures) is necessary so that the low density of excitations Il"-Y allow us to neglect their "collisions". These lead to a finite lifetime i;:r each excitation, and the concept of an elementary excitation loses its TI.eaning when the reciprocal of its lifetime becomes comparable to its '.,-~uency.

3..9. Classical real gases and weakly coupled classical plasmas '."c;e ideal gas model neglects the contributions to the thermodynamic "rnctions coming from the interactions between the particles of the sys;,,m_ However, interactions are present in most real situations for systems >f material particles and may also be important in gases of excitation X3Ilta. Within equilibrium statistical mechanics, the interactions be.,, een the particles give structure to the system and may induce phase c·:;nsitions (see Chapters 4-6). In a classical real gas with short-range interactions between the par:~:les (the interparticle potential v(r) must decay faster than 1/r 3 with =-·creasing separation r) and away from the critical point, the free energy ;'1.ift !'..F = F - F;J due to the interactions admits an expansion in in•erse powers of the temperature (cumulant expansion) and an expansion c powers of the density p = N / V (virial expansion) [12], [13]. Here "'"" = Nk 8 T[ln(p) - 1 - ln(2Jrmk 8 T/h 2 ) 3i 2 ] is the free energy of the _,Jeal classical gas. The expansion parameters are the strength of v (r) i.--id the number of particles involved in a collision, respectively. In deriv_og these expansion one is assuming that the thermodynamic functions .4ve no singular contributions, so that the results are not correct in the ~·eighbourhood of the liquid-vapour critical point (see Chap. 6). At leading order the virial expansion yields F = F;J + N pkn T B2(T) "''d therefore p = pknT[l + pB2(T)], with B2(T) = (1/2) dr[l -

J

34

Mario Tosi and Patrizia Vignola

exp(-,Bv(r))] arising from two-body collisions and being known as the second virial coefficient. fu Chap. 4 we shall see how the van der Waals

equation of state may be constructed by physical arguments from these results and used to give a mean-field theory of liquid-vapour coexistence. The above expansions do not exist for a classical plasma of charged particles interacting by the long-range Coulomb potential v(r) = e 2 /r. The plasma has an intrinsically collective behaviour, as was shown in the work of Debye and Hiickel [14]. For simplicity we shall consider a plasma of identical point charges neutralized by a uniform inert background of opposite charge at the same average density (the so-called OCP or one-component classical plasma [15], [16]). The crucial idea is to introduce the effect of screening by means of the Poisson equation. Let (r) be the self-consistent electric potential created in the plasma by any one of its ions, taken at the origin. The Laplacian of (r) is determined by the charge density: \'2(r) = -4n{(ion at the origin) +(background)+ (other ions)}

= -4n{e8(r) - ep

+ epg(r)}.

We approximate the probability g(r) of finding another ion at a distance r from the ion at the origin through a Boltzmann factor determined by the potential energy e(r), and furthermore we assume that e(r) is small compared with k 8 T: thus g(r) ~ exp[-e(r)/ k 8 T] ~ 1 - e(r)/ k 8 T. The Poisson equation reduces to \1 2 (r) = -4n(e8(r) - e2 p(r)/ k 8 T) and its solution is (r)

= (e/r) exp(-lff)

(3.25)

where (3.26) is the inverse Debye length. We see from Eq. (3.25) that l/K has the meaning of a screening length: it is the distance over which the electrical potential of the ion at the origin is cut down by a factor exp ( -1) from the rearrangements of the surrounding ions. We recall that according to Faraday's law a static electric field (here the field due to the charge of the ion taken at the origin) does not penetrate in the bulk of a conductor (here the surrounding plasma). The screening length 1/ K was originally introduced by Gouy and Chapman in their theory of how the ions in an electrolyte solution screen a charged planar electrode. In this problem l/K describes the thickness of the dipole layer generated at the interface between the ionic solution

35

Statistical mechanics and the physics of fluids

and the electrode, and hence determines the electrical capacitance of the interface per unit area. In order to obtain the thermodynamic functions of the plasma, it suffices to note from Eq. (3.25) that lim,__.o 2r0 with a depth which is small compared with k 8 T. Then B 2 (T)

'°' 2rr

1

2ro

2

r dr

+ 2rr

1"°

0

2

r {3v(r)dr

=b - a/(ksT)

(4.4)

2ro

where b measures the intrinsic volume of an atom and a is the (negative of the) space average of the attractive potential outside the "atomic diameter" 2r0 . This estimate of the free energy, F "" F;d + N p(k 8 Tb - a), includes only the contribution from two-body collisions and is thus valid only for pb « 1. We now try to estimate the effect of excluded volume (described ' by the quantity pb) to all higher orders in pb by using the approximation -ln(l - pb) ""pb + ... ,i.e. we write 1 F"" F;d - Npa- Nk 8 T 1n(lpb). The van der Waals equation of state follows from this expression: (p

+ a/)(l

- pb) = pk 8 T.

(4.5)

However crude it may be, the van der Waals equation of state contains the essential physics needed to yield a transition from the vapour phase to the liquid phase. The condensation of the gas is driven by the attractive interactions: owing to the attractive term described by a, the van der Waals equation yields a cubic equation for p as a function of p at given T. Thus, the isotherms at low temperature (where the cubic equation admits three real solutions) take the form indicated in Figure 4.3, showing that two phases of different density coexist at D and E. These two states are fixed by the condition that the two phases are not only at the same

I Why do we use the logarithm instead of any other function having the same linear tenn? Notice that an infinitely hard repulsion can contribute only an entropic term to the free energy.

45 Statistical mechanics and the physics of fluids

p

c

Figure 4.3. Showing a van der Waals isotherm below the critical temperature.

lemperature and pressure, but also at the same chemical potential µ,: this condition is fulfilled if the areas D - A - B - D and E - A - C - E are equal (Maxwell construction). In fact, using /LD = /LE and recalling that d µ, = v dp at constant T, from integration on the isotherm we find 0 =LE dµ, =LE vdp =LE (v - vA)dp

=LA (v - VA) dp

+ iE(v -

+ VA(PE

- PD)

VA) dp,

i.e. equality of the two indicated areas.

The fluid is thermodynamically unstable along BC on the isotherm, since in these states it would have a negative compressibility. Within the ''an der Waals theory, the critical point is most easily located by finding first the locus of points such as B and C: these are determined by the condition (8p/8V)r = 0 and lie on the curve p = ap 2 (1 - 2bp). The point of maximum on this curve is the critical point, given by V, = 3Nb, p, = a/(27b 2 ) and kB Tc = 8a/(27b) whence NkBT,/(p, Vcl = 8/3. The measured values for this quantity in real fluids are of order 375. The van der Waals equation of state thus provides a simple mean-field theory for the liquid-vapour critical behaviour. We return to view this topic beyond mean-field in Chap. 6.

4.6. Melting and Lindemann's law The phase diagram illustrated in Figure 4.2 shows the crystalline solid as coexisting with the vapour on the sublimation curve and with the liquid on the fusion curve. Across these lines there is a fundamental discontinuity, since they separate an ordered state from disordered states. It is impossible to go continuously through this structural change, which is associated with a discontinuous change of symmetry.

46

Mario Tosi and Patrizia Vignola

Of course, a static model for crystalline order is not correct. The atoms are always in thermal motion and in a crystal each of them is executing small-amplitude vibrations around a fixed point belonging to an ordered lattice. It is the average atomic positions which are ordered over long distances. From this viewpoint it is a very remarkable aspect of fusion that the melting point is so sharp for a crystalline solid. The cohesive forces maintain crystalline order in spite of the atomic vibrations, up to a temperature where the amplitude of the vibrations becomes so large that the solid melts. In 1910 Lindemann proposed that one could estimate the melting temperature by assuming that melting occurred when the amplitude of vibrational motions in the hot crystal exceeds a "critical" fraction of the atomic spacing. Let Xc be that vibration amplitude which leads to melting in the above viewpoint. If f denotes the force constant between one atom and its neighbour, then for harmonic motion the mean total energy during vibration will be fx~/2 and this can be set equal to ksTm where Tm is the melting temperature, since a linear oscillator has two degrees of freedom. Setting x, = ~a where a is the atomic spacing and ~ < I is a parameter, and relating f to the Young modulus Y of the solid as f = Ya, we get (4.6)

Empirically, this formula with~ "" 0.15 accounts for the measured temperature of melting of many solids at atmospheric pressure. The equivalent criterion for the freezing of a liquid at atmospheric pressure is related to the strength of its state of short-range order, as measured by the peak value Sp of its structure factor (see Chap. 5). For simple liquids this criterion, as proposed by Hansen and Verlet, is (4.7)

at Tm. In a liquid near freezing each atom may be viewed as rattling for times of the order of a ps inside the shell of its first neighbours before diffusing away, and the amplitude of the rattling is comparable to that of the oscillations of an atom around its lattice site in a crystal at melting.

4.7. Brownian motion in the liquid phase In this and the next section we turn to introduce some characteristic dynamical phenomena of the liquid state of matter. Here we consider the molecular thermal motions in the liquid phase. These motions are easy to visualize in a dilute atomic gas: the atoms fly around with a kinetic energy which on average is proportional to temperature, yielding speeds of

47

Statisticaf mechanics and the physics of fluids

the order of hundreds of meters per second at ordinary temperatures. During these motions the atoms collide with each other and with the walls of the container, exerting a pressure and performing a complicated pattern of zig-zag trajectories. In a tightly packed liquid, on the other hand, many of the atoms are so confined by their neighbours that each of them can only vibrate as if inside a cage. Almost as soon as an atom moves away from the centre of the cage, collisions from its neighbours reverse its velocity and send it back. Motions of this sort have frequencies of order 10 12 - 10 13 Hz, similar to those of the vibrational motions in a solid or of the internal 'ibrations in a molecule, and may last on average for time intervals of a few ps. However, the "cage" is not a rigid one but is made of other atoms, which are going through their own thermal motions. If it so happens that ilS neighbours move in some appropriate concerted way,· an atom may IDcceed in exiting from the cage and start on a diffusive type of motion "hich will ultimately bring it far away from its initial position. Each atom in the dense liquid is hopping along a zig-zag trajectory made of discrete microscopic jumps interspersed with rattling motions inside discrete sites cl residence. The diffusive motions of finely divided particles suspended in a liqi!id can be observed under an optical microscope, as was first done in 1827 by the English botanist Brown. In this so-called "Brownian mooon" particles with typical dimensions of 1 µ.m or less are seen to dance :miund in an irregular manner under the effect of random collisions by tt.e molecules of the medium. Again a zig-zag trajectory is observed for such a suspended mesoscopic i?'11icle. According to an argument first given by Stokes, a macroscopic particle W..>.ating in a fluid and experiencing collisions from its molecules only ~ls viscous friction against its direction of motion, since the collisions "':impensate each other in the other directions. The particle must be large '~°tlmpared with the mean free molecular path, so that it feels the buffeting ~ the molecules as if the fluid were a continuum. The resistive force J" felt by a solid sphere of radius a moving through a fluid with steady '"'!ocity v is obtained by simple dimensional analysis as F ex a11v,

where 1J is the shear viscosity ,js found to be for.

(4.8)

(see Chap. 8). The proportionality constant

As the size of the diffusing particle decreases, however, the probability D.-reases for an unbalanced collisional event which may deflect it into

I

I

li

I'

48

Mario Tosi and Patrizia Vignolo

a particular direction. Further collisions produce viscous friction as the particle proceeds in that direction towards the next deflection. This picture may be brought down to the microscopic level to describe the hopping motion of an atom in a liquid under the effect of collisions against its partner atoms, over time scales much longer than that of the localized rattlings inside the cage of first neighbours. In 1908 Laugevin described the diffusive motions of a mesoscopic particle in a fluid by partitioning the forces that it feels into the sum of a viscous force aud of a random collisional force. This yields the Langevin equation of motion (4.9) mx =-Ji +F, where m is the particle mass, x is the component of its displacement in any one of the three space directions, f is a friction coefficient, aud F, is the force due to random collisions. We multiply Eq. (4.9) by x and average over a large number of collisions, thus averaging away the random term as positive and negative values of F, are equally probable in the long run. We get m(xx) = - f(ix) or, with the identity xx = d(ix)/dt - i 2 aud taking m(i2) = k 8 T, m d(ix)/dt + f(ix) = k 8 T. This integrates to (ix) = (ks T / f) + A exp( - ft/ m) and in steady state, after the exponential term has decayed to zero, we find (ix)= k 8 T/f. Thus the meau square displacement of the diffusing particle increases linearly with time over long times:

Jim (x2) = (2ks T / f)t.

HOO

(4.10)

This behaviour is characteristic of diffusive motions and indeed Eq. (4.10) can be used to relate the diffusion coefficient D to the viscous coefficient

fas D=ksT/f.

(4.11)

Diffusion in liquids will be discussed in depth in Chap. 7. Here we remark that f = frrra~ according to Stokes' law (Eq. (4.8)) so that Eq. (4.11) yields at once the Stokes-Einstein relation between diffusion coefficient, particle size and shear viscosity, the latter transport coefficient being also determined by collisions. We also remark that, while Brownian motion results from spontaneous fluctuations, the diffusion coefficient D that these determine can be shown to be proportional to the particle mobility µ according to the Nernst-Einstein relation

D =ksTµ

(4.12)

(see Chap. 7). The mobility µ is accessible to measurements of driven transport under a constant external field, the most common case being that of charged particles in an electric field.

49

Statistical mechanics and the physics of fluids

4.8. Flow properties of dense liquids We have already emphasized that, while solids exhibit resistance to shear, a liquid flows under an arbitrarily small shear stress. This may be viewed as a collective or cooperative property on the macroscopic scale. Let us begin to discuss the flow properties of liquids at an elementary, though basic level (cf. the books by Tabor [23], [24]). Let us consider first the flow of an idealized liquid which is both incompressible and without viscosity. If the liquid flows in a continuous steady state, one can draw a line such that the tangent at any point gives the direction of flow of the particles. Such streamlines are smooth continuous lines throughout the fluid and cannot intersect, and no particle can flow across from one streamline to another. We take an imaginary tube in the liquid bounded by streamlines and impose energy balance during a short time interval dt, in which the pressure drives through the tube a volume dV = a 1 v 1dt of liquid at A and an equal amount dV = a2 v2 dt of liquid leaves at B: we have set the flow velocity as v 1 and the crosssectional area of the tube as a 1 at A (at height h 1) and similarly at B (at height h 2 ). The sum of the pressure work, the kinetic energy, and the gravitational potential energy at A must be equal to that at B, P1

1

2

+ 2PV1 + pgh1 =

1

2

P2 + 2PV2

+ pgh2

(4.13)

where p is the density of the liquid. This is Bernoulli's equation of flow for an ideal liquid. The assumption of incompressible flow is not permissible in handling flows where large pressure differences arise (e.g. flows through narrow channels) nor in treating the propagation of sound waves (see Chapter 8). Excluding such situations, Eq. (4.13) is applicable when the speed of flow vis small compared with the speed of sound c, the ratio l'/c being known as the Mach number. In particular Eq. (4.13) shows that an increase in flow velocity is accompanied by a drop in pressure. This accounts for aerodynamic lift: because the top surface of an aerofoil is larger, the air velocity in flight is higher and the pressure lower than over the bottom surface. In fact, real liquids experience a viscous resistance to flow. If the velocity gradient between two neighbouring planes is dvx/dz the force Fxz per unit area to overcome viscous resistance (or better the shear stress needed to maintain the velocity gradient in a steady state) is Fxz = 1}dVx/dz

(4.14)

,.·here 11 is the shear viscosity, with dimensions [11] = L - 1MT- 1• Fluids which obey this relation are known as Newtonian fluids (see Chap. 8

50

Mario Tosi and Patrizia Vignola

and 11 ). While the viscosity of gases is explained in terms of transfer of molecular momentum across the flow, in a dense liquid a simple molecular model explains viscous resistance by considering that the molecules are so close together that energy is expended in dragging one molecular layer over its neighbour. If the flow is steady and stable the work done is expended solely in overcoming viscosity and appears as heat. However, if the velocity of flow is too high or if there are some other unfavourable circumstances, vortices may develop and some of the work goes in providing their kinetic energy. Vortices often assemble in a boundary layer on a solid boundary. The condition for turbulent flow was first established by Reynolds. Consider a liquid of mass density p and viscosity ry flowing with velocity v along a channel of lateral dimension a. There is a critical velocity v, above which orderly streamlined flow gives way to turbulent motion. The Reynolds number Re is defined by purely dimensional analysis as Re= pav/ry.

(4.15)

It is dimensionless and the transition from streamlined to turbulent motion occurs when its value is around 1000 to 2000 (see Chap. 12). Turbulence is essentially the result of a condition of instability. The laminar flow pattern in a pipe becomes turbulent when primary eddies get out of control before viscosity may quench them, and they start generating strings of further eddies. Experiments show that above the critical Reynolds number the pressure gradient needed to drive a fluid through a pipe increases more rapidly than linearly with the rate of flow Q. Hydraulic engineers, who need to transfer fluids loosing as little pressure head as possible, may achieve values of the critical Reynolds number as high as 105 by taking special care in pipe construction and lay out.

Chapter 5 Thermodynamic fluctuations and liquid structure

i;

I,' ~

•1

In this Chapter we shall first discuss equal-time correlations between fluetnations of thermodynamic quantities and then turn to the concept of structure for a disordered system in terms of instantaneous correlations between density fluctuations on the microscopic scale. We have already introduced in § 3 the structure of a fluid as described by the radial distribution function g(r), which measures the probability of finding two particles at separation r.

5.1. Thermodynamic fluctuations and their correlations We wish to determine the probability distribution for the fluctuations of some thermodynamic quantity (x, say) in a subsystem at thermal equilibrium. We have already met this problem for the fluctuations of energy in § 1. 7, in connecting the statistical definition of entropy with me second principle of thermodynamics: we considered a closed sysi.em described by the microcanonical distribution, that we rewrote as the probability of partitions of the total energy among the subsystems, dw oc exp[S/ k 8 ]n;dECil with S = L; SCil(£Ul) =entropy of the closed system as a function of the energies of its subsystems. Similarly, a flue: mation in any thermodynamic quantity x in one of the subsystems takes !he system away from equilibrium and is all the less probable the larger ti the accompanying decrease in total entropy. We thus write the proba', t.ility dw(x) that the value of the thermodynamic quantity be in the range :. between x andx+dx as dw(x) oc exp[S(x)/ k 8 ]dx, with S(x) =entropy • Di the closed system as a function of x. We are implicitly assuming that we can exactly determine the "instan: llmeous" value of the entropy corresponding to each value of x. However, r iilf T is the scale for the variation of x in time, according to the Heisenberg pinciple the energy is determined within t;.E "" ft/r and the entropy :•ithin t;.S "" /i/(rT). We require t;.S « kn, i.e. knT » /i/r or

-

52

Mario Tosi and Patrizia Vignolo

»

r li/(k 8 T). Thus the present approach becomes invalid at very low temperatures or for very rapid variations in time. Assuming that these conditions are fulfilled, it is convenient to denote with x the deviation of the quantity of interest from its mean value (x) (i.e. we set (x) =0). Using the fact that S(x) is a maximum at x = 0, i.e. as;axlx~o = 0 and a 2 s;ax 2 1x~o < 0, we may write the probability distribution for fluctuations of limited amplitude as dw(x) = w(x)dx with w(x) = Aexp[-ax 2 /2k8 ]

(5.1)

where A= (a/2rrk8 ) 112, from normalization. The mean square fluctuation is (x

2

)

=

1:

2

x w(x)dx

= k8 /a.

(5.2)

Thus the probability distribution for fluctuations around the mean value in a subsystem at thermal equilibrium is a Gaussian with a width determined by the mean square fluctuation. More generally, however, we should consider the simultaneous fluctuations of several thermodynamic quantities, which may correlate with each other. Let x 1 , x 2, ... , Xn be the deviations from their mean values: their probability is determined by w(x1, x2, ... , Xn) ex exp[S(x1, Xi, ... , Xn)/ ks] =Aexp[-L:aijX;Xj/2k 8 ],

(5.3)

i,j

from an expansion of the total entropy around the equilibrium state. Since the coefficients of the expansion have the symmetry property aij = a;;, the quadratic form entering the sum in Eq. (5.3) can be diagonalized. It is physically more transparent, however, to proceed by introducing a set of variables X; that are thermodynamically conjugate to the variables x; (we shall need them later in dealing with transport). Their definition is

X;

=

-as;ax; = L:a;ixi. j

(5.4)

53 Statistical mechanics and the physics of fluids

I We find

J

00

(x;Xj) =A

dx1 ..• dXnXi L°'jlXl exp [- L°'kmXkXm/2ks]

-co

l

k,m

(5.5)

110-e see that (within our basic assumptions) a thermodynamic fluctuation ) iS simply correlated with its own conjugate variable and does not correlate • 0 and KT > 0 express the conditions that the system be stable against small-amplitude fluctuations in T and V. From the results in§ 5.1 we find

(t:,.Tt:,.V) =0,

2

((6T) 2 ) =ksT 2 /Cv,

((6V) ) = VKyksT. 12 2 12 We verify from these expressions that [((6T) )] 1 /T ex N- 1 and [((t:,.V)2)]112;v ex N-112. If instead we choose p and S as independent variables, we find from Eq. (5.6) 2

w ex exp[-(6S) 2/(2k 8 Cp) - V Ks(6p) /(2k 8 T)], 2 (Ks =adiabatic compressibility); i.e. (t:,.S /':,.p) = 0, ((/':,.S) )

(5.8)

= k 8 Cp

and ((/':,.p) 2 ) = k 8 T/VKs. 2 Finally, we return to the result given above for ((/':,. V) ), which was obtained by implicitly keeping constant the number of particles. Division by N 2 yields the mean square fluctuation in the density p = N / V, (5.9) ((/':,.p) 2 ) = n 2 Kyk T/V (n = (p)). 8

We may then get the mean square fluctuation in the number of particles in a given volume: ((/':,.N) 2 )

= Vn 2 KyksT = -n 2 ksT(oV /ap)T,N = nksT(aN /ap)y,v = ksT(aN /8µ,)y,v.

For a classical perfect gas this yields ((6N)

2

)

= (N).

55 Statistical mechanics and the physics of fluids

5.3. Spatial correlations of density flnctnations and liqnid structure The notion of autocorrelation ((/'>p) 2 ) of the thermodynamic fluctuation in density may he extended to the microscopic scale by considering the correlations between simultaneous density fluctuations occurring at different points in a fluid at thermal equilibrium. The "istantaneous" particle density at a point r is described by p(r) = L; o(R;-r), R; being the position of the i-th particle. Indeed, the density I= number of particles per unit volume) at point r is different from zero only if any one of the particles is within an infinitesimal volume centred on this point, and in such a case it is infinite. We introduce the correlation function

which depends only on lr1 - r21 from invariance under translation and rotation. For r 1 f= r 2 we write (p(r 1)p(r,))dV1dV2 = (ndVi)[ng(lr1 r2DdV2], thereby defining ng(r)dV as the probability that, given a particle at a point chosen as the origin, a second particle is found at the same time inside a volume d V at a distance r from the fitst (g (r) is the radial distribution function already met in § 3.9). The full expression for (p(r1)p(r2)) (including also the case r 1 = r2) is (5.10)

The function g(r) describes the short-range order in the liquid. It tends in the limit r ---+ oo, since the positions of two widely separated particles in a fluid will be completely uncorrelated. Figure 5.1 shows, from numerical simulation data, the shape of g (r) in a classical liquid Slllch as argon near freezing, where the particles interact via a short-range irepulsive potential supplemented by an attractive tail outside the atomic oore (the unit of interatomic distance r in the figure is the atomic diameter G. which is a few AJ. The main features of g(r) are an excluded-volume region, of extent essentially determined by the diameter of the atomic core, and a first-neighbour peak in approximate correspondence with the minimum of the attractive interatomic well. Such short-range order in the local structure around a particle in a liqmid is measured in diffraction experiments, using X-rays or neutrons of wavelength comparable to the mean first-neighbour distance. We may blk upon the distribution of particles around an "average particle" of llbe fluid as forming a spherical diaphragm. The diffraction pattern is delermined by the interference between the waves scattered by the central

ID unity

56

4.0

Mario Tosi and Patrizia Vignolo

'

3.0

~2.0 1.0

0

0.4

0.8

1.2

1.6

2.0

2.4

2.8

r/ a

Figure 5.1. Pair distribution fuuctiou g(r) for a fluid having the same repulsive forces as those of the Lennard-Jones potential, as calculated by Chandler and Weeks (1970). The results are compared with molecular dynamics data of Verlet and with the numerical Percus-Yevick solution for the full Leonard-Jones potential. u denotes the "atomic diatemeter".

particle and those scattered by the surrounding particles, whose average number is (4nr 2dr)(ng(r)) in each spherical shell of thickness dr at a distance r from the centre. The iuteusity I (k) of scattered radiation as a function of the modulus of the transferred wave vector k (given by k = (4n /A.) sine where A. is the wavelength of the probe and is ouehalf of the scattering angle) is

e

/(k) = Nlf(k)l 2 {1

+ (L j(f;)

siu[k· (rj -r;)])} k· (rj -r;)

= Nlf(k)l 2 S(k) (5.11)

where f (k) is the scattering amplitude of a single atom and siu(kr)/(kr) is the diffraction function of a spherical diaphragm. Iu Eq. (5.11) we have defined the structure factor S(k) = 1 +

("" (4nnr 2dr) siu(kr) [g(r) -

Jo

= 1+n

f

kr

l] (5.12)

dr[g(r) - 1] exp(ik · r).

S(k) describes (through its deviations from unity) the diffractive effects of the local structure: g (r) may thus be determined experimentally by Fourier inversion from a measurement of S(k). Figure 5.2 shows the measured structure factor S(k) of superfiuid liquid 4 He. The maiu peak iu S (k) reflects the short-range order which

· 57 Statistical mechanics and the physics of fluids

,.

,., Figure 5.2. Liquid structure factor S(Q) of superfluid 4 He, as determined in various experiments (from Woods and Cowley, 1973). The result of the Feynman theory is also shown at low momenta. is present in the liquid. We remark that the position of this peak at k "" 2 A- 1 corresponds to the position of the roton minimum in the dispersion curve for the elementary excitations in the superfluid (see Chap. 8). We should also notice that the spatial order described by S(k) is combined in superfluid 4 He with order in momentum space and is relatively weak, in the sense that the height of the main peak in S(k) is appreciably lower than that normally seen in a classical liquid such as liquid argon close to freezing (see the "crystallization criterion" for a classical liquid in Eq. (4.7)).

5.4. Connections between liquid structure and thermodynamics We return to the correlation function {p(r 1)p(r2 )) and notice its following property: J dVi J dV2{[p(r1 - n][p(r2) - n]) = nV 000 (4:n:r 2dr){8(r) + [g(r) l]} = n V limk-+O S(k). On the other hand, we have J dV1 J dV2{[p(r1 -

f

2

n][p(r2 ) - n]) = n{{f dV[p(r) - nJ} ) = (!'>.N) 2 • With the expression (5.9) for {(/1N) 2 ), we find for a classical monatomic liquid that the structure factor in the long-wavelength limit is related to the isothermal compressibility,

Jim S(k) = nksT Kr.

k-+O

(5.13)

Another (less fundamental) relation between liquid structure and thermodynamics is an expression for the internal energy of a classical fluid of

'1

~

)l 58

Mario Tosi and Patrizia Vignola

'l

:.i

particles interacting via a pair potential:

1

00

U / N = -kBT 3 2

+

(4:rcnr 2 dr)g(r)v(r).

0

This expression follows from the meaning of g(r) as counting pairs of particles at various separations. From the virial theorem we also find the pressure as p

= nKBT -

I -n 2

6

1

00

0

dv(r) (4:rcnr 2 dr)g(r)--. dr

(5.15)

We stress again that above we have considered the spatial correlations be- ] tween simultaneous density fluctuations at different points in a fluid. We :j have seen that they can be measured in diffraction experiments and how ] they are related to correlations between thermodynamic fluctuations. The·. extension to spatial correlations between density fluctuations at different times will be given in Chap. 8 and will provide us with the appropriate ~ language for a discussion of the collective dynamics of a liquid and its inelastic scattering spectrum. j

·.·I· .•·

1

:~

S.S. Inhomogeneous fluid

,!

We consider next a monatomic fluid at given temperature and chemical potential, which is subject to an external potential V(r). The external potential breaks the translational symmetry of the fluid, making the average density n (r) = (p (r)) a function of position. We first present below some general properties of such an inhomogeneous fluid. We shall then consider the special case in which the "external" potential becomes the interatomic potential v(r) generated by an atom taken at the origin in a classical fluid: in this case n(r) = ng(r). We define u(r) = µ, - V(r). Given the interactions between the component atoms, V (r) completely determines the Hamiltonian of the fluid and hence u(r) determines (in principle!) the equilibrium grand canonical ensemble w and the corresponding grand potential Q. More generally, given the Hamiltonian H and a grand ensemble w' (not necessarily the eqnilibrium one for that Hamiltonian) the grand potential can be calculated from Du[w']

= Tr[w'(H -

µ,N

+ ,e- 1 1n w'))

(5.16)

where Tr denotes the trace over the number of particles and over the quanta! states (see Sees. 1.5 and 2.7). From the fact that Q is at a minimum in the equilibrium state, we have the inequality (5.17)

··l I,

"

.~

59 Statistical mechanics and the physics of fluids

if w' is an ensemble different from w (and thus is a non-equilibrium ensemble at given u(r)). Since the equilibrium ensemble determines the equilibrium density profile n(r), we may say that u(r) determines n(r). The HohenbergKohn-Mermin theorem [25],[26] ensures that also the reverse is true: in principle we may go back from a given n(r) to the potential u(r) which determines it. The proof of the theorem proceeds by reduction ad absurdum. Suppose that two different potentials u(r) and u' (r), which determine two different equilibrium ensembles w and w', were to yield the same n(r). From the minimum property we would have rlu1[w 1] < rlu1[w]

= rlu[w] +

f

1

drn(r)[u (r) - u(r)].

By interchanging u and u' we would also have rlu[w] < Qu[w']

= rlu1[w 1] +

f

drn(r)[u(r) - u'(r)],

and by summing these two inequalities we would reach the absurd result rlu{W 1] + rlu[w] < Qu[w]

+ rlu1[w 1].

We conclude that there is a biunivocal relationship between u (r) and n (r), .,-hich allows us to treat these microscopic variables in the way as we treat ,wnjugate thermodynamic quantities. We have already seen above that Q is a functional of u(r) in the fluid n thermal equilibrium, meaning that its value is in principle known if die values of u(r) are known at all space points. By means of a Legendre transformation we can introduce a new thermodynamic quantity F, "'-hich is a functional of n(r): F[n(r)] = Q[u(r)]

+

f

drn(r)u(r).

(5.18)

, h is evident from this relation that F is the "intrinsic" Helmholtz free i mergy, i.e. not including the average interaction energy of the fluid with ' lii:le external potential V(r). We find from Eq. (5.18) the equilibrium '· c"l!lndition which determines n(r) at given u(r),

'

8F /8n(r) ;.is

= u(r)

(5.19)

well as the equilibrium condition which determines u(r) at given n(r),

8Q/8u(r) = -n(r)

(5.20)

I

60

Mario Tosi and Patrizia Vignola

The notation used· in these equations denotes the functional derivative, giving the change in F (or Q) due to a change in n(r) (or u(r)) at any point r. Let us consider for an illustration a classical ideal gas of atoms which do not interact with each other. Its equilibrium density profile is given by the Boltzmann distribution, ----.. n(r) =A. - 3 exp[,Bu(r)] where A. = [2irli 2 /(mk 8 T)] 112 is the thermal de Broglie wavelength. This result can be obtained from the appropriate equilibrium condition; which is 8F;d/8n(r) = u(r), if the ideal free energy functional is given by F;d[n(r)] = k 8 T If n (r)

J

drn(r){ln[A. 3 n(r)] - l]}.

(5.21)

=constant we recover the free energy of the homogeneous classi-

cal gas (see § 3.9). More generally, for a real fluid of interacting particles it is convenient to break F into the sum of its ideal part and of the "excess" part due to the interactions between the particles: F[n(r)]

= F;d[n(r)] + Fex[n(r)].

(5.22) .

If the fluid is classical F;d is still given by Eq. (5.21). Thus the equilib-

rium profile has the form n(r) =A. - 3 exp[,BuKs(r)]

(5.23)

where u K s (r) is the potential uKs(r) = u(r) - 8Fex/8n(r).

(5.24)

introduced by Kohn and Sham [27]. These two equations establish a mapping of the fluid of interacting particles into an ideal gas. Of course, uKs(r) is a functional of n(r) and a self-consistent evaluation of the equilibrium density is necessary, using a suitable approximate expression for the functional Fex.

5.6. Direct correlation function The higher derivatives of the functionals Q and Fex define two hierarchies of correlation functions. In particular, for their second derivatives we

61

Statistical mechanics and the physics of fluids

;!'

have H(rl, r 2)

= -{J =

_1

82Q

8u(r 1 )8u(~ = fJ

_ 18n(r1)

8u(r2)

(5.25)

([p(r1) - n][p(r2) - n])

and c(r1, r 2 )

= -{J

82 Fex

1 8u(ri) = --8(r1 - rz) - { J - - . 8n(r1)8n(r2) n(r 1) 8n(r2)

(5.26)

The identification of H with the two-body correlation function in the last step of Eq. (5.25) can be demonstrated from the expression of Q in the equilibrium grand canonical ensemble (an explicit proof is recommended as an exercise). The function c in Eq. (5.26) is the Omstein-Zemike direct correlation function and plays a special role in the theory of classical liquids. The first-order functional derivatives entering Eqs. (5.25) and (5.26) have a precise physical meaning: 8n(r 1)/8u(r2) describes the density response of the inhomogeneous fluid to a change in the external potential, and 8u (r 1) / 8n(r2) is its inverse. Treating them as matrices with indices r1 and r 2 , from their matrix product we obtain the Omstein-Zemike relation,

Finally we take the limit V (r) -> 0: we recover the homogeneous fluid, in which c(r1, r 2 ) = c(r 12 ) and H(r1, r 2 ) = n8(r 1-r2 ) +n 2[g(r12)- l]. The Ornstein-Zemike relation becomes (5.27) )

with h(r) = g(r) - l; namely, in Fourier transform, S(k) = [l - c(k)r 1 .

(5.28)

This result shows that a diffraction experiment on a classical liquid gives us directly its density response to a weak, static external potential varying periodically in space with a wavelength 2n / k. This is the microscopic generalization of the relationship found in§ 3.4 between the longwavelength structure factor and the isothermal compressibility.

62

Mario Tosi and Patrizia Vfgnolo

5.7. Liquid-structure theory for classical systems The objective of liquid-structure theory ~s to evaluate the radial distribution function g(r) from given interactiOns between the particles. We give here an illustration of two alternative (approximate) approaches to this problem for the simple case of a monatomic classical liquid whose particles interact via a pair potential v(r). (1) One way to obtain an approximate expression for the functional Fex uses an expansion of the inhomogeneous fluid around the homogeneous one at the same mean density n. With the notation L'>n(r) = n(r) - n, the expansion yields for the difference L'>F of the intrinsic free energies of the two fluids, at the same temperature and chemical potential, the expression

f ff

L'>F = (µ, - k 8 T)

- kBT

+ k8 T

dr L'>n(I')

dr

f

drn(r) ln[n(r)/n]

dr'c(Jr - r'J)L'>n(r)L'>n(r')

+ ....

The higher terms involve higher-order direct correlation functions of the homogeneous fluid. We use this expansion to treat the case indicated at the beginning of § 5.5: the "external" potential is the interatomic potential v(r) generated by an atom of the fluid taken at the origin. Since n(r) = ng(r) in this case, we find from the equilibrium condition g(r) =exp {-{Jv(r)

= exp(-{Jv(r)

+n

f

dr1 c(Jr - r'l)h(r')

+ h(r) -

c(r)

+ ... }

+ ... }.

In the last step we have used the Omstein-Zernike relation h(r) = c(r)

j

+ n dr c(Jr -

r'J)h(r').

The truncation of the expansion at the terms shown above is known as the hypernetted chain approximation (HNC): we get two coupled equations for g(r) and c(r), whose (numerical) solution allows us to calculate the structure of the fluid from the interatomic potential. The terms arising from the higher-order direct correlation functions are known as the "bridge function". Self-consistent schemes are available in the literature for their estimate. (2) An alternative approximate theory of liquid structure was developed by Percus and.Yevick [28]. It assumes the relation c(r) = g(r)(l - exp[{Jv(r)]}

63

Statistical mechanics and the physics of fluids

in combination with the Ornstein-Zernike relation. This theory has the advantage that it can be solved analytically in the case of a fluid of hard spheres [29]. [30]. If a is the hard-sphere diameter and ry = nna 3 /6 is the packing fraction of the fluid. the result for c(r) is c(r) = 0

\for r >a

c(r) =a+ b(r /a)+ c(r/a) 3

for r 0 provides a model for the orderdisorder transition between the paramagnetic and the ferromagnetic state. However, other order-disorder transitions may be described by the same type of Hamiltonian: (a) the paramagnet - antiferromagnet transition, when one takes J < O; (b) phase separation or chemical ordering of the components in a binary alloy (for J > 0 and J < 0, respectively) when one assigns to the values s; = ± 1 the meaning of indicating whether the i-th site is occupied by a type-A atom or by a type-B atom; (c) condensation from the vapour phase in a lattice-gas model (for J > 0) when .one assigns to the values s; = ±1 the meaning of indicating whether the i-tb site is occupied or empty.

I

I 1·

I II 1r

1, 1 ii

I,';

'' ii

6.3. Mean-field theory The mean-field method, though yielding only approximate and under some fundamental aspects incorrect results for critical phenomena, still

Ii ii

I I

i I I'

68

Mario Tosi and Patrizia Vignolo

has a useful role in statistical mechanics. In general terms, given an Hamiltonian H({;}) which depends on a set of microscopic variables q,, (such as the spins s; in the Ising model) the mean-field approach in its simplest version replaces these variables by a common value q, to be determined in a self-consistent manner. The Hamiltonian becomes an internal energy U() and the entropy is separately evaluated in order to determine the order parameter q, by minimizing the free energy. In the Ising model q, is the mean magnetic moment m per site. The numbers of up and down spins are (1 + m)N /2 and (1 - m)N /2, and each spin sees a mean field given by h + J vm, where v is the number of first neighbours of a site. The simplest approximation for the entropy is obtained by neglecting the spin-spin interactions, S =kn ln(N!/{[(l + m)N /2]![(1 - m)N /2]!}) ""(Nk8 /2)[2ln2-(l +m)ln(l +m)- (1-m)ln(l -m)], as already given in § 6.1. The equilibrium condition gives the equation m = tgh[(h + Jvm)/ knTJ

(6.7)

to be solved for m. At h = 0 this equation has only the solution m = 0 for T > T, = J v/ kn, but at T = T, a second type of solution emerges, yielding a spontaneous magnetization. For T < T, the soiution m f= 0 lies at a minimum of the free energy, 1 while the solution m = 0 still exists but corresponds to a maximum. It is left as an exercise to show that the spontaneous magnetization vanishes at the critical temperature like (T,-T) 112 , and that at T > T, andh f= Othe system is a paramagnet with susceptibility x ex (T-T,)- 1 diverging for T _,. T, (Curie-Weiss law). The mean-field approach thus predicts the order-disorder transition. As will become clear below, its results are incorrect under two main aspects: (a) the nature of the lattice enters only through the number of first neighbours and there is no trace of a dependence on its dimensionality; and (b) the values of the critical exponents, entering the laws that describe the behaviour of the physical properties of the system near the critical point, are not correct. The mean-field approach does not keep account of fluctuations, which are instead of crucial importance near the critical point and are indeed all the more crucial as the dimensionality of the system decreases.

1There are, of course, two solutions with m 1= 0, which are equal and opposite values of m~ Ensemble averages in the ferromagnet must be taken on a broken~symmetry ensemble corresponding to a chosen sign of the magnetization.

69 Statistical mechanics and the physics of fluids

The Landau theory of phase transitions still is a mean-field theory, but its nature is much less specific. Since the order parameter is small near the critical temperature. Tc, the theory assumes that the free energy can be expanded in powers of in its neighbourhood: F = ao(T)

+ a 2 (T)2 /2 + a.(T)4/4 +. · · -

h

(for T "" Tc and small). Having assumed that such an expansion exists, then at zero field it can involve only even powers of , from invariance under inversion of all the spins in zero field. Minimization of F shows that, at h = 0, is either zero or (if a2 /a4 < 0) = ±(-a2/a4) 112. Furthermore, if a4 > 0, the solution = 0 is stable for a2 > 0 while the solution i= 0 is stable for a1 < 0: i.e. starts taking a non-zero value (spontaneous order appears) when a 2 changes sign. Since the theory is limited to T "" Teo it suffices to take a2 "" c(T - Tc), with c a positive constant, and a4 "" positive constant: hence, 0< (Tc - T) 112 for T ->- Tc. For h i= 0 and T > Tc we find "" h/a2 and hence the susceptibility is x = l/[c(T - Tc)]. 13 Furthermore, at T = Tc we have a4 3 "" h, i.e. 0< h 1 . The Landau theory is easily extended to the case in which the field has a weak space dependence. One needs to add to the free energy a term (V )2 /2 as a contribution specifically associated with the gradient of the order parameter. Fourier transform yields F

= ao + L: Tc yields q ""hq/(a 2 + q ), i.e. the generalized susceptibility is (6.8)

The generalized susceptibility, that here we have evaluated at small but non-zero wave number, is the Fourier transform of the spin-spin correlation function G(r), by extension of the relation given in Eq. (6.5) between Xq~o and the volume integral of G(r). There is complete analogy between these properties of spin-spin correlations and those of the correlations between density fluctuations in a fluid as discussed in Chap. 5.

6.4. Ising magnetism in dimensionalities d = 1 and d = 2 It is easily shown that the Ising model does not admit an ordered state for a chain ·Of localized spins in zero field, except at T = 0. In the ground

70

Mario Tosi and Patrizia Vignolo

state at T = 0 all spins point in the same direction, the internal energy being U(T = 0) = -(N - 1)1. At T > 0 we ask what is the change in free energy if we start inverting spins. We define a partition point as the boundary between a region of spins + 1 and a region of spins -1. In the macroscopic limit the ordered state is destroyed as soon as the chain . is brought to a non-zero temperature: indeed, the creation of the first' partition point requires energy 21 and gives a contribution ks ln(N - 1) to the entropy, since the partition point can be created at any one of the ·· (N -1) interstitial sites. We have b.F = 21 -k 8 T ln(N -1)] and this is always negative in the macroscopic limit at any temperatnre T cl 0. This result is completely different from the prediction that one would make from a mean-field approach. Let us make the argument somewhat more precise. At finite temperature the chain will contain a fluctuating number of partition points, but if the temperature is not too high their average number N' will be much smaller than the number N of sites. We may therefore treat the partition points as an ideal gas: the single-particle energy is 21, the chemical potential is zero (since the number of partition points fluctuates) and the gas obeys Fermi statistics (since an interstitial site may be occupied by either zero or one partition point). Hence, N' / N = [exp(21 / kBT)

+ 1r1,

[U(T) - U(O)]/N = 21 /[exp(21 /k 8 T)

(6.9)

+

l].

(6.10)

As long as N' / N cl 0 (i.e. for T > 0) the spins do not point in a single · direction: the boundaries between regions with opposite spin orientations. can move freely, so that the average magnetization is zero. From Eq. (6.9) we can interpret ~ = exp(21 / k 8 T) as a measure of the mean distance ·· between partition points, in units of the lattice constant: at a given T the spins are ordered as long as N < ~, but partition points will appear if the j' chain is sufficiently long. The gas of partition points in d = 1 can be treated also at h cl 0, thus allowing an exact solution for the magnetic behaviour of the Ising chain of spins (see for instance the book by S. K. Ma cited in the preface, pp. 299-301). The. low-field susceptibility is

x=

(N/k 8 T)exp(21/k 8 T),

exponentially diverging for T --+ 0.

(6.11) \

71

Statistical mechanics and the physics of fluids

The instability of the ordered state in d = I at T > 0 comes from the fact that, after the creation of partition points, the regions of up and down spins can change in length without an energy cost, so that on average the numbers of the two spin populations are equal. This is no longer true in d == 2. The proof that in d = 2 (and in particular on a square lattice) the ordered state is stable over a finite temperature range is due to Peierls [31]. Starting from the ordered state at T = 0, inversions of spins at finite temperature generate partition points between pairs of opposite spins on neighbouring sites and these points are joined by lines embracing areas of spin inversion. If L is the length of such a line (i.e. the perimeter of a region of spin inversion, with L :::: 4) the energy associated with it is 2L J. The entropy may be estimated by noticing that at each point along the perimeter (except near its closure) we have three possible choices for the direction in which we continue tracing the perimeter: hence, the perimeter can be traced in"" 3L ways and the entropy is"" Lk 8 In 3. This estimate yields /'!,.F ""2LJ - Lk8 T In 3, so that the ordered state is stable as long as k 8 T is smaller than "" 2J /(In 3) "" l.SJ. The calculation by Peierls was more precise and led to an upper limit for the probability of spin inversions as a function of T, with the result that at k 8 T = J this probability is at most a few percent. The exact solution of the Ising model in d = 2 at zero field was later found by Onsager [32]. Further exact results have been obtained by Yang and Wu [33], [34]. The critical temperature T, for the order-disorder transition is given by the condition sinh(2J / k 8 T,) = 1, namely k 8 T, = 2.2691. The magnetization is almost complete up to T "" 0.6T, and vanishes at T, with the law M/N

) =

w2

+ (Dk2)2

(7.23)

'

an expression which is valid at low k and w. The half-width of the spectrum in this limit is Dk 2 . The deep physical meaning of this result is discussed in § 8.1. More generally, one may express the relationship between the selfdiffusion coefficient and the van Hove dynamic structure factor S,(k, w) as D

=~Jim {w2 k--+0 lim[S,(k, w)/ k2]} 2 tv--+0

.

(7.24)

This expression, which can be verified from Eq. (7.23) is one of the celebrated Kubo formulae relating transport coefficients to wavenumber and frequency dependent correlation functions.

Chapter 8 Viscosity, sound waves, and inelastic scattering from liquids

8.1. Hydrodynamic variables Thennodynamic equilibrium in a classical fluid is established and maintained by collisional events occurring with a time interval T which in many instances may be of the order of 10- 10 - 10- 13 s. We introduce the mean free path l = VT, where v "" (k8 T/m) 112 is the thermal velocity, with the meaning of the average distance travelled by a molecule between successive collisions. Consider now a disturbance of the equilibrium which varies periodically in time and space with frequency w and wave-number k: if wT « 1 and kl « 1, there are many collisions within each space-time cycle and we may assume that the fluid at each point in space is close to equilibrium at each instant in time. In fact, some degrees of freedom are sure to vary slowly in time at long wavelengths. We met an example of such a variable in treating mass diffusion in Chap. 7 (see especially § 7 .9). Particle conservation implies a continuity equation relating the time derivative of the density of particles to the divergence of their flux: as a result, the frequency range relevant to the variation in time of the particle density is Dk 2 , vanishing for k -+ 0 (see Eq. (7.23)). This argument extends to the densities of all conserved quantities, namely (for a one-component fluid) momentum and energy in addition to particle number. Thus, sound waves and thennal conduction in a simple fluid have characteristic inverse times which vanish as the wavelength goes to infinity. More generally, the time dependence of the density of each conserved quantity is detennined by the divergence of its current density and, for slowly varying disturbances, this is a local function of the fields which are thermodynamically conjugate to the conserved quantities. In Fick's law the particle flux is driven by the gradient in chemical potential, which is then taken to be proportional to the density gradient in defining the diffusion coefficient.

I /1

lI. 11

·I 111

90

Mario Tosi and Patrizia Vignola

The equations relating currents to fields are known as the constitutive relations and the parities under time reversal of each field and of the currents that it drives are crucial. A current has opposite parity to that of the associated density of conserved quantity, since the two are related by a continuity equation. Hence, a coefficient relating a current and a field having the same parity is purely reactive, while the relation between a current and a field having opposite parities is necessarily dissipative. Of course, the relations between currents and fields may contain both reactive and dissipative terms - an example being the propagation of sound waves. The term hydrodynamics is commonly used to refer to such dynamics at long wavelength and low frequency [44]. For a one-component fluid there are five conservation laws and five hydrodynamic modes. This number increases in liquid mixtures because of mass conservation for each component, and also in ordered systems with continuous broken symmetries. As an example of the latter we may cite a nematic liquid crystal, in which the molecules in the shape of short sticks line up along the so-called director. Since the director can point in any direction (in the absence of anchoring) it takes a vanishingly small energy to induce a slow continuous variation in this direction and hence the time rate of change of the variable that describes the broken symmetry must be small. For a nematic liquid crystal this argument leads to two additional hydrodynamic modes (see Chap. 11). In a snperfluid the additional hydrodynamic variable is the superfluid velocity: the continuous broken symmetry is in this case related to gauge invariance and, since the superfluid velocity can be expressed through the gradient of a phase, there is just one extra hydrodynamic mode (see Chap. 9). In this Chapter we shall be concerned with momentum as a conserved quantity in an isotropic one-component fluid and shall defer to Chap. 9 an account of energy conservation. We shall then see that the hydrodynamic theory presented here for sound-wave propagation is correct when energy fluctuations are decoupled from density fluctuations: this occurs in the limit when the specific heat ratio Cµ/Cv becomes unity.

8.2. Stresses in a Newtonian flnid and the Navier-Stokes equation In this section we first introduce the Newtonian law relating the stress in a viscous fluid to the local gradients in fluid velocity, and then derive the basic equations governing the variation of the momentum density in time. Consider a fluid flowing along the x 1 direction with velocity v1 (x 2 ) corresponding to a uniform velocity gradient dvi/ dx 2 • In a microscopic

91

Statistical mechanics and the physics of fluids

view the molecules in any given layer are moving more slowly than those in the layer above it. Newton assumed that in such circumstances there is a shearing drag force between adjacent layers, directed along x 1 and acting in the plane orthogonal to x 2 • Denoting the stress (force per unit area) as a 12 , then Newton's law of viscosity gives its magnitude as (8.1) at low rates of shear, ry being the shear viscosity. This law obeys two basic requirements: (i) the stress changes sign when the flow is reversed; and (ii) if the velocity gradient vanishes everywhere, there is an inertial frame in which the fluid is at rest and the shear stress vanishes. In such a frame only the diagonal elements of the full stress tensor of the fluid are different from zero and are given by the pressure as a;1 = - p8u. As a first step towards extending Newton's law to three-dimensional flow, let us consider the case where a velocity gradient dv2fdx 1 is also present. The quantity (8.2)

describes the local rate of rotation of the fluid. No stress can be associated with such vorticity, since local rotation does not change the separation between any two neighbouring points inside the fluid. The stresses a 12 and a 21 must be equal to each other and proportional to the average of the two velocity gradients - or otherwise each volume element of the fluid would be subject to a couple about the x 3 axis and hence to an angular acceleration. The conclusion is that stress in a fluid is a symmetric second-rank tensor and may thus be expressed as the sum of an isotropic term and of an anisotropic symmetric term. The most general form of the stress for a Newtonian fluid in the hydrodynamic regime thus is

where ry and ry' are known as the first and the second coefficient of viscosity. The quantity \7 · v is the divergence of the velocity field. Particle conservation gives the continuity equation ap/at = -\7 · (pv), with p being the particle density and pv the particle current density j. The continuity equation shows that \7 · v is related to the total time derivative of the density p, as it can be rewritten as \7 · v = -(Dp/ Dt)/ p where (D/Dt) = (a/at)+v·\7. Thus,forincompressibleflowwesetV'·v = 0. We may also define the mean pressure p as the trace of the stress tensor,

92 Mario Tosi and Patrizia Vignolo

+ 2,, /3 is the bulk viscosity. This enters to determine the attenuation of sound waves (see § (8.7)). We now write the equation of motion of the fluid in the form of a continuity equation relating the time derivative of the momentum density . · g;. = mpv; to the divergence of the momentum current density JC;j, .,

ft =

(cr11

+ 1: for instance, in Selenium the Prigogine-Defay ratio lies in the range 1.2 - 2.4. The implications of the non-equilibrium value of the Prigogine-Defay ratio in the glass transition were emphasized in the early work of Gold. stein [89]. He pointed out that in this case on the free energy surface G(p, T) of a liquid there is an infinite set of points, each of which defines a distinct glass. There is no "ideal" glassy state having a unique structure: rather, the structure of a glass is whatever the liquid structure happens to be when the liquid solidifies at Tg. This leads us in turn to the vivid picture of a disordered free energy landscape for the glassy state, having its conceptual precursors in Goldstein [89] and Anderson [90]. As emphasized in a review article by Stillinger [91], in order to understand the phenomena involved in supercooling and glass formation it

131

Statistical mechanics and the physics of fluids

is useful to adopt a topographic view of the potential energy function (x 1 , ... , XN). We can imagine a 3N-climensional map showing the "elevation" at any "location" R = (x 1, ••• , XN) in the configuration space of the N-particle system. Such a -scape presents maxima ("mountain tops") and minima ("valley bottoms") as well as saddle points ("mountain passes"). A minimum corresponds to a mechanically stable arrangement of the N particles in space and is enclosed in its own "basin of attraction", containing all configurations that are connected to it by strictly downhill motions. The lowest-lying minima are those that the system would select if it were cooled to absolute zero slowly enough to maintain thermal equilibrium. Higher-lying minima represent amorphous packings, which may be sampled by the stable liquid above melting. Transition states correspond to saddle points through which the system may pass in migrating from a minimum to another. A schematic view is shown in Figure 10.3 (left panel).

L,.:._J

----

.

__

Figure 10.3. Schematic map of the potential energy hypersurface in configu-

ration space for a many-particle system (left) and interbasin transitions corresponding to /J-type and a-type relaxation in a fragile glass-former (right). Redrawn from March and Tosi [18]. Such a -scape picture lends itself to a description in terms of a freeenergy landscape on account of the role of temperature. The equilibrium state at any temperature T corresponds to preferential occupation of basins having an optimum depth = 0 up to"" 0.494) to a crystalline state (on the branch from "" 0.545 to ,::; 0.74, the latter corresponding to close-packing). The disordered dense hard-sphere state is not represented by any of these lines, but as shown in Figure 10.5 is a metastable extension of the fluid branch, ending into a state of random close-packing (RCP) at "" 0.644. The RCP can be precisely defined [99] as having the largest packing fraction over all ergodic isotropic ensembles at which the first neighbour distance equals the hard sphere diameter. In the context of the hard-sphere phase diagram, the fundamental problem in creating metastable dense systems is to make sure that they are

111

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Mario Tosi and Patrizia Vignolo

ToRCP~!

,:

Metastable

,' ~

: ToFCC (Cl""'

Packed)

branch~~

,, MeH(ng

VOiume Fraction (t)

0.49 0.64 0.55

0.74

Figure 10.5. Phase diagram of the hard-sphere system, including the metastable amorphous branch (dashed curve). Redrawn from March and Tosi [18].

truly random. While there is no perfect measure of order or disorder, Rintoul and Torquato [100] quantify the degree of local order in their computer realizations of hard-sphere assemblies through an order parameter which is a rotationally invariant average over all bonds and is nonzero in the presence of any type of crystallization. This affords a precise quantitative means of creating dense random systems that lie on the metastable amorphous branch of the phase diagram. On these systems Rintoul and Torquato determine key statistical properties, with special attention to the statistics of voids and to mean pore sizes. The pore length scale determines transport properties, such as the mean survival time of Brownian particles diffusing in a system of traps. A random assembly of non-overlapping or even overlapping hard spheres provides a model matrix inside which to study the equilibrium and transport properties of fluids adsorbed in disordered porous materials [101]. Some important questions concern the phase transitions that may occur in such heterogeneous fluids and how are they related to those in the bulk fluid. A great deal of attention has been given to the phenomenon of capillary condensation, i.e. the shift in the bulk liquid-gas transition due to confinement, and to surface phase transitions (layering and prewetting) from interactions with a solid matrix.

10.6.2. Supercooled water The lowest temperatures down to which water remains a liquid in ordinary time-scale observations have been determined in cloud-chamber

137 Statistical mechanics and the physics of fluids

experiments, in which the behaviour of large numbers of droplets can be studied. At standard pressure such small droplets freeze by homogeneous nucleation at about -40 C. The vitrification temperature at standard pressure is estimated to lie at about -135 C from extrapolations of data on glass-forming aqueous solutions. Many properties of supercooled water have been measured in great detail [102]. These include thermodynamic properties (density and expansivity, vapour pressure, heat capacity, compressibility and sound velocity) transport properties (diffusivity, viscosity, electric conductivity, dielectric relaxation, sound absorption, nuclear and electron spin relaxations) and spectroscopic properties (Raman and infrared spectra, proton magnetic resonance).

10.6.3. Metallic glasses Amorphous metallic alloys were first discovered in 1960 by P. Duwez, who showed that Au75 Si25 could be frozen into an amorphous state by rapid cooling from the melt. Metallic glasses have acquired considerable commercial importance, e.g. in the production of high-strength and wearresistent materials or of soft magnetic materials. Applications thus range from protective coatings to tape recorder heads and other magnetic devices. Bulk and surface devitrification may be used in tailoring material properties. The realization of a metallic glass requires ultra-rapid cooling. A common laboratory technique is by melt-spinning, in which a fine stream of molten metal alloy is allowed to fall onto a copper wheel in fast rotation and solidifies into a ribbon a few millimeters wide. This method gives cooling rates of the order of 105 - 106 degrees per second. A variety of other techniques can be used and a large range of alloy types and compositions can be brought to a glassy state, as listed for instance in the review article by Lindsay Greer [103]. Examples of glass-forming alloys include (i) transition metals in combination with metalloids, (ii) early and late transition metals, and (iii) aluminium-based, alkaline-earth-based, lanthanide-based and actinide-based systems. Rapid cooling from the melting (liquidus) temperature Tm down to T8 is evidently easier when the interval between these two temperatures is small. Tm can be a strong function of composition while T8 usually is not: therefore, glass formation can be expected to be favoured near deep depressions at eutectics in the liquidus curve. A general rule is that the glass forming ability is promoted by stabilizing the liquid relative to the solid: this has been exploited by adding solutes and by increasing the number of alloy components. The presence of a multiplicity of components, and

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especially their different sizes, inhibits crystallization. The glass forming ability of an alloy such as Zr4J.2Ti13.sCu12.s Ni10.0Be22.s approaches that of oxide glasses. However, metallic glasses are much less stable: they crystallize on heating or may devitrify into a quasicrystalline phase of icosahedral symmetry.

10.6.4. Superionic glasses Ionic glasses are often realized from a network former such as Si02 or Al20 3 and a network modifier such as Na2 0 or Li20. The modifiers lower the glass transition temperature and open up the network structure by introducing non-bridging oxygens, while the alkali ions are mobile and can diffuse through the glassy network. The decoupling of conduction modes from viscous modes is involved in the generation of ionconducting glasses below T8 and is quite distinct from the (a, {3) bifurcation [104]. The modes of small low-charged cations become anharrnonic at temperatures far below those of other species and such cations can easily escape from their initial sites and wander through the glassy structure. A characteristic feature of singly modified ionic glasses is that the ionic conductivity rr rises sharply with the content x of the mobile alkali species [105], with an effective power law rr c* is somewhat steeper than the Jaw n ex c 2 predicted by the mean-field theory. A scaling argument showing that n cx c 914 is given immediately below. Scaling concepts have proved very useful in understanding the physical behaviour of polymeric systems [129], [130], [131]. On purely dimensional reasons, the osmotic pressure can be written as a function of the segment concentration c and the number N of monomers as (11.25) where a is an effective segment length and f denotes. some function of the indicated arguments. We consider a scaling transformation defined by N -+ N / '!.., c -+ c / '!.. and a -+ a'!.. v, which corresponds to grouping together'!.. segments at constant radius of gyration R 8 . We take this unit of'!.. segments as representing a new segment and impose that the osmotic pressure should be unchanged. This condition reduces the form (11.25) to

n = ~k8 T f (~u diffusion

tttt

L*

Figure 12.3. Illustrating convection and diffusion in a laminar boundary layer over a flat plate (from Tennekes and Lumley, 1972). Figure 12.3 illustrates laminar flow over a flat plate with no-slip boundary: a characteristic length L and a characteristic velocity U have been attributed to the flow, so that its characteristic time is t "" L j U. The viscous term on the RHS of Eq. (12.12) describes transport of momentum by molecular processes across the main flow: for a fixed time scale, this process defines another length scale L * "" (vLj U) 112 = LjRe 112 • With characteristic velocity fluctuations of order u as indicated in the figure, we can write t "" L *ju and hence we have the scale relation L*jL""ujU.

(12.13)

The viscous length L * in laminar flow represents the transverse thickness of the boundary layer, inside which the molecular transport of momentum occurs away from the solid surface. We consider next the case of turbulent flow. The momentum transfer is now dominantly effected by turbulent eddies, and we may estimate the thickness of the boundary layer by again equating the time scale for turbulent diffusivity to the convective time scale, that is L *ju "" Lj U. This merely states that .iu an imposed flow the turbulence, being part of the flow, must have a time scale commensurate with that of the flow. Thus Eq. (12.13) still gives the scale relation for the boundary layer. However, the small eddies which are responsible for dissipation have much shorter time scales, tending to make them statistically independent of the main flow.

12.6.1. Reynolds stresses As discussed in detail in § 8.2, the viscous term in Eq. (12.12) arises from the Newtonian stress tensor as determined by the gradient of the velocity field via intermolecular collisions. In the regime of fully developed turbulence its role is taken over by turbulent stresses arising from fluctuations in the velocity field, which are known as Reynolds stresses. Let us consider steady flow at mean velocity U and let u = (u 1 , u 2 , u 3 ) be a fluctuating velocity field with zero time average. The x1-component

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Statistical rriechanics and the physics of fluids

of the momentum flux per unit area across the faces dx 2 dx 3 and dx 1dx 3 of a fluid volume element are mp(U1 + u 1) 2 and mp(U1 + u 1)(U2 + u 2), with average values mp(Uf + (uf)) and mp(U1U2 + (u1u2)) where the brackets denote a time average. The equivalence between momentum flux and stress follows at once from Newton's second law, so that the fluctuating terms in the momentum fluxes above correspond to stresses a11 = '-mp(uf) and 0"12 = -mp(u1u2) on the faces of the volume element. Notice that (i) a 11 is compressive since (uf) is positive, and (ii) the rate at which the xrcomponent of momentum passes through the face dx 2 dx 3 leads to a shear stress a 21 = a 12 on that face, just as with viscous stresses. The conclusion thus is that in turbulent flows, even though the relative root-mean-square fluctuations of the velocity field may be of the order of a percent, the mean motion is not determined only by viscous forces. In a mainstream of fully developed turbulence, as already discussed in § 12.5, the Reynolds stresses are in fact much larger than the viscous stresses. If one then tries to account for the Reynolds stresses by deriving from the original Navier-Stokes equatfon additional equations for the velocity autocorrelation functions, one runs into what is known as the closure problem of turbulence: uukuown correlation functions such as (u 1 u~) are generated by the convective term. This problem is typical of all nonlinear stochastic systems. Many attempts have therefore been made in the literature to find an approximate reduction of the Reynolds stress tensor to a form similar to that of the Newtonian stress tensor involving the velocity gradient, by introducing an eddy viscosity. There are some special cases in which the turbulent diffusivities depend simply on the velocity and length scales of the flow, as in the Couette flow between a fixed wall and a moving wall. In general, however, there are two profound differences between turbulent stresses and viscous stresses: (i) turbulent stresses are continuous whereas molecules collide only at intervals, and (ii) the dimeusious of turbulent eddies are not small relative to those of the flow. More recently, great progress has been made in understanding turbulence through the solution of the uonlinear Navier-Stokes equation by direct numerical methods [175]. The so-called lattice Boltzmann method has been progressively refined and extended to the point where it is a competitive technique to treat a variety of nontrivial flows.

12.6.2. Lattice Boltzmann computing The structure of the Navier-Stokes equation is quite independent of the details of the underlying microscopic dynamics, which only determine

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the numerical values of the transport coefficients. This universality motivates the use of microdynamical models which, while giving up as much irrelevant detail as possible, still retain the basic aspects of the physics of fluids. Lattice gas models are within such a class of models. Their aim consists in the definition of a simplified microworld which allows one to recover in the macroscopic limit the equations of fluid dynamics. A proper choice of the symmetry of the lattice is crucial for this purpose [176], [177]. Once the correct symmetries of the lattice are chosen, there are two possible ways of defining the evolution rules. These are the lattice gas automata and the lattice Boltzmann method. In lattice gas automata the variables are the boolean populations and the evolution is defined by a set of collision rules. In particular, one can consider the Boltzmann approximation to the collisional dynamics, by making the same assumption which leads to the Boltzmann equation in continuous kinetic theory: particles entering a collision are uncorrelated. The development of lattice Boltzmann (LB) methods [178], [179], [180] was originally motivated by the need to beat the statistical noise problem plaguing the lattice gas automata method. The main LB equation has a Chapman-Enskog form for the evolution of a particle population distribution f; (r, t) occupying at time t the node r of the lattice along a direction specified by a velocity variable c1 (with i = 1, ... , b): b

J,(r + c;, t

+ 1J -

J;(r, 1)

=I: Aij