Statistical Mechanics Volume 2

A Review of the Recent Literature published up to July 1974

By K. Singer

The Royal Society of Chemistry

Copyright © 1975 The Chemical Society
All rights reserved.
ISBN: 978-0-85186-760-1

Contents

Chapter 1 Theory of Time-dependent Correlations in Simple Classical Liquids By P. Schofield, 1,
Chapter 2 Thermodynamic Behaviour in the Critical Region By D. W. Wood, 55,
Chapter 3 Equilibrium Theory of Electrolyte Solutions By C. W. Outhwaite, 188,
Chapter 4 Statistical Mechanics of Surfaces By S. Toxvaerd, 256,
Chapter 5 Equilibrium Statistical Mechanics of Molecular Liquids By C. G. Gray, 300,
Author Index, 324,

CHAPTER 1

Theory of Time-dependent Correlations in Simple Classical Liquids

BY P. SCHOFIELD

1 Introduction

The tremendous increase in our understanding of the dynamical properties of the liquid state in recent years has its roots in three closely related areas of development : the use of radiation and particle-scattering techniques to study spectral lineshapes which can be related directly to dynamical correlations, the use of computer simulation methods to model the properties of real fluids, and the development of the theory of time-dependent correlation functions as a means of describing the behaviour of the fluid at the microscopic level. This chapter is mainly concerned with reviewing the present status of the last of these topics with particular regard to the application to the class of liquids known to theoreticians as ‘simple’ in the sense which we define below.

The renaissance of the theory of liquids (in the broad sense covering both static and dynamical or time-dependent properties) dates from the middle 1950s and coincided with the advent of slow neutron scattering as a means of probing the density fluctuations in liquids on scales of distance comparable to the interatomic spacing and of the corresponding time taken by an atom to traverse this distance. The formulation by van Hove of the neutron scattering cross-section in terms of the space-time correlation function, therefore, forms a convenient starting point to this review. It can be argued that previous to this time very little convincing progress had been made in the theory since the work of Enskog on dense gases, of which the book by Chapman and Cowling remains the best and most complete account despite its cumbersome notation. The reason for this lack of progress was in part the lack of detailed information about the atomic motion which would support or destroy the hypotheses on which the theory was based, but in part also the inability to test whether the success or failure of a theory to predict, for example, the viscosity of a liquid, was due to the theory or due to the data on the interaction potential, pair distribution function, etc., used in the computation. It is here that the molecular dynamics method of computer simulation has had a decisive impact on theory. By studying the dynamics of a model system of particles with a given interaction one may not only compute the transport properties of the model to compare with theory, but by more detailed analysis of the computation one may examine whether or not the theory is soundly based. It would be possible to review the early theories in this way, but since this would prove a largely destructive exercise, we prefer to concentrate on more recent developments.

Formalism follows fashion. It is the main purpose of this chapter to underline the physical content of the theory of the density and momentum fluctuations in a simple liquid and to review recent work in this light. Nevertheless it is useful to have a framework for the discussion and we shall use the ‘memory function’ formalism for this purpose. The seminal paper for this approach is the linear response theory of Kubo, from which are derived expressions for the transport coefficients of diffusion, viscosity, thermal conductivity, etc., as time integrals of equilibrium correlation functions, now known generally as Kubo relations. Historically the earliest of these, for the diffusion coefficient of a particle as the time integral of the velocity auto-correlation function, was derived by Einstein. It is remarkable that not until 1954 were the equivalent expressions for viscosity and thermal conductivity obtained by Green and later by Mori. Even at this stage these authors used the coarse-graining and time-smoothing arguments then in vogue; the realization that the results were independent of these operations and that the transport coefficients were related to the time integrals of the true microscopic correlation functions marked a major development in the understanding of non-equilibrium statistical mechanics.

In this introduction we have referred to a few key papers on which the time-dependent correlation theory is built. To those we should add that of Kadanoff and Martin, who discussed the general mathematical structure of long-wavelength fluctuations and its relation to hydrodynamics. Their approach, based on linear response theory, is completely equivalent to the approach adopted here which follows more closely that of Kubo and Mori.

A complete theory of liquids would lead to a unified treatment of both static and dynamic correlations. Such a synthesis has not yet been achieved. It is customary in dealing with time-dependent properties to assume that one knows the static correlation functions. The problem presents itself as the determination of the dynamical correlation functions in terms of the static correlation functions and the interatomic potential (also assumed known). The theory of the static properties was reviewed in the first two chapters of Volume 1, and reference is here to these Reports in support of assertions concerning these properties. Similarly we shall refer to the Report by Gubbins in the context of transport coefficients.

The form of this chapter is as follows. The following two sections deal with the basic formalism of time dependent correlation theory, Section 2 being mainly concerned with definitions and Section 3 with perturbative methods. Section 4 treats the velocity autocorrelation function, which demonstrates both the principles and methods currently invoked in the discussion of more general correlation functions. Section 5 is devoted to the hydrodynamic limit and Section 6 to recent theories of density and current correlation for general wave vector and frequency. Section 7 gives a brief account of the powerful methods based on kinetic equations for fluctuations in phase space.

In concluding this introduction, we wish to stress our concern with the theory of liquids. For this reason we omit detailed reference to the extensive and important work on low-density fluids, in particular that related to expansions in density. The reader is referred to the report of Gubbins.

2 Definitions and Elementary Properties of Correlation Functions

In this section we review the elementary properties of the time-dependent functions. We consider the statistical mechanics of an ensemble of particles whose evolution is described by a classical Hamiltonian depending only on the co-ordinates and momenta of the centres of mass of the particles. This is our definition of a simple fluid. Much of the formalism is of more general applicability. However, the applications reviewed will be restricted to the ‘simple’ case.

Definitions. — We write the time correlation function of two quantities a(t), b(t’) as . a(t) is to be understood as an abbreviation of a function aN({ri(t),p,(t)}) of the 3N co-ordinates and momenta of N particles at time t, ri (t), Pi(t), and the angular brackets denote a thermodynamic average over a grand canonical ensemble in a large volume V in which the mean number of particles is N and the mean density n. The Hamiltonian is assumed to have the form

[MATHEMATICAL EXPRESSION OMITTED] (1)

where Mi is the mass of particle, i, and ΦN({ri}) is a translational invariant potential energy function [ΦN({ri}) = ΦN({ri + r})].

The time evolution of the system is given by the operator [??]N(t):

[MATHEMATICAL EXPRESSION OMITTED] (2)

We shall assume that the potential energy, ΦN({ri}), is a differentiable function of the coordinates, in which case it can be written in terms of the Liouville operator [??]N:

[MATHEMATICAL EXPRESSION OMITTED] (3)

(α represents a Cartesian index, and the summation convention is used). Thus the time derivative of aN({ri, pi}) is given by [MATHEMATICAL EXPRESSION OMITTED] which, from Hamilton’s equations of motion is equal to [??] NaN.

A factor of i is often introduced into the definition of [??]N so that it is Hermitian rather than anti-Hermitian and [??]N written exp(I [??] Nt); there seems no virtue in this. The restriction on the potential is important since it rules out discontinuous potentials, in particular rigid-core interactions where the potential is finite outside some diameter [alpa] and infinite for shorter distances. Realistic potentials do not have this property. Nevertheless the rigid-sphere model is known to be a good zeroth approximation for thermodynamic and transport properties. We shall return to this point throughout this chapter, but remark here that the rigid core can be treated as a limiting case of a differentiable potential as the steepness of the short-range repulsion, or the ‘hardness’ as we shall call it, is increased.

It is desirable to normalize correlation functions to be finite in the infinite volume limit. Thus we define the correlation function Xab(t) of a(0),b(t) in a thermodynamic state specified by a temperature, T, and chemical potential, μ, by

[MATHEMATICAL EXPRESSION OMITTED] (4)

with β= (kT)-1 where ri abis a normalizing factor conventionally chosen in such a way as to give a finite limit, and [??](T,μ) is the grand canonical partition function. Note that since [??] N is a constant of the motion, the correlation function is independent of the initial time t0. We use the convention that the variables in which the time is unspecified have the values at the instant the thermodynamic average is taken. One might consider also the canonical and microcanonical ensemble averages. In the infinite volume limit the results are the same. The advantage of the grand canonical ensemble is that for large volumes one may treat a large subvolume and the total volume on the same footing. The choice of ensemble is a non-problem as far as the physics is concerned ; however, it does raise some difficulties in the interpretation of simulation experiments with a few hundred particles. These will not be discussed here.

We list now the variables and correlation functions in which we are interested. 10 The notation is conventional. For a fluid of identical particles we define a ‘single particle density’ by the Dirac delta-function.

ni (R, t) = δ {LAN[SpanishR]LAN – ri(t)} (5)

The integral of ni(R, t) over any volume of the fluid has the value unity if the particle is within the volume and zero otherwise. The correlation function

[MATHEMATICAL EXPRESSION OMITTED](6)

has the property that Gs (R’,R,t) d3R‘ is the probability that the particle is within a volume d3R‘ at time t, given that it is at R at time zero. This is the ‘self correlation function’ of van Hove. In the infinite volume limit it is a function of (R’ — R), Gs(R‘ — R, t). At t = 0, Gs(R,t) is S(R) and in a fluid it tends to zero at large times.

The total microscopic density is given by

[MATHEMATICAL EXPRESSION OMITTED] (7)

Its expectation value n, and its correlation function

[MATHEMATICAL EXPRESSION OMITTED] (8)

gives the mean density of particles at R’, given that there is a particle at R at t = 0. At t = 10,

G(R,0) = δ(R) + n g (9)

where g(R) is the pair distribution function of the fluid. At large t, G(R,t) tends to the mean density n. We may define also ITL Gd(R, t) the difference between G(R,t) and G.(R,t), as the function which gives the distribution of particles other than that originally at the origin. This is the ‘distinct’ correlation function.

It is with the properties of these and the related current-correlation functions defined below that this Report is mainly concerned. However, it is more convenient to work with their Fourier transforms. Thus we define the ‘density fluctuations’ of wave-vector [??]:

[MATHEMATICAL EXPRESSION OMITTED] (10)

[MATHEMATICAL EXPRESSION OMITTED] (11)

and define the ‘self’ and ‘density’ correlation functions

[MATHEMATICAL EXPRESSION OMITTED] (12)

[MATHEMATICAL EXPRESSION OMITTED] (13)

In a uniform system the correlations of n[??] and n[??] etc. vanish unless [??]’ = — [??]

The normalization is such that [??]s([??],0) is unity while

[??([??], 0) – S([??]) – 1 + h([??]) (14)

where S([??]) is the static structure factor measured in diffraction experiments with h([??]), the pair correlation function,

[MATHEMATICAL EXPRESSION OMITTED] (15)

One may similarly define correlation functions of the momentum fluctuation by the following sequence. The α-Cartesian component of the momentum density is

[MATHEMATICAL EXPRESSION OMITTED] (16)

from which we obtain

(17) [MATHEMATICAL EXPRESSION OMITTED]

and the current correlation functions

[MATHEMATICAL EXPRESSION OMITTED] (18)

where [??] is a unit vector in the direction of [??]. The form of the second line follows from symmetry in an isotropic fluid ; L and T denote the longitudinal and transverse components wit h respect to t he direction of [??]. The normalization is such that

[MATHEMATICAL EXPRESSION OMITTED] (19)

In later sections, where we consider the kinetic equation approach, we deal with the more general ‘phase-space’ correlation functions, in particular that of the single particle phase space correlation function. For this we have the following sequence of definitions:

[MATHEMATICAL EXPRESSION OMITTED] (20)

[MATHEMATICAL EXPRESSION OMITTED] (21)

with the normalization

[MATHEMATICAL EXPRESSION OMITTED] (23)

where m(P) is the Maxwell distribution of momentum

[MATHEMATICAL EXPRESSION OMITTED] (24)

The Fourier transform in R-space of F is proportional to the probability that at time t one will find a particle at R wit h moment um P’ given a particle at the origin at time zero with momentum P.

Clearly

[MATHEMATICAL EXPRESSION OMITTED] (25)

[MATHEMATICAL EXPRESSION OMITTED] (26)

In the same way the correlation of any functions of the form u(pi) exp{i[??] ·ri} can be obtained by quadrature if [??]s ([??],P,P’;t) and Fd]IDL ([??],P,P’;t) and [??]d ([??],P,P’;t) are known. An alternative method of evaluating such correlations would be in terms of a complete orthonormal set of momentum correlation functions Un(Pi) of which products of the Hermite polynomials [MATHEMATICAL EXPRESSION OMITTED] or the related Sonine polynomials used by Chapman and Cowling are convenient choices.

Properties of Correlation Functions. — There are certain properties of the correlation functions which follow from the stationarity of the ensemble, the condition that <a(t0)b(t + t0)) is independent of t0. Differentiation with respect to t0 yields

[MATHEMATICAL EXPRESSION OMITTED] (27)

(i) From the condition that [MATHEMATICAL EXPRESSION OMITTED], where the asterisk denotes complex conjugate, be greater than or equal to zero for real λ, one has the Schwartz inequality

[MATHEMATICAL EXPRESSION OMITTED] (28)

Thus if the autocorrelation function <a*(0)a(t)) is real, then its modulus is less than its initial value

[MATHEMATICAL EXPRESSION OMITTED] (29)

This applies in particular to the density-density correlation functions defined above: Gs.d(R, t) are even functions of R hence their Fourier transforms are real.

(ii) If the Hamiltonian is differentiable, then the autocorrelation functions have a Taylor expansion about t = 0:

[MATHEMATICAL EXPRESSION OMITTED] (30)

where

[MATHEMATICAL EXPRESSION OMITTED] (31)

Further since [??] is an odd function of momentum a2s + 1 = 0, and <a(0)a(t)> is an even function of t.

(iii) Thus, applying the stationary condition (27),

[MATHEMATICAL EXPRESSION OMITTED] (32)

with

[MATHEMATICAL EXPRESSION OMITTED] (33)

where a(s) denotes the s’th time derivative of a(t ) at t= 0. From the Schwartz inequality (28) applied to a(s) and a(s + 2)]ITl, we find

[MATHEMATICAL EXPRESSION OMITTED] (34)

The radius of convergence of the Taylor expansion is given by a time τ’c,

[MATHEMATICAL EXPRESSION OMITTED] (35)

Sum Rules. The terms ‘sum-rules’ or ‘moment relations’ relate to the values of the coefficients A2sin the Taylor expansion of the autocorrelation functions. We define the spectral function of a correlation function [??]ab(t) by its Fourier transform

[MATHEMATICAL EXPRESSION OMITTED] (36)

Then, as is well known, the value of the s‘th frequency moment of the spectral function, where it exists, is given by the s’th time derivative of Xab(t) taken at t = 0

[MATHEMATICAL EXPRESSION OMITTED] (37)

This is the s’th moment relation for the correlation of a and b. The term ‘sum rule’ is also applied to identities obtained by different ways of evaluating the time derivative, as a result of the stationary condition. Thus

[MATHEMATICAL EXPRESSION OMITTED] (38)

[MATHEMATICAL EXPRESSION OMITTED] (39)

etc. The equalities here, when evaluated explicitly, give the sum rules associated with the s‘th moment. For example, the second sum rule for the density-density correlation function gives the identity relating the pair distribution function to the derivative of the potential and the triplet distribution function.

Mathematically, the sum rules in a classical system are derived by integration by parts of the phase integrals in equation (4), using

[MATHEMATICAL EXPRESSION OMITTED] (40)

where q is a component of position or momentum. In this way any moment relation can be reduced to a form in which all the momenta are integrated out (since these have a Guassian distribution) and all first derivatives of the potential are removed. For a system with pair interaction only this gives the form requiring knowledge of the least number of multi-particle distribution functions.

(Continues…)Excerpted from Statistical Mechanics Volume 2 by K. Singer. Copyright © 1975 The Chemical Society. Excerpted by permission of The Royal Society of Chemistry.
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