Title: Fluctuations in systems far from equilibrium
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Title: Fluctuations in systems far from equilibrium
Physical Description: vii, 207 leaves : ill. ; 28 cm.
Language: English
Creator: Marchetti, Maria Cristina, 1955-
Publication Date: 1982
Copyright Date: 1982
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Subject: Irreversible processes   ( lcsh )
Statistical mechanics   ( lcsh )
Hydrodynamics   ( lcsh )
Shear flow   ( lcsh )
Physics thesis Ph. D
Dissertations, Academic -- Physics -- UF
Genre: bibliography   ( marcgt )
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Statement of Responsibility: by Maria Cristina Marchetti.
Thesis: Thesis (Ph. D.)--University of Florida, 1982.
Bibliography: Bibliography: leaves 203-206.
Additional Physical Form: Also available on World Wide Web
General Note: Typescript.
General Note: Vita.
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Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
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Resource Identifier: alephbibnum - 000334705
oclc - 09483361
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FLUCTUATIONS IN SYSTEMS FAR
FROM EQUILIBRIUM


BY

MARIA CRISTINA MARCHETTI


A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


1982


















Digitized by the Internet Archive
in 2009 with funding from
University of Florida, George A. Smathers Libraries



A Teresa e Piero M.


http://www.archive.org/details/fluctuationsinsy00marc









ACKNOWLEDGEMENTS

I would like to thank Professor James W. Dufty for his constant

guidance and support in the course of this work. He suggested this

problem to me. It was exciting and challenging to work with someone who

has such a contagious enthusiasm for physics. I have learned much from

him and he has also been a good friend.

My gratitude also goes to all those who have helped me, both as

colleagues and as friends. In random order they are: Greg, Mark, Bonnie

and Pradeep, Robert, Vijay, John, Annalisa M., Annalisa P., Martin,

Carlos, Simon, Bob, Teresa, Jackie, Attilio R., and many others.

Without the love and support of my parents I may not have had the

courage to venture to the United States. A special thanks goes to them

for having always been near me during these three years.

I would like to acknowledge the support of a Fulbright-Hays Travel

Grant, which allowed me to come to the University of Florida, and an

Educational Award from the Rotary International for the year 1981-82.

Finally, I thank Viva Benton for her quick and accurate typing of

the manuscript.










TABLE OF CONTENTS

Page

ACKNOWLEDGEMENTS .....................................................iii

ABSTRACT ................................................................vi

CHAPTER

I INTRODUCTION .....................................................1

II NONEQUILIBRIUM STATISTICAL MECHANICS
OF CLASSICAL SYSTEMS ............................................17

1. The Nonequilibrium Distribution Function:
Average Values and Fluctuations ..............................17
2. Reduced Distribution Function Formalism ......................24
3. Generating Functional for Nonequilibrium
Averages and Fluctuations ....................................28

III HYDRODYNAMICS ...................................................34

1. Generating Functional for Hydrodynamics
and Nonlinear Navier-Stokes Equations ........................34
2. Correlation of Fluctuations ..................................46
3. Equal Time Fluctuations ......................................53
4. Discussion ...................................................57

IV KINETIC THEORY ..................................................60

1. Generating Functional for Phase
Space Fluctuations ...........................................60
2. Low Density Limit and Boltzmann Equation .....................66
3. Correlation of Fluctuations ..................................70
4. Equal Time Fluctuations ......................................75
5. Hydrodynamic Limit ................. .........................81

V TAGGED-PARTICLE FLUCTUATIONS IN SHEAR FLOW .......................91

1. Definition of the Problem ....................................91
2. Transformation to the Rest Frame .............................97
3. Two-time Velocity Autocorrelation Function ..................100
4. Equal-time Velocity Fluctuations ............................103
5. Results and Discussion ..................................... .105

VI DISCUSSION......................................................113





APPENDICES

A DERIVATION OF THE X-DEPENDENT NONLINEAR NAVIER-STOKES
EQUATIONS............................................. .. ....... 118

B EVALUATION OF THE SOURCE TERM FOR
HYDRODYNAMIC FLUCTUATIONS......................................137

C DERIVATION OF THE X-DEPENDENT BOLTZMANN EQUATION ................145

D SCALING METHOD FOR A LOW DENSITY CLOSURE
OF THE HARD SPHERES BBGKY HIERARCHY.............................155

E HYDRODYNAMIC "NOISE" FROM KINETIC THEORY ........................187

F DETERMINATION OF R.. FOR MAXWELL MOLECULES ......................191
ij
REFERENCES..................................... .... ........ ... .......203

BIOGRAPHICAL SKETCH............................... ..................207










Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy


FLUCTUATIONS IN SYSTEMS FAR
FROM EQUILIBRIUM

by

Maria Cristina Marchetti

August, 1982

Chairman: James W. Dufty
Major Department: Physics

A unified formulation of transport and fluctuations in a

nonequilibrium fluid is described at both the kinetic and the hydro-

dynamic levels. The method is based on the analysis of a generating

functional for the fluctuations of the set of observables of interest

around their average values. It is shown that the first functional

derivatives of such a generating functional obey an inhomogeneous form

of the macroscopic regression laws (nonlinear Boltzmann equation or

nonlinear Navier-Stokes equations). From this equation the dynamics of

multi-space and -time fluctuations may be obtained by suitable

functional differentiation. In particular the equations for the second

order correlation functions of fluctuations at different and equal times

are obtained to illustrate the method. The dynamics of a nonequilibrium

fluid is governed by three sets of coupled equations: (1) nonlinear

equations for the macroscopic observables, (2) linear homogeneous

equations for the two-time correlation functions, and (3) linear






inhomogeneous equations for the equal time correlation functions. The

description obtained constitutes a precise statement of the

generalization to nonequilibrium states of Onsager's assumption on the

regression of fluctuations. The results apply to general nonstationary

nonequilibrium states, either stable or unstable. The description of

nonequilibrium fluctuations obtained rests on the same theoretical basis

as the macroscopic description of the system (Boltzmann kinetic theory

for a low density gas and hydrodynamics for a general fluid).









CHAPTER I
INTRODUCTION

A large class of macroscopic irreversible processes in many-body

systems is adequately described in terms of relatively few (compared to

the number of degrees of freedom) macroscopic variables, changing contin-

uously in time.1 This description applies when the time scale of

interest is large compared to a microscopic time scale characteristic of

the problem and the system considered. The macroscopic variables can be

identified with the quantities measured in an experiment, and their

values at a time t completely characterize the state of the system at

that time. Examples of such a set of observables are the hydrodynamic

densities in a fluid and the concentrations describing a chemical

reaction. On this macroscopic scale static and dynamic properties of

the system are expected to be governed by a closed set of macroscopic

laws. In particular the time dependent properties should obey a set of

differential equations of first order in time. It is then a

deterministic description in the sense that the equations assign fixed

values to all the macroscopic variables, once the values at t=0 have

been given. The completeness of a set of macroscopic observables for

the description of a particular problem on a particular time scale is

difficult to determine a priori and is established only by the

derivation of the macroscopic equations themselves.



*The complete set of observables often coincides with the set of
conserved variables in the system. However nonlinear coupling between
the hydrodynamic modes in a fluid can lead to the appearance of memory






2

The role of statistical mechanics consists in providing a micro-

scopic basis for the macroscopic description, defining its limit of

applicability, and establishing a precise connection between the

dynamics of the observables and the time evolution of the positions and

moment of all the particles in the system.

A derivation of the macroscopic laws from the microscopic dynamics,

governed by the Liouville equation, can be accomplished by introducing

the idea of several well separated time scales in nonequilibrium fluids.3

A system in an arbitrary nonequilibrium state is considered. On a very

short time scale, of the order of or smaller than the duration of a

collision, the dynamics of a system of N particles is very complex and a

large number of variables (the 6N coordinates and moment of the part-

icles in the system or, alternatively, all the reduced distribution

functions) is necessary to describe it. However, after a microscopic

time, of the order of the duration of a collision, or mean collision

time, Tc, the system relaxes to a kinetic regime, where the dynamics

may be described more simply in terms of the average density in the

single particle phase space.4,5 Therefore, when the time scale of

interest is large compared to T a contraction of the formal descrip-

tion of the dynamics of the N-body problem to a description in terms of

the one-particle distribution function is obtained.




effects in the macroscopic equations of motion: this indicates that the
chosen set of variables is not complete and needs to be enlarged (for a
discussion and references on this point see Ref. 2).

*For intermolecular forces with a finite and nonvanishing range, t is
defined as the ratio of the force range, a, and the mean thermal speed
of the molecules, v ,-i.e. T =o/v .
o c o





3

At larger time scales (for gases larger than the time between

collisions, or mean free time, tf, ) a further contraction of the

description takes place. The system relaxes to a state near local

equilibrium and its dynamics is described in terms of an even smaller

number of variables, identified with the local conserved densities,

e.g., mass density, energy density, flow velocity.6 This is the

hydrodynamic regime, which describes variations on time scales of the

order, for example, of the time required by a sound wave in a fluid to

cross a region of macroscopic interest .

The macroscopic regression laws governing the relaxation of a

system displaced from equilibrium can be derived in principle by averag-

ing the microscopic equations of motion and introducing approximations

suitable for the time scale of interest. For example, in the limit of

times large compared to a collision time (i.e., the kinetic regime), the

nonlinear Boltzmann equation for the one-particle distribution function

is obtained by a low density closure of the BBGKY hierarchy.5,8 The

Chapman-Enskog expansion provides then a systematic method for deriving

the hydrodynamic equations for the average densities in the gas, on the

longer time scale.8'9 Alternatively, the hydrodynamic equations can be

obtained directly by averaging the microscopic conservation laws,

6,10
bypassing consideration of the kinetic stage .

The contraction of the description of the dynamics of a many-

particle system in terms of few average variables naturally introduces

the concept of fluctuations. From a microscopic point of view in fact





*Defined as t = Z/v where Z is the mean free path of the molecules.
For liquids, t can Be of the order of T and the separation of time
scales is no longer useful.







4

the dynamical variables fluctuate instantaneously around their average

values. The physical origin of the fluctuations can be different for

different systems. In general, however, the amplitude of their space

and time correlations controls the adequacy of the macroscopic

description and can be measured in a variety of experiments. Thus the

statistical mechanics of irreversible processes in a many-particle

system should also provide a framework to describe fluctuations in an

arbitrary nonequilibrium state.



The description of fluctuations in equilibrium systems is well

established from both phenomenological considerations,1 and first

principles calculations for some limiting cases (e.g., low density

gases).12 A precise connection between the spontaneous fluctuations in

the equilibrium state and the macroscopic response of the system to a

small external perturbation is established by the fluctuation-

dissipation theorem.13,14 This result is concisely expressed in

Onsager's assumption on the regression of fluctuations,15 which states

that spontaneous fluctuations around the equilibrium value decay in time

according to the same linear laws governing the relaxation of a

macroscopic state close to equilibrium. In other words the linearized

regression laws apply regardless of whether the nonequilibrium condition

is prepared or occurs spontaneously. The adequacy of this assumption is

a consequence of the fact that the size of the equilibrium fluctuations

is limited by their thermal origin. Furthermore the space correlation

of the equal time fluctuations can be calculated directly from the

equilibrium ensemble.





5

The relationship of correlation functions and response functions to

a great number of measurable properties of macroscopic systems in equi-

librium is also well understood and a wide variety of experiments is

available for comparison.16 The interpretation of Rayleigh-Brillouin

light scattering experiments in fluids17 constitutes a well known

example of the success of the equilibrium fluctuations theory. The

detailed evaluation of time dependent correlation functions remains in

many cases a difficult many-body problem, but the theoretical formu-

lation can be stated clearly and is founded on the basic principles of

statistical mechanics.



The corresponding theoretical and experimental study of fluctua-

tions in a nonequilibrium system is much less complete and several new

problems arise, even from a purely theoretical viewpoint, in addition to

the difficult computational aspects.

The systems considered are in general open systems, in contact with

several reservoirs, which are not in equilibrium among themselves.3'6

These reservoirs impose external forces and fluxes on the system,

preventing it from reaching thermodynamic equilibrium and keeping it in

a nonequilibrium state, which is stationary if the external forces are

time independent. In contrast to the equilibrium system, which is

specified by a unique thermodynamic state, a large class of nonequili-

brium states must now be considered. The first problem encountered when

dealing with systems out of equilibrium is therefore the adequate

characterization of the particular nonequilibrium phenomenon of

interest.

Furthermore, the formulation of the macroscopic description of the

system and the derivation of the nonlinear regression laws for a






6

complete set of observables, such as the nonlinear hydrodynamic equa-

tions, presents several difficulties. At the hydrodynamic level the

evaluation of the form of the macroscopic fluxes as functionals of the

thermodynamic forces is required. These constitutive relationships are

highly nonlinear and only a formal expression can be provided in the

general case.10,19 An explicit evaluation has only been performed in

specific model cases (shear flow0',19-24) or under precise limiting

conditions (Burnett coefficientsl820) On the other hand, in near

equilibrium situations the formalism of linear response theory provides

expressions for susceptibilities and transport coefficients in terms of

the correlation functions of equilibrium fluctuations (Green-Kubo

relations).3'6 Nonlinear transport phenomena also can be related to

nonequilibrium fluctuations by introducing the concept of nonlinear

response functions. The understanding of the dynamics of fluctuations

in the nonequilibrium state is therefore intimately related to the

description of energy, momentum, mass and charge transport in systems

far from equilibrium.

Other characteristic nonequilibrium phenomena where fluctuations

play an important role are the appearance of instabilities, bifurcations

and turbulences, and the related onset of nonequilibrium phase tran-

sitions.25 A system macroscopically displaced from equilibrium and

brought outside the region of applicability of the linear response

theory (into a region where the macroscopic regression laws are non-

linear) evolves through a succession (or branch) of nonequilibrium states

as the magnitude of the parameter measuring the strength of the external

perturbation increases. At some critical value of this parameter the

state of the system may become unstable. The system will then make a





7

transition into a new branch of states. The occurrence of such

instabilities may lead to a change of symmetry of the stable state of

the system. In this sense the phenomenon is analogous to a phase

transition in a system in thermodynamic equilibrium. Therefore, as the

mechanism of equilibrium phase transition can be understood by recog-

nizing the importance of the fluctuations near the transition point, the

occurrence of bifurcations (and other nonequilibrium phase transitions)

is intimately related to the fluctuations in the system at the

instabilities.

A large variety of physical systems shows this general behavior.

The most popular hydrodynamic example is the Bdnard instability, occur-

ring in a liquid layer heated from below in the gravitational field.

Other examples are found in nonlinear optics, autocatalytic reactions

and biological systems. The formulation of a theory of nonequilibrium

fluctuations which does not impose restrictions on the size of the

fluctuations and applies near instabilities is an essential starting

point for the understanding of all these nonequilibrium phenomena.

However, even for the simplest case of hydrodynamic fluctuations,

it is not obvious how Onsager's assumption should be generalized for

such nonequilibrium states. Because of the nonlinear character of the

average regression laws, the dynamics of the macroscopic observables is

no longer the same as that of their fluctuations. Furthermore a non-

trivial problem is the determination of the source of the fluctuations,

which may no longer be simply thermal in origin and whose knowledge is

required for the specification of the equal time correlation functions.



There have been several recent studies of fluctuations in nonequi-

librium stationary states that are similar in spirit and objectives to






8

the present discussion. It therefore seems instructive to put the

present work in context by presenting a brief overview of these

approaches. They can be classified in general into two broad categor-

ies: stochastic approaches,26-33 where the microscopic fluctuating

variables are assumed to define a random process, characterized by a set

of probability densities, and microscopic approaches, 46 which attempt

to derive from first principles the dynamics of nonequilibrium

fluctuations.

The stochastic formulations are essentially based on a generaliza-

tion of the Onsager-Machlup regression hypothesis to the nonequilibrium

case. The time-dependent properties of the system are governed by a set

of equations for the probability densities (Fokker Planck equations) or

for the stochastic variables themselves (Langevin equations).

A stochastic theory of hydrodynamic fluctuations has been developed

by Tremblay, Arai and Siggia26 by assuming that the fluctuating

hydrodynamic equations proposed by Landau and Lifschitz4 '8 can be

applied to describe not only fluctuations in the equilibrium state, but

also small fluctuations around a nonequilibrium stationary state. The

time evolution of the fluctuating variables is then governed by a linear

Langevin equation whose coefficients depend on the nonequilibrium state

of the system. The correlation of the random forces is chosen to have

the same form as in equilibrium, except with local thermodynamic

variables for the nonequilibrium state. The characteristic feature of

the Langevin theory is the separation of the time variation into a

slowly varying and a rapidly varying part. The basic assumption is that

the correlations of the fast variables are determined by a local

equilibrium ensemble. This Langevin theory has been successfully






9

applied by several authors2649-51 to the evaluation of the scattering

of light from a fluid with a temperature gradient and/or in shear

flow. However the limitation on the size of the fluctuations clearly

restricts its applicability to regions far from instabilities and

critical points. Furthermore in Ref. 26 the example of a case where the

theory fails is given: electrons scattering off impurities to which a

potential difference is applied. Presumably this is a case when the

local equilibrium assumption for the fast variables does not apply.

A formal theory of nonequilibrium fluctuations and nonlinear

irreversible processes has been developed by Grabert, Graham and

Green.27,28 Their approach generalizes to the nonlinear regime the

functional integral expression for the transition probability between

two macroscopic states introduced by Onsager and Machlup'l15 as an

extension to the time dependent domain of Boltzmann's relationship

between entropy and probability. The basic assumption of the theory is

that the probability of a given fluctuation from one state to another is

measured by the minimum increase in action associated with the change

among the two states considered. This minimum principle provides the

possibility of constructing a path integral expression for the

conditional probability which appears as a natural generalization of the

Onsager-Machlup functional. However this minimum principle is purely an

assumption and has not been justified on any microscopic basis. Only a

posteriori can it be seen that the results obtained agree with those

derived through different formulations. In particular the conditional




*It should be pointed out that the results of Ref. 50 for the Brillouin
scattering in temperature gradient are not in agreement with the results
of the other calculations 6.





10

probability can be shown to satisfy a Fokker-Planck equation. This

equation has also been applied by Grabert28 to evaluate the spectrum for

light scattering from a fluid in a temperature gradient. His results

agree with those derived by other methods.

Within the framework of this generalized Onsager-Machlup theory,

Graham25 has also analyzed in detail problems associated with the

stability and the breaking of symmetry in nonequilibrium stationary

states.

Several authors29-32 have formulated studies of nonequilibrium

fluctuations based on a master equation in an appropriate stochastic

space. With the aim of generalizing Landau-Lifshitz's fluctuating

hydrodynamics to the nonlinear region, Keizer30 has rewritten the

microscopic conservation laws in a fluid in the form of master equations

by describing the fluid in terms of elementary molecular processes. His

results support the Langevin theory proposed in Ref. 26. Coarse grained

master equations for inhomogeneous systems have been assumed as the

starting point to describe phase space fluctuations by Onuki31 and

Ueyama.32 For systems near equilibrium the linearized Boltzmann-

53
Langevin equation5 is recovered. Furthermore by using the Chapman-

Enskog expansion method fluctuating hydrodynamics can be derived.

Finally, van Kampen29 has applied the method of expansion in the size of

the system to the solution of master equations for several problems,

from the Boltzmann equation to the rate equations governing chemical

reactions. The principal limitations of the master equation approaches

are the necessity of assuming (instead of deriving from first



*This is in effect a generalization of an analogous master equation
proposed by Logan and Kac52 for homogeneous systems.






11

principles) the basic equation, and the strong dependence of this

equation on the model considered.

Finally, a generalization to the nonlinear regime of the linearized

Boltzmann-Langevin equation53 has been proposed33 to describe

nonequilibrium phase space fluctuations in low density gases. This

method is similar in spirit to the fluctuating hydrodynamics of Ref. 26,

since the correlation of the random forces is again assumed to have a

local equilibrium form.

The microscopic studies are based on several different techniques.

First of all the methods of the kinetic theory of gases have been

used34-38 to describe phase space fluctuations. To make any progress

with the formal theory the limitation of low density has in general to

be imposed. This restriction, however, allows the introduction of

controlled approximations. Furthermore, the kinetic theory is more

general than the hydrodynamic description since it describes fluctua-

tions on shorter space and time scales and it incorporates the latter.

In particular Ernst and Cohen37 have applied the methods of the nonequi-

librium cluster expansion to derive kinetic equations for the equal and

unequal time two-point correlations of phase space fluctuations for a

gas of hard spheres. The meaning and adequacy of their expansion will

be discussed in Chapter IV and Appendix D, where an alternative method,

leading to equivalent results, is presented.

An alternative microscopic theory of fluctuations in nonequilibrium

steady states has been developed by an M.I.T. group.39-42 For states

not too far from equilibrium (up to second order in the parameters

measuring the deviations from equilibrium) they apply the method of

nonlinear response theory to express nonequilibrium averages and






12

correlation functions in terms of higher order equilibrium correlation

functions.40 These correlation functions are then evaluated (in the

hydrodynamic case) by assuming a separation of time scales in the

system, implying that the correlations involving the dissipative fluxes

decay on a time scale shorter than the macroscopic time scale of inter-

est. Their results are also generalized to nonequilibrium stationary

states arbitrarily far from equilibrium by employing the projection

operator techniques .41 They conclude that in general a simple

generalization of the equilibrium fluctuation-dissipation theorem to a

local equilibrium form does not hold. The formalism used however

obscures many of the physical hypotheses entering the derivation.

Furthermore the separation in fast and slow variables is not well

founded for states arbitrarily far from equilibrium.

Finally Kirkpatrick, Cohen and Dorfman43-46 developed a

hydrodynamic theory of nonequilibrium fluctuations in stationary states

based on the use of projection operator techniques and on the Kadanoff-

Swift mode-coupling theory for the evaluation of the equal time

correlation functions. They also have applied their description of

hydrodynamic fluctuations to the problem of light scattering from a

fluid subject to a temperature gradient, for both the cases of small and

large gradient.

The formulations briefly described here suffer from several limita-

tions. Their applicability is often restricted to nonequilibrium

stationary states and to small fluctuations around the states. The

latter condition implies that the system has to be far from insta-

bilities and critical points. A precise connection between the

stochastic and the microscopic approaches, as is possible for the case






13

of equilibrium fluctuations, has not been established. In the

stochastic methods it is often not clear how the macroscopic information

on the nonequilibrium state, as specified in an experiment, enters the

problem. In general there is a need for developing an exact treatment

of nonequilibrium systems whose validity extends to both the kinetic and

hydrodynamic regimes and that establishes a precise connection between

the theoretical and the experimental description.



The objective of the present work is to provide a unified

formulation of the description of nonequilibrium fluctuations founded on

a microscopic basis and constituting a precise statement of the general-

ization of Onsager's assumption on the regression of fluctuations to

arbitrary nonequilibrium states. The problems mentioned above are

addressed explicitly. An open system in contact with external

reservoirs is considered. The reservoirs prepare or maintain the system

in a general nonequilibrium state. The results derived apply to

nonstationary states, either stable or unstable.

The method rests on the definition of a generating functional whose

first functional derivatives are directly related to the complete set of

macroscopic nonequilibrium observables relevant to the problem consider-

ed. Higher order functional differentiation generates the correlation

functions of the fluctuations of the chosen dynamical variables around

their average values. The first advantage of this approach is that it

makes it possible to discuss the problems of average behavior and

fluctuations in a unified context. Characterizing the nonequilibrium

state amounts to providing a statistical mechanical derivation of the

macroscopic regression laws. This can be accomplished both at the






14

kinetic and hydrodynamic level in such a way that the properties of the

generating functional are preserved.

A set of nonlinear equations for the first functional derivatives

is first obtained: these equations have the same form as the macroscopic

regression laws, the only difference being the appearance of additional

contributions arising from transient effects in the system whose life-

time is short compared to the time scale of interest. By functionally

differentiating these equations, equations for the correlation functions

are obtained.

After defining precisely in Chapter II the class of problems

considered and the general form of the associated generating functional,

the specific cases of hydrodynamic and kinetic regimes are analyzed in

Chapter III and Chapter IV, respectively.

The appropriate macroscopic variables for the hydrodynamic descrip-

tion are the five average conserved densities. For simplicity attention

is limited to a simple fluid whose nonequilibrium state is adequately

described by the nonlinear Navier-Stokes equations. The set of equa-

tions for the first functional derivatives can be derived by using the

nonlinear response theory, as shown in Appendix A. The equations are an

inhomogeneous form of the nonlinear Navier-Stokes equations. Additional

contributions appear as extra terms in the irreversible heat and

momentum fluxes. By functionally differentiating these generalized

Navier-Stokes equations, a set of linear equations for the correlation

functions is obtained. The additional contributions to the irreversible

fluxes survive only in the equations for the equal time correlation

functions, derived by a limiting procedure on the equations for the

multitime fluctuations.





15

A similar procedure is carried through at the kinetic level in

Chapter IV. Here the macroscopic variable of interest is the one-

particle distribution function, interpreted as the average of the

microscopic phase space density. A low density kinetic equation for the

first functional derivative of the associated generating functional is

derived in Appendix C by following closely the nonequilibrium cluster

expansion used to obtain the Boltzmann equation from the BBGKY hier-

archy. Again the equation obtained differs from the nonlinear Boltzmann

equation by the presence of extra short-lived terms that only contribute

to the equations for the equal time correlations. The equations for the

correlation functions are obtained by functional differentiation of this

generalized Boltzmann equation.

At both levels of description, kinetic and hydrodynamic, the same

structure is obtained: nonlinear equations for the averages and linear

equations for the correlation functions. The coefficients in the linear

equations depend on the solution of the nonlinear problem: the dynamics

of the fluctuations is then entirely governed by the nonequilibrium

state. The amplitude of the noise, or source of fluctuations, enters in

the form of an inhomogeneous term in the equations for the equal time

correlation functions and is derived without introducing any assumptions

other than the ones entering in the derivation of the macroscopic

equations.

The formulation presented here starts from the Liouville equation

for the system, but all the equations considered for the macroscopic

description involve only averaged quantities, either observables or

correlation functions. The information on the nonequilibrium state

enters through boundary and initial conditions for such macroscopic

variables: these are the parameters controlled in an experiment.






16

The linear character of the equations for the correlation functions

appears as a general result in the present discussion, not as the

outcome of a linearization around the nonequilibrium state, applicable

only when the fluctuations are small in size. In this sense the results

obtained here agree with the conclusions of the M.I.T. group: the

dynamics of the fluctuations is, as in equilibrium, determined by the

state of the system in a form that naturally generalizes Onsager's

regression hypothesis.

As an application of the general description of nonequilibrium

fluctuations obtained by the generating functional method, the velocity-

velocity autocorrelation function of a tagged particle in a fluid in

shear flow is evaluated in Chapter V, for arbitrarily large shear

rate. Substantial simplifications occur in the problem as the result of

three conditions: (1) low density, (2) uniform shear rate and (3)

Maxwell molecules. In fact it is well known that the irreversible part

of the stress tensor may be evaluated exactly from the nonlinear

Boltzmann equation for a gas of Maxwell molecules in uniform shear

flow.54 Consequently, the hydrodynamic problem can be solved exactly to

all orders in the shear rate.19

Here the velocity-velocity correlation function is evaluated by

solving the appropriate set of coupled kinetic equations. The velocity

correlations do not decay to zero at long times, since the asymptotic

velocity of the particle is equal to the velocity of the fluid at the

position of the particle. Nonvanishing position velocity correlations

are then present in the fluid at large t. Also, due to the presence of

the shear in the fluid, equal time correlation functions which are zero

in equilibrium are now nonvanishing.









CHAPTER II
NONEQUILIBRIUM STATISTICAL MECHANICS OF CLASSICAL SYSTEMS

II.1 The Nonequilibrium Distribution Function:

Average Values and Fluctuations

To prepare or maintain a system in a nonequilibrium state external

forces have to be introduced in general; nonequilibrium statistical

mechanics involves the description of a system in interaction with its

surroundings in addition to specified initial conditions.3'6 The external

forces are assumed to have the character of reservoirs, in the sense

that they can be specified independently of the state of the system. In

the case of a fluid, for example, the most general situation is repre-

sented by interaction with particle, energy and momentum reservoirs.

In classical mechanics a system of N interacting particles enclosed

in a volume V is described in terms of the canonical coordinates,

ql...qN, and the canonical moment, pl, .. N, of the particles. The

state of the system is represented by a point in the 6N-dimensional

space of the canonical coordinates.

In the language of statistical mechanics the state of the system at

the time t is described by the N-particle distribution function,




PN(x ...,xN,t) = PN(xl(-t) ...,x (-t)) (2.1)



where x. = (qi,pi) Here xi(t) are the canonical coordinates of the

i-th particle as evolved from their initial value x.(0) x., according
to the dynamics of the system interacting with its surroundings. The
to the dynamics of the system interacting with its surroundings. The







18

distribution function pN represents the probability that the system lies

in a neighborhood dr = dxl...dxN of the phase space point

F = (xl,...,xN) at time t. The time evolution of the distribution

function is governed by the Liouville equation for a system in

interaction with its surroundings,3,6



S+ }PN PN 0 (2.2)
pn


where here and in the following summation over repeated indices is

understood. The operator C in Eq.(2.2) is



C = L(x ,...,xN) + P -I- (2.3)
bPn


where L is the Liouville operator for the isolated system, defined as



LpN = {PNH} (2.4)



Here (..*} indicates the classical Poisson bracket and H is the Hamil-

tonian of the system. P is an external force representing the effect
n
of the reservoirs on the n-th particle of the system: it depends in

principle on all the degrees of freedom of the system and reservoirs and

on the details of their interaction. The force n is in general
n
nonconservative and time dependent, and can be assumed to be applied

everywhere at the boundary of the system. This assumption does not

constitute a serious restriction since it corresponds to most


experimental situations of interest.







19

Equation (2.2) can be integrated formally when supplemented by an

initial condition at t=0,



pN( ...,xN,0) = PN,0(X1(0) ...,x (0)) (2.5)



In general pN, will be assumed to represent an arbitrary nonequilibrium

state. Also, the distribution function pN is normalized at all times,



I f dx ...d NPN(x, ...XN,t) = 1 (2.6)
N=0


As already mentioned in Chapter I, a macroscopic description of a

many-body system only involves a set of relatively few (compared to the

number of microscopic degrees of freedom) conveniently chosen variables.

The appropriate set of observables depends on the system considered and

on the macroscopic space and time scales of interest: the one-particle

distribution function is the object to consider in the kinetic theory of

gases and the five average conserved densities are the appropriate se:

of variables to describe hydrodynamics in a simple fluid. In any case

the choice is guided by the requirement that the macroscopic description

is closed on some time scale when given in terms of the variables

considered. A discussion of this point can be found for example in

Ref. 1.

Within the formalism of statistical mechanics the macroscopic

measurable quantities are defined as ensemble averages of corresponding

microscopic dynamical variables, {A (x1,... ,xN, s, functions of the

phase space coordinates,








A (t) = (2 .7)



where <...;t> indicates an average over the nonequilibrium distribution

function at time t,



= J dr A (x,..., N)pN(xl ,...,xNt) (2.8)



and f dr = f dxi ...dxN. Here and in the following a caret
N=0
indicates the microscopic variable as opposed to the averaged one.

An alternative expression of Eq.(2.8) is obtained by observing that

the time evolution of the dynamical variables {A } is governed by the

equation of motion,



} A(r,t) = 0, (2 .9)



where the operator has been defined in Eq.(2.3). Equation (2.9) can

be formally integrated with the result,



A (r,t) = T(t,0)A (F) (2.10)



The time evolution operator T(t,t') is the solution of the equation,



- } T(t,t') = 0 (2.11)



with initial condition T(t',t') = 1. It also has the property,


(2.12)


T(t,t')T(t',t") = T(t,t").






21

Equations (2.2) and (2.9) differ because in a nonconservative

system a region of phase space does not maintain a fixed volume as it

evolves in time. This can be seen explicitly by evaluating the Jacobian

of the transformation of an element of volume in phase space as it

evolves in time,



J = {x( )J (2.13)



For a conservative system, J=1. Here J satisfies the equation,


oF
+ C }J = n (2.14)
opn


and therefore its change in time accounts for the difference in the time

evolution of the distribution function and the dynamical variables.

The average values of the dynamical variables A can then be

written



= f dF A (xl,...,xN,t)pN(xl,...,xN,0) (2.15)



The possibility of shifting the time from the distribution function onto

A will be useful in the following to display some properties of the

time-correlation functions.

At this point the dynamics of the macroscopic set {A } is deter-

mined through the time evolution of the distribution function. In this

sense Eq.(2.8) has only the meaning of a formal definition.

On a microscopic scale A fluctuates instantaneously around its

average value. The microscopic fluctuations are defined as








6A (F;t) = A (F)
(2.16)



A complete description of the system should incorporate a systematic

analysis of the fluctuations and of their space- and time-correlation

functions of any order.

The second order equal time correlation function is defined as



M (t) E <6A A ;t>



= f dr aA()6A (r)pN(r,t) (2.17)



and, in general, the k-th order correlation function is



M (t) = <6A 6A ...6A ;t> (2.18)
a"'.Va a1 C2 ak


Again, as for the average quantities, the time evolution of the equal

time correlation functions is determined through the ensemble.

The amplitude of the equal time correlation functions is a measure

of the width of the probability distribution of the fluctuations of a

variable around its average value. The fluctuations can be viewed as

the result of a stochastic force which represents explicitly the effect

of instantaneous molecular collisions not accounted for in the macro-

scopic description and averages to zero over a time long compared to the

microscopic collision time. In the language of stochastic processes the

fluctuations are then the result of the "noise" in the system and the

equal time correlation functions measure the size of this noise. It

should be pointed out however that, even if here and in the following






23

some of the terminology of stochastic processes is used, the description

presented here is entirely macroscopic and always deals with averaged

quantities.

In equilibrium, away from critical points, the probability of fluc-

tuations is Gaussian and only second order correlations are important.55

Their amplitude is proportional to kBT, where T is the temperature and

kB is the Boltzmann constant, as suggested from the fact that in the

equilibrium state spontaneous fluctuations can only be thermal in

origin. In systems driven out of equilibrium new sources of noise may

become available, in a way strongly dependent on the particular state

considered. Furthermore in a nonequilibrium state equal time

fluctuations are dynamical variables: the clear separation of static and

dynamic properties that appears natural in equilibrium does not apply to

this case.

The correlation of fluctuations at different times are measured in

terms of multitime correlation functions. The most general definition

of a two-time correlation function is



M a(tl,t2;t) = <6A (tl)6A (t2);t> (2.19)



By using Eqs.(2.8) and (2.15) to translate the time arguments, this can

also be written as



<6A (t1 )6A (t2);> = <6A (tl-t)6A (t2-t);0> (2.20)



Therefore the two-time correlation function really depends on two time

arguments and on the initial time. In the following the dependence on







24

the initial time will not be indicated explicitly, unless needed. An

average over the initial nonequilibrium ensemble will simply be written



E d f dr A(r)pN(r,0) (2.21)
N=0


The k-time correlation function is then


A A A
M(t ...t) = <6A l(t )6A 2(t2)...6A k(tk)> (2.22)



The macroscopic variables and the correlation functions of their

fluctuations around the average value have been expressed here as

nonequilibrium averages of the corresponding microscopic quantities. In

the next chapters equations describing the dynamics of the observables

here defined will be derived in two specific macroscopic conditions.



II.2 Reduced Distribution Function Formalism
A
In the special case when the dynamical variables (A } are sum of

single particle functions, two-particle functions, etc.,


N N N
A (xl,...,xN) = a (x) + b (xi,x) + ... (2.23)
i=l i=1 j=1
i j

their averages and correlation functions are conveniently represented in

terms of reduced distribution functions. Many of the physical variables

of interest in real systems, such as the microscopic mass, energy and

momentum density in fluids, have the form (2.23).

A set of microscopic phase space densities is defined as







N
fl(x ,t) = 6(x1-xi(t))
i=l

N N
f2(x1,x2,t) = I 6(x1-xi(t))6(x2-xi(t)) (2.24)
i=1 j=1
i j

etc.



The reduced distribution functions usually defined in kinetic theory,56'57


s N!
n fs(xl...,xs t) = (N! f dx ..dxN N(x ...,xNt) (2.25
s (N-s)! dXs+l1'* PN(l N):,
N>s


can be interpreted as averages of the phase space densities over the

nonequilibrium ensemble,


s^
n f (x ,...,x ,t) < (x ,...,x ,t)> (2.26)



In particular the one-particle distribution function is defined as



n fl(xl't) =



S (N-)! f dx2 ...dxN PN(x 2,...,''xNt) (2 .2)
N>1


The reduced distribution function fs represents the probability density

that, at the time t, the positions and moment of particle 1 through s

lie in a neighborhood dx ...dxs of the point {x1,...,xs1, regardless cf

the positions and moment of the other particles in the system.

The average of a variable A of the form given in Eq.(2.23) is then

written as







26

(A ;t> = n f dx a (x )fl(xl,t)



+ n2 / dxdx2b (x1,x2)f2(x ,x2,t) + ... .(2.28)



For simplicity only the expressions for the correlation functions of

variables that are sums of single particle functions will be derived

here. The extension to more general cases is straightforward.

The correlation functions of the {A } can then be expressed in

terms of the correlation functions of the fluctuations of the one-

particle phase space density around its average value, Cs, with the

result,



M a(t) = f dx dx2a (x)a (x2)C2(xlt;x2,t) (2.29)



and

Ma(tl,t2) = dx1dx2 a (xl)a(x2) C2(xl,tl;x2,t2) (2.30)



In general the k-th order correlation function is given by


k
.M (t. ...,tk) = J...f i dxia .(xi)}Ck(Xl,tl;...;xk,tk) (2.31)
1 k i=1 1


Equivalent expressions for the correlation functions of more

general variables involving also two-particle functions, etc., will

involve correlations of higher order phase space densities.

The equal time correlation functions of phase space fluctuations

introduced in Eqs.(2.29-30) are defined as









27
A A
Cs(x1,t;...;Xs,t) = <6fl(x ,t) ...6fl(xs,t)> (2.32)



where

6fl(x,t) = fl(x,t) (2.33)



It is immediately seen that these correlation functions can be expressed

in terms of the reduced distribution functions f

Similarly, the multitime correlation functions introduced in

Eq.(2.31) are given by



Cs(x1,tl;...;xs,ts) = <6fl(xl,tl)...6fl(xs,ts)> (2.34)



The one-particle distribution function and the fluctuations in the phase

space density are the objects of interest in kinetic theory and will be

analyzed in detail in Chapter IV.

The expressions given in Eqs.(2.29-31) for the averages and the

correlation functions of a general set of observables in terms of the

reduced distribution functions are obtained by carrying out a partial

ensemble average over a reduced number of particles. They provide a

connection between macroscopic observables and kinetic theory and a

formal reduction of the many-body problem to an effective one-, two-,

... s-body problem. The reduced distribution function method is

particularly useful when dealing with gases at low or moderate density.

In this limit closed kinetic equations for the lower order distribution

functions can be derived.







28

II .3 Generating Functional for Nonequilibrium

Averages and Fluctuations

To describe the dynamics of a nonequilibrium system it is

convenient to introduce a generating functional G, from which averages

and correlation functions of any order can be generated through

appropriate functional differentiation.

The same idea has already been used by Dufty58 to derive kinetic

equations for equilibrium multitime correlation functions. The advan-

tage of this formalism is mainly that once a single set of equations for

the first functional derivatives of G has been derived, the equations

for the correlation functions are simply obtained by functional

differentiation.

If (A } is the set of dynamical variables chosen to describe the

system, the corresponding generating functional is defined as



G[{k}1] = ln (2.35)



where

U[{x }] = exp Jf dt A (r,t)X (t) (2.36)



and summation over the index a is understood. The test functions {kx}

in Eq.(2.36) have no physical interpretation. The only requirement

imposed at this point is that they are sufficiently localized in time

for the integrals in Eq.(2.36) to exist.

The first functional derivative of the generating functional is

given by








A (tX) <= a (2.37)
a


When evaluated at X =0, Eq.(2.37) reduces to the nonequilibrium average
a
of A,'



a (t|X=0) = A (t) (2.38)



Similarly, by successive functional differentiation,


(2)
M (tlt2 (2) G[XJ (2.39)
ap '2 T ta 1 t I a =01


and in general,


(k)
(t V=. t 6 G[X1 (2.40)
Mal ...ak(t ,...,tk) 6xaal ) 6xk (tk a=02.40)


Here and in the following the times are chosen to be ordered according

to t > t > *** ts. This implies no loss in generality since the

dynamical variables commute in classical mechanics. As desired,

averages and correlation functions of any order can be generated from

G. The objective is now to derive a set of equations for the time

evolution of the functionals IAa(tIk)}. When evaluated at

X=0, these should reduce to the macroscopic regression laws for the

system considered. Furthermore equations for the correlation functions

of any order can be obtained by functional differentiation. In this

sense the generating functional method provides a unified description of

the system, since all information can be derived from one single set of

equations.






30

The set of macroscopic regression laws describing, at long times,

the nonequilibrium state of a many-particle system (such as the Boltz-

mann equation and the nonlinear Navier-Stokes equations which apply at

the kinetic and hydrodynamic level respectively) can be derived by

averaging the microscopic equations of motion, at least in the case of

low density gases. The nonlinear Boltzmann equation for the one-

particle distribution function is obtained by performing a low density

closure of the BBGKY hierarchy. The Chapman-Enskog expansion provides

then a systematic method for deriving the hydrodynamic equations.

Alternatively, the nonlinear hydrodynamic equations can be obtained

directly by averaging the microscopic conservation laws over a

nonequilibrium ensemble which is a formal solution of the Liouville

equation and by evaluating the constitutive equations expressing the

irreversible part of the fluxes as functionals of the thermodynamic

gradients.

The same well established methods can be applied to the generating

functional to derive a set of equations for the X-dependent functional

{A (tXh)}. Furthermore the derivation preserves the properties of the

generating functional, in the sense that the set of equations so obtain-

ed may be differentiated to derive equations for the correlation

functions. This program is carried out explicitly in Chapter III and

Chapter IV for the hydrodynamic and kinetic limit respectively. The

theory of nonequilibrium fluctuations so derived is justified in the

same well understood limits leading to the macroscopic description.

The convenience of introducing a generating functional or charac-

teristic function to describe fluctuations has often been recognized in

the literature.59 In particular Martin, Rose and Siggia60 defined a







31

time ordered generating functional, involving both the dynamical

variables of the system and a set of conjugated operators describing the

effect of small perturbation in the variables. Also, in Eq.(52) of Ref.

31, Onuki defines a characteristic function analogous to the one

proposed here, from which equal time correlation functions of phase

space fluctuations can be generated. The definition given in Eqs.

(2.35-36) is however more general and it can be applied to a larger

class of statistical mechanical systems.

The next two chapters will be dedicated to the explicit derivation

of equations for the averages and the correlation functions describing

the properties of a fluid at the hydrodynamic and kinetic level.

However, before carrying out this program in detail, it is instructive

to present the results. The macroscopic description of the dynamic

properties of a many-particle system have the same structure at both

levels of description (kinetic and hydrodynamic).

The relaxation of the macroscopic variables {A } is described by

nonlinear equations of first order in time (to be identified with the

nonlinear Navier-Stokes equations for the average densities and the non-

linear Boltzmann equation for the one-particle distribution function),



A(t) + N [{A(t)}] = 0 (2.41)



where N represents a general nonlinear functional of the (A The set

of Eqs. (2.41) has to be solved with the appropriate initial and

boundary conditions, constituting an experimentally appropriate

definition of the macroscopic nonequilibrium state considered.






32

The dynamics of the correlations of fluctuations around the non-

equilibrium state is described by a set of linear homogeneous equations,



M (tt) A L[ l(tI) ,t = 0 (2.42)
t a 1' 2 ay yp 12


where
6Na[(A(t1)}]
-L (tl) 6A. (t- (2.43)
ap 1 6A P(t


Equation (2.42) applies-as long as the separation tl-t2 is large

compared to an appropriate microscopic time, tm,-of the order of the

lifetime of the transient before the adopted macroscopic description of

the system applies. Specifically, t = z the duration of an
m c
interparticle collision, for a kinetic description, and t = tf, the
m
mean time between collisions, for a hydrodynamic description. The

linear functional Lp in Eq. (2.42) depends on the solution of the

nonlinear equations, Eqs.(2.41). In this sense the dynamic evolution of

the two-time fluctuations is entirely determined by the macroscopic

nonequilibrium state.

The correlations of fluctuations at the same time, needed as

initial conditions for the set of Eqs.(2.42), are the solutions of a set

of bilinear inhomogeneous equations,



6 '(t) + L r[A(t)}]M (t) + L [{A(t)]4 (t) = FA(t)] .

(2.44)



The specific form of the inhomogeneity F depends on the problem con-

sidered. To interpret these equations, it is instructive to compare















33

them with the corresponding equilibrium results. In this case Onsager-

Machlup's hypothesis applies and the decay of the mean values and of the

fluctuations is governed by the same set of linear equations, obtained

from Eqs.(2.42) by evaluating Lap at equilibrium,


A(tl) Ao(t )

ap 1 2 yp 1 2

where the superscript "zero" indicates the equilibrium value or that the

average has to be understood as an equilibrium average. Also the fact

that in equilibrium, in virtue of time translation invariance, the two-

time correlation function depends only on the time difference has been

indicated. The equal time fluctuations are given by equilibrium sta-

tistical mechanics as a property of the ensemble. Equations (2.44)

become then identities: the inhomogeneous term is identified with the

amplitude of the noise in the system (this appears evident by using a

Langevin description). The inhomogeneity in Eqs.(2.44) can then be

interpreted as the noise term. Indeed it originates from microscopic

excitations in the system whose lifetime is short compared to the time

scale of interest.









CHAPTER III
HYDRODYNAMICS

III.1 Generating Functional for Hydrodynamics and

Nonlinear Navier-Stokes Equations

In this chapter the generating functional formalism outlined in

Section 11.3 is applied to derive a set of equations describing macro-

scopic properties and fluctuations at the hydrodynamic level. For

simplicity, attention is limited to the case of a simple fluid whose

nonequilibrium state is adequately described by the nonlinear Navier-

Stokes equations. The method used here is easily extended to more

general systems and other nonequilibrium situations, as will be seen in

the next chapter.

In general a hydrodynamic process is one which is adequately de-

scribed in terms of the averages of the locally conserved quantities in

a many-particle system.7 This contracted description applies to time

scales large compared to a characteristic microscopic time scale, the

mean free time between collisions. The hydrodynamic equations are

derived by averaging the microscopic conservation laws and by closing

the macroscopic equations so obtained with constitutive relations for

the irreversible part of the fluxes. This procedure can be carried out

in detail by applying, for example, the methods of the response theory

and provides a precise statement of the validity of the hydrodynamic

description for processes varying on space and time scales large com-

pared to some microscopic space and time scales characteristic of the

system.6,10








35

The microscopic conserved densities for a simple fluid are the
A A
microscopic mass density p, the total energy density E, and the three
A
components of the momentum density gi, respectively given by


N
p(r) = m 6(r-q) ,
n=1 ,. .

N
g(r) = n 6(rqn
n=l

N -
g(P)= Pni6(r-qn) (3.1).
n=l


Here N is the total number of particles in a volume V and m is the mass

of the particles; r represents a point in the fluid. The function e is
n

the energy of the n-th particle: for a central pairwise additive

interaction potential, V(qnn,), with qnn' = n n n is given by,


2



n'nn

In the following, for convenience, the set of five conserved densities

will be indicated as




{ca(W)} = (p(r),^(r),gi(r)) (3.3)



or
N
-
,(r) = a (x)6(r-q) (3.)
n=1


with








^a (xn)} = (mnPn i) (3.5)



Also, here and in the following, the indices n,n',... are used to label

particles and the indices i,j,k,l,... are used to label the components

of vectors and tensors. Greek indices, a,p,... run from 1 to 5 and

label sets of hydrodynamic variables.

The microscopic conservation of mass, energy and momentum is

expressed by the set of five conservation laws,



+ a A ^ .
(rt) +- i(r,t) = 0 (3.6)



The set of Eqs.(3.6) constitutes the definition of the microscopic

fluxes Y i, explicitly given by



{Y ()} = (gi()'s,(),tij (r)) (3.7)


A A
where s. is the energy flux and t.. is the momentum flux. Explicit

expressions for the microscopic fluxes can be found for example in Ref.

7, Eqs.(4.6). It should be noted that in writing Eqs.(3.6) no external

sources have been taken into account, even if a general nonequilibrium

system in interaction with its surroundings is considered here. The

adequacy of Eqs.(3.6) as the starting point to describe properties in

the interior of the system and the possibility of incorporating the

effect of the boundaries entirely through the thermodynamic parameters

is discussed in Appendix A.

The macroscopic quantities of interest are the nonequilibrium
averages of the
averages of the I{d>},








S(r,t) = <(r~();t> (3.8



and the correlation functions of their fluctuations around the average,



Ma l. k(rl 'tl;';rk'tk) = (3.9


where

6da(r,t) = pa(r,t) <(p(r);t> (3.10


)


)


)


For a one-component- fluid the average densities are explicitly given by


(3.11)


a (r',t) + = (p(r,t),E(r,t),p(r,t)vi(rt))


where p, E and v are the average mass and total energy density and the

macroscopic flow velocity, respectively.

The macroscopic conservation laws are obtained by averaging

Eqs.(3.6) over the nonequilibrium ensemble,19 with the result,



S +r = 0 (3.12)
1


The hydrodynamic equations are obtained from Eqs.(3.12) when these are

supplemented by an equation of state, relating the thermodynamic var-

iables, and by constitutive equations for the macroscopic fluxes as

functionals of the average densities.

The intensive thermodynamic variables, {y(r,t)}, such as temper-

ature, pressure, etc., are defined in the nonequilibrium state by

requiring that the nonequilibrium average densities are the same





38

functionals of temperature, pressure and flow velocity as they are in

equilibrium.6 This is obtained by choosing



<@a(r);t> = <+ (r);t>L (3.13)



where <...;t>L indicates the average over a local equilibrium ensemble

at time t,



L = X f dr A(D)pL(t) (3.14)
N=O


The local equilibrium ensemble for a one-component fluid is given by



pL(t) = expt-qL(t) f dr y (r,t)a (r)} (3.15)


where qL(t) is a constant determined by the requirement that PL(t) is

normalized to one. The right hand side of Eq.(3.13) is an explicit

functional of the {y }: this equation constitutes the definition of the

local thermodynamic variables conjugated to the ( I}. Explicitly,

r 1 2

{y (r,t)} = (-v +1 Pv2, ', -,vi) (3.16)


-1
where vp- is the chemical potential per unit mass and P = 1/k T, where

T is the Kelvin temperature and kB is Boltzmann's constant. Equations

(3.13) do not imply a limitation to states near equilibrium or local

equilibrium. Other definitions are possible and sometimes desirable.

However, the present definition has the advantage that the functional

relationships of all nonequilibrium thermodynamic parameters are the

same as in equilibrium.






39

The main problem in closing the hydrodynamic equations is the

specification of the irreversible part of the fluxes,



S(r,t) < i(r);t>L, (3.17)



as functionals of the thermodynamic gradients. These expressions can be

evaluated to first order in the gradients by using the linear response

theory, with the result,6


ajy (r"t)
Yi(r,t) = L (rt {}) (3.18)



where L is the Onsager's matrix of transport coefficients. For times

large compared to the mean free time between collisions, after the

initial transient describing the complicated behavior of the system

before hydrodynamics applies, the time evolution of the macroscopic

state is then described by the set of nonlinear Navier-Stokes equations,

given by


S a + ijy (r,t)
r. ___ ( t;{()} + Ly
(r,t) + {E(r,t{}) + L (, ) r = 0,
i O
(3.19)



where E represents the contribution from the nonlinear Euler
a
equation. Explicitly, the nonlinear Navier-Stokes equations are



p + V(pv) = 0 (3.20a)


av
S + + (3.20b)
(-- + v'V)u + hNVv = -f.q tij (3.20b)
ij br.








at
(_ + .v 1 p ij
(- +v* (iP + ] (3.20c)
+t v i p br. r
2 j

1 2
where u is the average internal energy density, u = e 2 pv h is the

enthalpy density, h = u+p, and p the pressure. To Navier-Stokes order
*
the irreversible parts of the heat flux, qi, and the stress tensor,

tij, are given by the usual Fourier and Newton laws as



qi(r,t) = -K(p,u) r. (3.21)


vk
j(,tP) = {Ip,u)Aijkl + C(p,u)6ij6kl} r, (3.22)



where A. =6 6. + 6. 6 2 6. 6. The coefficients of thermal
ijkl ij kl ik jl 3 ij k
conductivity, <, and of shear and bulk viscosity, n and C, respectively,

are in general functions of the thermodynamic variables, as indicated.

The generating functional G for hydrodynamic averages and fluctuations

is given by Eq.(2.35), with



U[(~}] = exp f_ dt f d'r (r,t)x (r,t) (3.23)



The first functional derivatives of G are



+G (,t X) 6GlXI
6X a(r,t)

<( (r,t)U[h]>
a (3.24)



where % often will be used to indicate the set of five test functions,

{Xa}. When evaluated at X=0, Eq.(3.24) reproduces the average densities







41

given by Eq.(3.8). Similarly, higher order functional derivatives

generate the correlation functions (3.9), as shown in Eq.(2.40).

The explicit form of Eq.(3.24),


) d -Lt U[XI
aN=O ar e- []> PNO)
N=O


S< (r;t>, (3.25)



where L is the Liouville operator of the isolated system (Eq. (2.4)) and

the time evolution operator is defined in Appendix A (Eq. (A.51)),

suggests that the same statistical mechanical methods used to derive the

hydrodynamic equations from the macroscopic conservation laws can be

applied to describe the time evolution of the set of functionals

{(a (r,tl)}. In fact, the time evolution operator is the same in both

cases and the difference in the equations can be incorporated as a

modification of the initial condition for the nonequilibrium ensemble.

The initial condition becomes dependent on the test functions

a through the functional U. This program is carried out explicitly in

Appendix A, where the methods of nonlinear response theory are applied

to derive a set of equations for the functional {a (r,tX)} The

derivation is identical to the derivation of the nonlinear Navier-Stokes

equations and preserves the properties of the generating functional.

In particular, a set of h-dependent thermodynamic variables,

{ya(?,tl|)}, is defined in analogy with Eq. (3.13) by requiring,



4 (r);t>=
kL' (3.26)





42

where <...;t>kL indicates the average over a X-dependent local

equilibrium distribution functional pL(tlx), given by
SL t


pL(tx) = exp{-qL(tXl) f dr ya(r,ti k) (r)} (3.27)


The constant qL(t X) assures the normalization to one of the distribu-

tion functional (Eq. (A.7)). The functional dependence of pL(tlX) on

the set t{y(r,tlX)} is the same as indicated in Eq. (3.13) for the

case X=O. The form of the thermodynamic equations relating {Iq(r,tlk)}

and {ya(r,tlX)} is therefore preserved and is again the same as in the

equilibrium case. The comments made when imposing the equality (3.13)

apply here. The thermodynamic variables {ya(r,tlX)} depend on

the {k } only through the X-dependent densities, { (r,t l)}.

The set of generalized Navier-Stokes equations for the five func-

tionals {1 [X]} derived in Appendix A are formally identical to the

usual nonlinear Navier-Stokes equations. In a matrix form they are

given by



-- Sa(r,t 1) + {E (r-,t;{4[X]})

= ((rrt)
+ L'(rt;{c{d]}) } = y (rtlX),
Sr. Dr. ai NS
(3.28)


where the Euler matrix, E and the matrix of the transport

coefficients, L %, are respectively defined as



E (r,t;{ [k]}) i );> (3.29)
a ai XL









and

L (r,t;{cp[]}) = lim ft dt <{eL(t-)^ () [> (3.30)
tt o ai j3.3)
t>>tf


where j[X] is the total (volume integrated) projected flux defined in

Eqs. (A.23) and (A.57), and the average in Eq. (3.30) is over the

X-dependent equilibrium ensemble with local thermodynamic variables,

defined in Eq. (A.54). Both E and L are nonlinear functionals of the
a ap
densities {[( (r,tl )} and depend on the test functions {( } only through

the {~}a themselves.

Additional contributions to the irreversible fluxes, not present in

the Navier-Stokes equations, appear in Eqs. (3.28). They originate from

the X-dependence of the initial condition for the nonequilibrium
**
distribution functional. The subscript NS to y i indicates that the

right hand side of Eqs. (3.28) has to be evaluated to Navier-Stokes

order in the thermodynamic gradients and in the limit of time t long

compared to the mean free time tf, i.e.


S (r* Yir' ) lim [ y (r,t )]NS (3.31)
br. ai NS r ai I )lNS
1 t>>t I


where [A]NS indicates the result of an expansion of the function A to

second order in the gradients of the thermodynamic variables. The
**
explicit form of y is derived in Appendix A, with the result,



S(,t|) = L (3.32)

Here the nonequilibrium distribution functional has been written

Here the nonequilibrium distribution functional has been written







D(tl() (3.33)
PN(tlI) = PL(tJX)e (3.33)
with
-Lt
D(tlX) = e D (X) + D'(tlX) (3.34)



and
t + L(t-T) y (a"TX) +
D'(tlX) = .- dr J dr e-(X
S ( |

+ rai (1} (3,35)





.D o( ) = D(O i) :


Uin ] PL(0)
= In (3.36)
(U[XJ>kL PL(OIX) '


where ai and a are the projected fluxes and the normalized densities

given by Eqs. (A.23) and (A.21), respectively, when the local

equilibrium average is substituted with an average over PL(tlX). When

evaluated at X=0, the additional contributions to the irreversible

fluxes vanish identically, since D (X=0) = 0, and Eqs. (3.28) reduce to

the nonlinear Navier-Stokes equations (3.19). However, when the

generalized equations for the functionals { (r,tJX)} are considered,

these extra contributions have to be kept and will be shown to be

directly related to the strength of the noise in the system.

The Navier-Stokes limit indicated in Eq. (3.31) is consistent with

the approximations introduced in the left hand side of Eqs, (3.28), but

can only be taken after functionally differentiating y i and evaluating

the result at X=0. The evaluation of this term will be needed when







45

deriving the equations for the correlation functions, and is carried out

explicitly in Appendix B.

The only assumption introduced in the derivation of Eqs. (3.28)

regards the form of the initial state, which has been chosen to be local

equilibrium. This choice does not imply any serious restriction,

because the deviations from initial local equilibrium are expected to

decay in a time of the order of the mean free time, tf, in states

leading to hydrodynamics.3,10 These deviations are only responsible for

the initial short lived transient in the system.

Using the same notation for the functionals and the corresponding

average densities, the X-dependent Navier-Stokes equations are

explicitly given by



p[X] + V((p[I]v[I]) = 0 (3.37a)
T4


[ I v ,* v .i [ ,]
(-- + v[]. V)u[] + h[X]--v[-] = -V.q [r] -tij ] -- (3.37b)
St V1 Or

+ 1 8p[A] 1tij []
( + v[.V)vi[] = + ) (3.37)
1 j

4
where the dependence on r and t of the functionals has been omitted to

simplify the notation. The irreversible parts of the heat flux and the

stress tensor are now given by




qi[O] = -[]NS
q[] = -[ --8 +q [INS (3.38)
1

]vk[i] **
t.[] X] = -{[X]Aijkl + [x]6j } r + tij[X]NS (3.39)







46

The transport coefficients in Eqs.(3.38-39) are functionals of X

through the X-dependent densities, {( (r,tlX)}. The additional

contributions to the irreversible fluxes are defined as

** D(t^) D'(t jX)

qi (r,tlX) = kL (3.40)

= XL

t..(r,t|x) = t (r)[e- e; (3,41)



where q.(r) is the microscopic heat flux, q.(r) = s.(r) v.t. (r).

Finally it should be stressed that although the extra contribution

to the irreversible-fluxes may seem'similar to the ones suggested by

Landau and Lifschitz47 to construct hydrodynamic Langevin equations (and

in effect their physical origin is analogous), there is a basic differ-

ence in the present formulation. All the equations here are equations

for averaged variables, not stochastic equations. In this way the

description obtained is directly related to the experimental situation.



111.2 Correlation of Fluctuations

The fluctuations of interest are those of the set of microscopic

densities, {((r,t)}. In particular the second order correlation

functions are defined as



HB(rl,tl2 = <6 (rltl)6 (r2,t2)> (3.42)



Using Eqs. (3.23-25), such correlation functions can also be expressed

as the first functional derivatives of the X-dependent densities,








6
M ( tl ;r2 ,t2 ( =0} (3.43)
6s (r2,t2) a


A set of equations for the two-time correlation functions can then be

obtained by functionally differentiating the generalized Navier-Stokes

equations, derived in the proceeding section, Eqs. (3.28). In a general

nonequilibrium state no symmetry properties relating the different

correlation functions can be identified a priori. The equations for the

set of twenty-five functions M (for a=i,...,5 and 6=1,...,5) are in

general all coupled together.

By functionally differentiating Eqs. (3.28) and remembering that

the X-dependence occurs in these equations only through the {4 (r,tjk)}

(or, equivalently, the thermodynamic variables ({y(r;tlX)}) and through

the additional terms in the irreversible fluxes, yai, the following set

of equations for the correlation functions is obtained


6 + +' + 4
t IMap(rltl;' r 2 t2+ + ,o(r lt1; {}aom(rltl;r2t2
4 4


= I ,(rl',tl;r2,t2) (3.44)



where ..
E a(r,t;{(}) aLl (r,t;{0}) ay (r,t)
a aao 0
C (r't;({ ) a +
Sr. + + r.
i (r,t) Y P (r,t)

ij oy a by(r ,t)
+ L(rt;{}) ~ y (3.45)
j 84 (r,t)


and
(+ [ 6y _(r ,t, 1)
I ,rl,t l;,t ) = lim [br ] (3.46)
tl >>t li 6s (r2 t2) 2=O NS






48
**
The fluxes y.i are defined in Eq. (3.32) and [A]NS indicates, as in

Section III.1, the Navier-Stokes limit of the function A. The differ-

ential operators in Lap operate on everything to their right, including

the operand of p itself. In deriving the set of Eqs. (3.44) the

assumption that the local equilibrium correlation functions appearing in

the operator f a are localized in space has been introduced. Therefore

these equations apply only if the system is away from critical points.

In the general case the operator Lap is nonlocal and the derivatives in

Eq. (3.45) have to be substituted with functional derivatives.

The inhomogeneous term on the right hand side of Eqs. (3.44) is

evaluated in Appendix B. For t -t2 tf, it is given by


t L(t -t +T) ,
I (r;,t ;r 2,t) = lim 2 d <[e i(r )]j>0



H 0) (3H.
x po(r2,;r (3.7)
ij


where <***>0 indicates an average over a reference equilibrium ensemble

with local thermodynamic variables evaluated at the point (rl,tl) (the

definition is given by Eq. (A.54), evaluated at X=0), and is the

volume integrated flux, defined by Eq. (A.57), at X=0. Also,
H + +
G (r 2,T;rl,0) is the Green's function for the Navier-Stokes equations,

defined in Eq. (B.17).

The time correlation function in Eq. (3.47) is the correlation

function appearing in the matrix of the transport coefficients, given by

Eq. (3.30) at X=0. Its lifetime is of the order of the mean free time

tf, and therefore much shorter than the macroscopic time scales of inter-

est in hydrodynamics. Consequently the term I does not contribute to the






49

hydrodynamic equations for the two-time correlation functions if



tl>>t ,



t -t2>>tf (3.48)



In this limit Eq. (3.44) becomes




-t M ,(rltl;r2,t2)+ oa(rl'tl; t )Ma(rltl;r2,t2) =0 (3.49)



This set of coupled equations describes correlations over time scales

large compared to a mean free time, as desired in a hydrodynamic

description. The equations for the correlation of fluctuations at

different times are linear, although coupled to the solution of the

nonlinear Navier-Stokes equations through the dependence of C a on

the {4 }. The linear character of the equations is a general result and

does not restrict their applicability to small fluctuations around the

nonequilibrium state.

For the case of small fluctuations around a nonequilibrium steady

state, Eqs. (3.49) agree with the results of others.26,39-42,46 The

equations derived here represent therefore a generalization of these

results: they provide a hydrodynamic description of fluctuations, valid

to Navier-Stokes order in the gradients, for nonstationary states and

fluctuations of arbitrary amplitude. A detailed comparison with

previous work will be presented in Section III.4.

For the special case of equilibrium fluctuations, the operator

fC reduces to that of the linearized Navier-Stokes equations. The






50

equations for the fluctuations at different times are then exactly those

suggested by Onsager's assumption: the fluctuations decay according to

the macroscopic linearized regression laws governing the dynamics of a

nonequilibrium system near equilibrium.

The explicit form of Eqs. (3.49) for a one-component fluid is

conveniently written in terms of the correlation functions



Mp(rl,t ;r2't2) = <6p(rltl )6 (r2,t2)>



M I (rl,tl;r2,t) = <6u(r ,t )6( (r2,t2



Mgi rltl;r2,t2) = <6gi(rl,tl)6 (r2,t2)> (3.50)



3=1,2,...,5. It is convenient to write the hydrodynamic equations in

terms of the correlation functions MuP instead of M -, where u is

defined by


N
u(r) = En(n-m(q n))6(r-q) (3.51)
n=l


The two correlation functions are related by



u Pl ;r2t2 = M p(r ltl;r2t2) i(rlt )mgi( ,l; 2)


+2 [v(rl,tl)]2 M (rltl;r2,t2) (3.52)



A coupled set of equations for the correlation functions (3.50) is

obtained from Eqs. (3.49) by respectively setting a=p,u,gi,







S + M = 0 (3.53)
at1 pp +r i gip


S + + + M u + ( hM + hM)
Svi ri) Mup + rh li Mvi v I 3 r, r prp u U
Sli 1 ri li


S {iK (TpMpp + TuM p) + (KM + KMU ) Kr }
ri rli p pp u up p pp r

av 6v
arj r p Vijkm + jm)Mpp + p ( i +jkm + S ijm)Mup
lj 1rlm

b V k 6
2V. + M ) = 0 (3.54)
ijkm + ijkm Or Or vi.p
1wm 1 1


M + (v M + py.M
SI ip g Ir j gip 1 v p


+ M p + p) + i jk + rvi
IJ


arli (pMpP + u up) + j iqj km ij 6m) im Mk}
Ir lj IF


+ [(n A .. + 5..k )M
Br j r -m p ijkm + p i p m)pp


+ (u Ajk.. + 6.ij km)M ]} = 0, (3.55)


where

P(rl't )i p(rl,t l;r2t2) Mg p(r,,t ;r2,t2)


+ -+
v i(rl,t )M (r ,tl;r2,t2) (3.56)


The space and time arguments of the correlation functions and the

hydrodynamic variables have been omitted in Eqs. (3.53-55) to simplify

the notation. The dependence of the correlation functions is the same

indicated in Eqs. (3.50) and the hydrodynamic variables are evaluated at






52

the point (rl,tl). Also, the thermodynamic derivatives of a function

A = A(p,u) have been indicated as



A [A(P'u)]u
p p '(3.57


A [A(pu)] (3.57)
u bu p


By comparing Eqs. (3.49) with Eqs. (3.53-5), the matrix


identified as





S(r,t) =





where



up

gip



uu
giu
giu


t (rt) is


pgi

ug.

gig
ii I


pgi br. '
1


=-h
1r.
1


V.

P
1
p


(-) I + h D
ri P P


2
(C -V) D6ij]


D.j [2D.i.+
13 1 3 j


1
2


I \
br.


8 T + (IT )K
r. p or p]


+ ( 2 - ) D6.] i
3 i p


= v. + h D
i or. u
1


1 1


T T
T + (T )' ]
1


1 2
2 Dij [2Dij + (C -j)D j],j


(3.58)


+ 2[2nD ij
ij i







53
1 1 + u 2 1
= -- 2[2nD + ( -rj)D 6.
ori p p r i ij 3 ij ri p


gip 8r. p
1


---- v.v.
Or. i j



gi u or u


S8 2
2 j lD + (Cp -- ) D
br. p i + r. p 3 P


8 8 A ,1k
r. Aijkm +' ij -) r
3 m


2 uD.. + (~ -2 ) D,
J 1


where


gigj 13 jr k Or i r k Aijkm m+ ~i r p








3 1


3.59)


3.60)


D D.. = Vv. (3.6
11



The differential operators in Eqs. (3.59) act on everything to their

right, unless otherwise indicated by the presence of a parenthesis.



111.3 Equal time fluctuations

Second order equal time correlation functions, defined by setting

t1=t2 in Eq. (3.42), i.e.



M(lt;r2 t) = <6((r ,t)6 (r2,t)>, (3.6


1)


2)


ugi








54

are needed as initial conditions for the solution of Eqs. (3.49).

However, equations for the equal time correlation functions cannot be

obtained directly by functional differentiation of the equations for

{<( (rl);t> }, as was done for the equations for the correlation of

fluctuations at different times, because the functional differentiation

does not commute with the time derivative. Instead, they can be

determined from the limit,


6 .* .6,
SM(r,t;r2,t) = [lim (+P P) (r);t+
a--t p 1 ( + 12 ap 66 2 ]=0 '
2t (3.63)



where the operators P12 and P p permute rl,r2 and a, B, respectively.

By using the results of the proceeding section, (Eq. 3.44), and

observing that the e-0 limit can be taken immediately on the left hand

side of the equation, an equation for Map is obtained in the form



t Mr(rl,t;r2't) + (1+P12P ) C(rl',t1;{ )Mo(r1lt;r2,t)



= FQ(rlr2;t) (3.64)


where the operator is defined in Eq. (3.45), and



p(rlr2;t) = lim (1+P 2P a >a (l,t+E;r2,t) (3.65)



The inhomogeneous term r a is evaluated in Appendix B, with the result






55

r (r ,r2;t) = (1+P2P ) L(r ,t;{ 6(r a-r)

(3.66)

where L is the Onsager's matrix of transport coefficients.
ap
Equations (3.64), together with the set of Eqs. (3.19) and the

equations for the two time correlation functions, give a closed descrip-

tion of transport and fluctuations (up to second order correlations) in

a hydrodynamic system whose macroscopic state is described in terms of

the nonlinear Navier-Stokes equations.

Again, the interpretation of Eq. (3.64) is clarified by comparing

with the case of fluctuations around the equilibrium state. Onsager's

assumption applies then and, as seen in the proceeding section, the time

evolution of the two-time correlations is governed by the linearized

hydrodynamic equations. Similarly, Eq. (3.64), evaluated at equi-

librium, becomes



(I+P12 P ) C(r ,{( })M (r ;r 2)



= (1+P P )L' o}) 6(r( 2) (3.67)
12 ap ap ri ar 2j 1 2
ii 2j


where the superscript zero indicates the equilibrium value and 0 now

depends on r1 only through the differential operators. Equation (3.67)

is an identity or, equivalently, constitutes a statement of the

fluctuation-dissipation theorem relating equilibrium fluctuations and

transport coefficients.

The description of nonequilibrium fluctuations derived here

indicates clearly how Onsager's assumption should be modified for

systems out of equilibrium. First, it is no longer true that the






56

regression laws for the fluctuations are the same as those for the

average values. They are instead linear equations, whose coefficients

depend on the solution of the nonlinear regression laws (they can be

interpreted as a linearization about the nonequilibrium macroscopic

state, {(a(r,t)}, at each instant t). In this sense knowledge of the

average dynamics still allows determination of the dynamics of fluctu-

ations. The two sets of equations form a bilinear set that must be

solved simultaneously. Secondly, the fluctuation-dissipation theorem

must be abandoned in favor of Eqs. (3.64) and (3.66) which, for

nonstationary states, indicate that even the equal time fluctuations are

dynamical variables.

In equilibrium, rTa is determined by the left hand side of Eq.

(3.67), since the equal time fluctuations can be calculated directly in

the Gibbs ensemble. In the nonequilibrium case, however, r a must be

provided independently. Its calculation requires the analysis of the

initial transients due to microscopic degrees of freedom not

incorporated in the Navier-Stokes equations. In this sense the source

function F p can be interpreted as a noise amplitude.

The explicit form of Eqs. (3.64) for a one-component fluid can be

obtained by using Eqs. (3.58-59) for the elements of the matrix rC.

The inhomogeneous term PF (r,2r ;t) is diagonal in the labels a and
atp 1 2'
p for a and p = p,u,g.. Its explicit form is given by Eq. (3.66), with



L (r,t) = 0,



L (r,t) = lim ft d- <[eLT sir)]S>0
t>>Cf





57
ij L t (>
L (r,t) = lim dr <[e tk r)] (3.68)
gkgl t>>t ok 0 0


where s!(r) and t!.(r) are the projected energy and momentum fluxes,

given by



s!(r) = s.(r) gi(r) (3.69)
p(r,t)
and

t() = tij(r) 6ij[p(r,t) + (-) u() + () p(r)] (3.70)
13 1j 13j u 6P
p u


Equations (3.68) are the Green-Kubo expressions for the transport

coefficients, or



LJ(rCt) = 6ij.T2 (r,t)K(p,u) (3.71)



ij +
L (r,t) = KrT(r,t)[ri(p u)Aijkl + C(p,u)6j Sk] (3.72)



The transport coefficients depend on r and t through the average mass

and internal energy density.



III.4 Discussion

Several authors26'27,39-46 have recently formulated a description

of fluctuations in nonequilibrium hydrodynamic steady states which are

adequately described by the nonlinear Navier-Stokes equations. It seems

therefore appropriate to compare these approaches with the one used here

and to discuss differences and similarities.

Tremblay, Arai and Siggia26 have proposed a Langevin theory of

fluctuations in nonequilibrium steady states. They assumed that the

Landau-Lifschitz theory of hydrodynamic fluctuations can be applied not





58

only for equilibrium fluctuations (in a regime where the macroscopic

regressions laws are linear), but also for small fluctuations around the

nonequilibrium state. The equations governing the time evolution of the

fluctuations are obtained by linearizing the nonlinear macroscopic

regression laws around the nonequilibrium state. The correlations of

the random forces are assumed to have the same form as in equilibrium,

with local thermodynamic variables. This description is supported by

the work of Keizer.30 It is however phenomenological in character and

restricted to small fluctuations around the nonequilibrium state. The

results of the present formulation basically confirm the Langevin

hypothesis. There are however some important differences. First, the

equations for the fluctuations are always linear, independent of the

size of the fluctuations. Secondly, the form of the noise spectrum is

derived here on the basis of a microscopic description, that clearly

places the source of the fluctuations in the transients present in the

system before the macroscopic description applies. Furthermore the

approach used here is not stochastic, but describes the system entirely

in terms of macroscopic variables (averages and correlation functions),

as required to make a precise and immediate connection with experiments.

Microscopic or semi-microscopic formulations of the description of

fluctuations in nonequilibrium steady states, similar in spirit and

content to the present one, have been developed by Kirkpatrick, Cohen

and Dorfman (KCD),45,46 and by an MIT collaboration,39-42 as indicated

in Chapter I. The results of KCD agree with the ones obtained here,

when the latter are specialized to the case of nonequilibrium steady

states. The inhomogeneous term in their equation for the equal time

correlation functions has apparently a different form from the one in

Eq. (3.66). The correlation of interest in Ref. 46 is the deviation of




















59

the equal time correlation function from its local equilibrium value.

The noise term there originates from the action of the Euler part of the

hydrodynamic equation over the local equilibrium correlation function,

as it is shown in Ref. 41. The form given in Ref. 46 and the one

derived here, Eq. (3.66),.are however equivalent. The hydrodynamic

equations obtained by Kirkpatrick, Cohen and Dorfman are therefore

identical to the ones derived here.

Instead, the results obtained by the MIT group (and, it seems, also

the Langevin theory) differ from those obtained through the generating

functional approach because the second term in the evolution matrix ,P

[see Eq. (3.45)], containing the thermodynamic derivatives of the trans-

port coefficients, is neglected there.41 This term is often small, but

is still of first order in the fluctuations around the nonequilibrium

state, and therefore cannot be neglected even when fluctuations of small

amplitude are considered.










CHAPTER IV
KINETIC THEORY

IV.1 Generating Functional for Phase Space Fluctuations

The hydrodynamic description of nonequilibrium fluctuations

developed in the previous chapter is restricted to states adequately

described in terms of the nonlinear Navier-Stokes equations and rests on

an estimate of the lifetime of certain correlations functions.

Within the framework of kinetic theory, it is possible, at least in

the limit of low density (where a small expansion parameter is

available), to derive an exact description of nonequilibrium fluctua-

tions. Furthermore the kinetic description is more general than

hydrodynamics. It describes macroscopic processes varying on shorter

space and time scales (of the order of or smaller than the mean free

path and the mean free time between collisions) and it incorporates the

hydrodynamic limit .3,8 Also, as shown in Section 11.2, the averages and

the correlation functions of the dynamical variables of interest in the

study of a large class of systems are simply related to the reduced

distribution functions and to the correlations of phase space

fluctuations.

The generating functional method is applied here to describe phase

space nonequilibrium fluctuations. No restrictions are imposed on the

nonequilibrium state or on its stability other than the well understood

assumptions entering in the derivation of the Boltzmann equation.4,5,57

The description of fluctuations obtained is justified in the same limits

60







61

which apply to the macroscopic description of the state. In particular

no extra assumptions are introduced to evaluate the amplitude of the

"noise" in the system, which contributes to the equations for the equal

time fluctuations. Finally, in the last section, the hydrodynamic limit

is recovered. Again, the results are restricted to nonequilibrium

states in low density gases. However all of the known qualitative

dynamics of fluids in general are also exhibited by gases. In

particular, the nonlinear hydrodynamic behavior is identical.8'9

The system considered is the classical system of N interacting

point particles in a volume V introduced in Chapter II. Since the

internal degrees of freedom of the molecules are neglected, the discus-

sion is limited to monoatomic gases.

The time evolution of the distribution function of the system is

governed by the Liouville equation for a system in interaction with its

surroundings, as given in Eq.(2.2). For pairwise additive central

interatomic forces the Liouville operator of the isolated system has the

form


N N
L(x ,...,xN) = L(x.) 9(xi,xj) (4.1)
i=1 i

with +
p.*V+
i q
L (xi.)= (4.2)



and



e(x ,x.) = [ V(q .)].( + - ) (4.3)
qi q Pi j






62

Equation (4.3) applies for continuous interatomic potentials, V(qij),

with qij = qi qj The potential V is also assumed to be short

ranged (of range o) and purely repulsive. The first term in Eq.(4.1)

represents the free streaming of the particles and the second term

contains the effects of the collisions.

Kinetic theory describes phenomena whose space and time variations

occur over scales large compared to the force range, a, and the duration

of a collision, Tc. The derivation of the kinetic theory from the

microscopic N-particle dynamics rests on the assumption that, for a

particular class of nonequilibrium states, a contraction of the

description takes place over a time of the order of T For times large

compared to T all the properties of the system depend on time only

through the one-particle distribution function. This is the idea

proposed by Bogoliubov to derive the Boltzmann equation from the

Liouville equation.4 The special states to which this macroscopic

description applies are identified through the requirement that the

initial correlations decay on times of the order of the time required

for a collision.5 The kinetic theory of nonequilibrium fluctuations

derived here will be restricted to the same class of states.

As mentioned, the macroscopic variable of interest is the one-

particle distribution function, fl(x,t), defined in Eq.(2.27) as the

nonequilibrium average of the phase space density, f (x,t). The

corresponding generating functional is given by Eq.(2.35), with



U[X] = UN[X] = exp jf+ dt f x fl(xt);(x,t) (4.4)


The first functional derivative of G is then given by







6G[X]
6W(x,t) (4


and reduces to the one particle distribution function when evaluated at

6G[=O]



[(xt] =o = f1(x,t) (4.6)


To obtain a kinetic equation, it is convenient to define the

functional f (x,t [) as



n f (x,tl ) for t > t (4.7)
1 o


where to is an arbitrary parameter introduced to assist in ordering the

times obtained on functional differentiation. Again, evaluating Eq.

(4.7) at X=0 and choosing to=0, the functional fl[X] reduces, at all

times t > 0, to the one-particle distribution function, as in Eq. (4.6).

The dependence on the parameter to can be introduced explicitly by

choosing the tests functions to be nonzero only for t < t i.e.



X(x,t) = X'(x,t)8(to-t) (4.8)



where 0 is the unit step function. In this way only time correlation

functions involving time arguments smaller than t can be generated. The

ordering chosen in Eq. (2.40) can be obtained by appropriate choice of

to. The correlation functions of phase space fluctuations, defined in



*A similar approach has been used in Ref. 58 to derive kinetic
equations for multitime equilibrium fluctuations.








64

Eqs.(2.32) and (2.34), are then immediately obtained as higher

functional derivatives of G, as shown in Eqs.(2.40). In the following

the dependence on the parameter to will be indicated explicitly only

when needed.

As already done at the hydrodynamic level, it can be argued that

the detailed form of the external forces and of their interaction with

the particles of the system should not appear in the equations governing

the time evolution of the reduced distribution functions (or

functionals).657 The dependence on the surroundings can be entirely

incorporated through appropriate boundary conditions to be used when

solving the kinetic equations. This is a consequence of the fact that

the lower order reduced distribution functions are localized quantities

describing properties at a particular point in the gas. Therefore, as

long as properties in the interior of the system are considered, at a

distance from the boundaries large compared to the force range, the

Liouville equation for the isolated system can be used to describe the

time evolution of the dynamical variables. Equation (4.5) can then be

rewritten as


-Lt UfX]
n fl(x,t |) = N / dx ...dxN e U > PN(0) (4.9)



This form suggests that fl(x,tlX) can be identified as the first member

of a set of functionals defined by



nsf (x X) N! dx dx e-Lt U[,]
... t ) = dx ..dx() .
s s' (N-s)! s+1 N N(O)
Nls








65

When evaluated at X=0, fs(xi,...,xs,tI) reduces to the s-particle

distribution function defined in Eq.(2.25). The operator governing the

time evolution of the set of functionals {fsA[]} is the Liouville oper-

ator of the system and does not depend on the test function X. This

dependence only enters through the initial condition. By differentiat-

ing Eq.(4.10) with respect to time it can then be shown that the

functionals f [X] satisfy the BBGKY hierarchy. In particular, for s=1,



{-- + Lo(x1)}f(x1,tIX) = n f dx29(x1,x2)f2(xx ,tlX) (4.11)



Again, as done in the hydrodynamic case, the standard methods of sta-

tistical mechanics which allow closure of the hierarchy and derivation

of a kinetic equation for the one-particle distribution function5 can be

applied to the X-dependent functionals. By assuming that the functional

relationship between fl[X] and K, as defined in Eq.(4.7), may be

inverted, it is possible to express, at least formally, f2[\] as a func-

tional of fl[X]. A formally closed equation for fl[X] is then obtained,



{- + Lo(xl)}fl(x,tlX) = n f dx2o(x1,x2)H(x1,x2,tlfl[X]) (4.12)



The functional H is highly nonlinear but it simplifies considerably when

evaluated at X=0. However, in order to preserve the properties of the

generating functional, the evaluation of H at nonvanishing K has to be

attempted. This evaluation is carried out in the next section in the

limit of low density.






66

IV.2 Low Density Limit and Boltzmann Equation

A systematic evaluation of the functional H defined in Eq.(4.12) is

possible in the limit of low density, by using the nonequilibrium

cluster expansion developed by Cohen5 as a generalization of the

technique proposed by Mayer for equilibrium systems. In this way the

two-particle distribution function for a dilute gas can be expressed in

terms of the one-particle distribution as a power series in the reduced
3
density no When this result is substituted in the first equation of

the BBGKY hierarchy, an expression for the rate of change of the single

particle distribution function in the form of a density expansion is

obtained. The terms in the expansion depend successively on the dynamics

of clusters of two, three, etc., isolated particles. To lowest order in

the expansion parameter, corresponding to the Boltzmann limit, only

binary collisions are retained.

The cluster expansion for the functionals f [X] is formally iden-

tical to the one for the distribution functions and is outlined in

Appendix C. To lowest order in the density, the result is



f2(x1,x2,t I) = t(x ,x2)f (x1,tX)f (x2,t X)



+ R(x1,x2,tlfl[X]) (4.13)



where the streaming operator is defined as




t(x1,x2) = S_t(x ,x2)St(x1)St(x2) (4 .1)



and St(xl,...,Xs) is the operator governing the time evolution of a






67

dynamical variable in a system of s isolated particles. It is given by


tL(x ,.. .,xs)
St (x, ...,X) = e (4.15)



where L(xl,...,xs) is the s-particle Liouville operator (see Eqs.(4.1)).

In other words St(xl,...,xs) describes the dynamics of an isolated

cluster of s particles. Finally, R(xl,x2,tlfl []) is a nonlinear func-

tional of fl[k], depending on the initial correlations in the system.

Its explicit form is given in Appendix C. For times long compared to a

collision time the streaming operator $- reaches a time independent

value,57



5(xl,x2) = lim S (x1,x2) (4.16)
t>>
c

Furthermore it is shown in Appendix C that if attention is restricted to

the time evolution of initial states of the system where the particles

have only short-range correlations (i.e. the s-particle distribution

function factorizes for interparticle distances large compared to the

force range), a part of the contribution to Eq.(4.12) from the function-

al R(xl,x2,tfl [\]) vanishes for times long compared to the collision

time.

By inserting Eq.(4.13) into Eq.(4.11) and by making use of the

above results, an equation describing the time evolution of the one-

particle functional f (x1,tXl ) is obtained. The equation only describes

variations over time scales large compared to T and is given by



-- + Lo(xl)}f(x,tl) = n 7[fi[kl,fl[k]] + W(x1,t fl[k]), (4.17)







for t > t where
o


J[A,B] = dx2 K(x1,x2)A(x1)B(x2)



and

K(x1,x2) = 9(x1,x2) -5(x1,x2)



The second term on the right hand side of Eq.(4.17) is given by



W(x1,tf 1[X]) = lim n f dx29(x ,x2)St(x1,x2)
t>>f
c

x {U2[Xlx2 X]U -1[x IX]UI -[x21 ] -1



x St(x1)St(x2)f (xlt )fl(x2,t 1),


(4.18)


(4.19)


(4.20)


where U [x ,... ,x ] is the low density limit of the s-particle
s 1 s
functional analogous to the N-particle functional defined in Eq.(4.4).

It is given by



Us[Xl,...,xs ] = exp fJ' dt St(xI,...,x2) (xi,t)
i=l


where


Sexp{ [ f dt X(xS(t),t)} ,
i=l



s
x.(t) = S (x ,..,x )x (0)
1 t s )xi()


(4.21)





(4.22)


A precise justification of the use of Eq.(4.21) as the low density form

of Us is given in Appendix C.





69

By evaluating Eq.(4.17) at X=0 and observing that

Us[x1,...,xs l=0] = 1 for all values of s, it follows that



W(xl,tlfl[%=O]) = 0 (4.23)



identically. Choosing to = 0, a homogeneous equation for the one-

particle distribution function is obtained,



{ -+ Lo(x )}f1(x1,t) = n 3[fl,fl] (4.24)



The right hand side of Eq.(4.24) is a nonlinear functional of the one-

particle distribution function. When the further approximation of

neglecting the difference in position of the colliding pair is intro-

duced (this implies that the resulting equation will only describe space

variations over distances large compared to the force range), the

operator J assumes the familiar form of the nonlinear Boltzmann oper-

ator, given by

IP -P21 2 + +

J[A,B] = f dp2 i db b J d4 A(qI ,P)B(ql p
A 2qI0 m 1 i2


A(qP )B(ql' 1 (4.25)



where b is the impact parameter for the binary collision and d is the

azimuthal angle measuring the orientation of the scattering plane. The

prime over the moment indicates their values after the collision. The

nonlinear Boltzmann equation for the one-particle distribution function

is then recovered. In the following both the operators J and J will be

referred to as the nonlinear Boltzmann operator.






70

Equation (4.17) has therefore the form of a generalized Boltzmann

equation. The extra term on the right hand side of the equation is a

complicated functional of X and its behavior at long times cannot be

analyzed before setting X=0. As just shown, it does not contribute to

the dynamics of the macroscopic variables of the system. However it has

to be retained in the equation for the functional fl(x,t|X) for the

purpose of deriving equations for the correlation functions.

Equations for the correlations of phase space fluctuations are

derived in the next two sections. As in the hydrodynamic limit, the

contribution from W to the equation for the multitime correlation func-

tions vanishes on a time scale of the order of t However, when
c
correlations of equal time fluctuations are considered, this term does

not vanish, but is responsible for the appearance of an inhomogeneous

part in the equations.



IV.3 Correlation of Fluctuations

The multitima correlation functions of phase space fluctuations

defined in Eq.(2.34) are obtained by functional differentiation of

G[k], as indicated in Eq.(2.40), or, equivalently,


s
Cs(xl,tl...;xst) f(x^,tk
k=1

(s-l)f ( lt I

[65(x2 t 2) ...6(xs ,ts ) to=t2 ,=0 '



with the time ordering t > t2 > .. t Equations for the time

evolution of Cs are then obtained by functionally differentiating

Eq.(4.17) and evaluating the resulting equation at k=0.






71

It is shown in Appendix C that the inhomogeneous term

W(x1,tl X) vanishes at long times (t >> T ) if to is chosen to

satisfy t t > T This choice corresponds to the one needed to
o c
generate multitime correlation functions in a Boltzmann description. In

this sense it immediately appears that W does not contribute to the

equation for the two-time correlation function, as long as tl-t2 >c

Here this result is also explicitly shown to follow from the analysis of

the functional derivative of W. This will make evident the difference

between the unequal and the equal time cases.

The equation for the two-time correlation function (s=2) is


S6W(x ,tI If I[ ])
--a + L(xl'tllfl ) C2(xltl;x2't2 [ 6(x2,t2) to=t,k=0 '

(4.27)

where

L(xl,tlfl) = Lo(xl) n A(xl,tllf1) (4.28)

and

A(x1,t lf1) = f dx3( (x1,x3) (,x3)(1+ P13)f (x3,tl) (4.29)



By functionally differentiating Eq.(C.26), the term on the right hand

side of Eq.(4.27) can be evaluated explicitly as a functional of the

one-particle distribution, with the result,


6W(x ,tl X)
[6X(x2,t2) -=o0 = n dx3e(x1,x3)Stl(x1 ,x3){[St 2(1,x3)


St(x )S t(x )][6(xl-x2) + 6(x3-x2 ]



x St (x)St (x3)f (x1,tI)f (x3,tl) (4.30)
1 1











6W(x ,tl X)
[ 2x ) ]
6X(x2,t2)


=n fdx39(x,x3){[ St (xX3) t (x ,Xj)

Sx St -t 1-t2 + 6(B-X

1( )S (x3)[6(x1-x2) + 5(x3-x2)i
2 1 2 1


x (x x3 )f( ,t)f ,t
1


where the choice t = t2 is understood. The streaming operator

St reaches a time independent limit for t>>T as shown in Eq.(4.16).

Therefore the difference between the two streaming operators in

Eq.(4.31) vanishes if



tl >> c


t t2 >> c (4.32)



In the above limit the right hand side of Eq.(4.26) vanishes,


lim
t -t>> c
1 c


6W(x ,tI X)
[6(xt) ]t 2,=0 = 0 ,
61(x t ) ti2t,X=0


(4,33)


and the time evolution of the two-time correlation function is governed,

by a homogeneous equation given by



( + L(x1,t1f1)}C2(x ,t1;x2,t2) = 0 (4.34)
l


(4.31)






73

The conditions imposed in Eqs.(4.32) imply that Eq .(4 .34) can only be

applied to describe correlations over time scales large compared to a

collision time. Not only the times t, and t2 have to be large compared

to T but also their difference. This is the natural outcome of con-

sidering the Boltzmann limit.

As in the proceeding section, when the difference in position of

the colliding pair of particles is neglected, Eq.(4.34) becomes


--I + LB(xl'tl fl)}C2(x'tl;x2,t) = 0 ,



where,

LB(xl,tllfl) = LO(x1) nl(x1,tjf) ,


(4.35)





(4.36)


I(x1,t1f1)A(x1) = f d3 f0 db b J d4 3 [ + P(pP3)]



x fl1,p'3',tl)A(ql'P ) fl1' 3t)A(ql1p .
(4.37)



The permutation operator P(pl,p3) interchanges only the moment of the

two particles. When the one-particle distribution function in Eq.(4.37)

is evaluated at equilibrium (i.e. coincides with a Maxwell-Boltzmann dis-

tribution), the operator I reduces to the linearized Boltzmann

operator.8

The equation for the correlation function is linear. This is an

exact result (in the Boltzmann limit considered here), not the outcome

of a linearization around the state. No restrictions have been imposed





74

on the size of the fluctuations. The operator L(x,tifi) depends on tha

nonequilibrium state of the system, characterized by the distribution

function fl(x,t), which is given by the solution of the nonlinear Boltc--

mann equation. In this sense the dynamics of the fluctuations is

entirely determined by the macroscopic nonequilibrium state of the

system.

To solve Eq.(4.34) specification of the initial condition, repre-

sented by the equal time correlation function C2(x1,t;x2,t), is also

needed. The derivation of an equation for the equal time correlation is

the subject of the next section.

Finally, the method described here can be extended to evaluate

higher order multitime correlation functions. In particular the

equation for the three-time correlation function is given by



b + L(x1 ,t 1f1)}C3(x1,t ;x2,t 2;x3,t3)
{i + L(l'l l)}C3(xl 2'tx"3')



= n J dx4K(x1,x )(i + P14)C2(Xl,t ;x2,t2)C2(x4,tl;x3,t3) (4 .3)



where the operator K(x1,x4) has been defined in Eq.(4.19). The equation

for the three time correlation function is again linear, but is coupled

to the lower order correlations through the inhomogeneous term on the

right hand side of Eq.(4.38).

In general, the equation for the s-time correlation function is a

linear inhomogeneous equation. The homogeneous part nas, at any order,

the form given in Eq.(4.34) and depends therefore on the nonequilibrium

distribution function. The inhomogeneous part couples the s-th order

equation to all the lower order ones. As s increases, the equ-aicns

become therefore very complicated, but always conserve linearity.







75

Furthermore the solution of the equation for the s-th order multi-

time correlation function requires the knowledge of all the correlation

functions obtained by progressively setting tl=t2, t=t2=t3, ...,

tl =t2 = ...=ts, in Eq.(4.26). Equations describing the time evolution

of such correlation functions can be obtained by applying an appropriate

limiting procedure on the equations obtained by functional different-

iation of Eq.(4.17). The second term on the right hand side of

Eq.(4.17) will only contribute when at least two of the time arguments

in the correlation function are equal, as will be shown in the next

section.



IV.4 Equal Time Fluctuations

In order to derive equations for the correlation functions of equal

time fluctuations, defined in Eq.(2.32), a limiting procedure on the

equations for the unequal time correlation has to be used. In this way

it is possible to circumvent the problem arising from the fact that the

time derivative and the functional derivative can only be interchanged

when involving different time arguments. As in the previous section,

the calculation will be carried out explicitly for the second order

correlation function. The rate of change of C2(xl,t;x2,t) can be

written in the form,




C2(x1,t;x2,t) = lim{C2(x1,t+E;x2,t) + .C2(x2,t+e;xl,t)} (4.39)



An equation for the equal time correlation can then be obtained from the

equations for the unequal time correlations, Eq.(4.27), with the result,







76

1 + (1 + P12)L(xl,tlf)}C2(xi,t;x2,t)


6W(x ,t+E X) 6W(x 2,t+E X)
= lm[ 6x(x2,t) ]t =tI=0 6+(x ,t) t =t,X=0
EO2 o 1 0


On the left hand side of the equation the limit E-+O has been taken,

The operator L(x,tlfl) has been defined in Eqs.(4.28-29). By using

Eq.(4.31) the right hand side of the equation can be written,


r(xl,x2,t)


6W(x ,t+ I ) 6W(x2,t+ I )
S!+1([- 6(x ,t) ]t =t,X=O + L 6(x ,t) ]to=t,=0


= li+(1+P12)n f dx3 (xi,x3) S (x1,x3)[1 St(x,x3)]

Ox e(x)S1- +
x S_E(xl)SF(x3)[6(x1-X 2) + 6(x2-x3)]


(4.41)


or, taking the limit,



P(x1,x2,t) (+P12) n f dx39(x1,x ){[S(:x-x2) + 6(X3-X2)] St(x1X3



s(x1,x2 )[6(x1-x2) + 6(x-x3)]}f (xt)f(x,t) .

(4.42)


By using the form (4.3) of the interaction operator O(x1,x3), it can

easily be shown that, for an arbitrary function F(xl,x3), the followi'

identities hold:


x St+e(XX3 )fl(x1,t+c)fl(x3,t+eV ) ,





77

(1 + P12) f dx38(x1,x3)6(x2-x3)F(x1,x3) = e(x1x2)F(x1,x2) (4.43)


(1 + P12) f dx3 e(x1,x3)8(x1-x29F(X1,x3)


= 6(x1-x2) f dx3O(x1,X3)F(x1,x3)


(4.44)


By introducing the above identities and considering the limit t >> c'

Eq.(4.42) can be written in the form,



F(xl,x2,t) = n 9(x1,x2) m(x1,X2)f1(x1,t)f1(x2,t)


+ 6(x1-x2)n f dx3e(xl,x3) m(x1,x3)f 1(x1,t)fl(x3,t)


- n[A(x1,tcf 1) + A(x25t If I )]56(x 1-x 2)f1(x15t)


(4.15)


or, from Eqs.(4.18) and (4.19),


r(x1,x2,t) = n K(x1,x2)f (x1,t)fl(x2,t) + n6(x1-x2)7[f,f]1



n[A(x1,tlfI) + A(x2,tifl)]6(xl-x2)f (xt) (4.46)


The equation for the equal time correlation function is then given by


(4.47)


ja + (1+P12)L(xitlf )IC2(x1,t;x2,t)= r(= IX 0


By neglecting again the difference in position of the colliding pair,

the various terms in Eq.(4.46) can be identified with Boltzmann-like






78

operators, as defined in Eqs.(4.25) and (4.37). In this limit, the

inhomogeneity F is given by



FB(xl,x2,t) = n 6(q-q 2)K(p 2 f(xl,t)fl(x2,t)



+ n6(x1-x )J[f 1,f i



n[I(x,tlfl) + I(x2,tlfl)]6(x-x2)f (xt) (4.48)



where K(p`1,2) is the kernel of the nonlinear Boltzmann operator,

defined by



f dp2K(pl'P2 )f (X,t)f (x2,t) = J[flf ] (4.49)



The dynamics of the equal time correlation function is determined by the

same operator that governs the time evolution of the two-time correla-

tion and again depends on the solution of the nonlinear macroscopic

problem. The linear equation for the equal time correlation function

however has now a inhomogeneous term, which is also a function of

fl(xi,t), i.e. of the state of the system.

The inhonogeneity in Eq.(4.47) assumes a familiar form when the

equilibrium case is considered for comparison. Equation (4.47) reduces

then to an identity, given by



(l+P12)n[I(Pi) + I o(2)]C2 (x;x2) = o(xl,x2) (4.50)


with







79

o(x1,x2) = n[I(1) + I(02)]6(x1-x2)(p1) (4.51)



where 4(p) is the Maxwell-Boltzmann distribution and I (p) is the usual

linearized Boltzmann operator. An identity equivalent to Eq. (4.50) is

obtained when equilibrium fluctuations are described through a

linearized Boltzmann-Langevin equation.53 The right hand side of the

equation is then identified with the amplitude of the noise in the

system. In a similar way the right hand side of Eq. (4.47) can be

interpreted as a measure of the noise in the system. It is a

complicated function of the nonequilibrium state through the one-

particle distribution function. This is a reflection of the new sources

of fluctuations, besides the thermal one, present in the nonequilibrium

state. As the random force in a Langevin description, the inhomogeneous

term r originates from microscopic excitations whose lifetime is shorter

than T which sets the time scale of the macroscopic description.

An alternative, and perhaps more popular, approach to the study of

phase space fluctuations in low density gases is based on a hierarchy

method analogous to the one used to derive the Boltzmann equation. Sets

of equal time and multitime distribution functions are defined as the

nonequilibrium averages of phase space densities .3 Hierarchies of

coupled equations for these distribution functions, formally identical

to the BBGKY hierarchy, are then derived from the Liouville equation.

The correlation of phase space fluctuations of equal and different time

argument are simply related to these distribution functions. A properly

reduced density can then be used as the expansion parameter to close the

hierarchies and derive kinetic equations for the correlation functions.






80

34-37
This approach has been used by several authors 7 In particular

Ernst and Cohen37 have derived kinetic equations for the nonequilibrium

correlation functions in a gas of hard spheres at low density by
3
performing an expansion in the parameter no and by retaining, to each

order in the density, only terms involving collisions among a limited

number of particles. The results derived in Ref. 37 agree with the ones

obtained here by using the generating functional method, but the

identification of terms of different order in the density is certainly

not transparent for space scales greater than a. The reason for this

lies in the choice of the expansion parameter. Ernst and Cohen applied

the method used when deriving a kinetic equation for the one-particle

distribution function. To close the first equation of the hierarchy

information on the two-particle distribution function, f2(x1,x2,t), over

distances smaller than or of the order of the force range a is needed in
3
this case. An expansion of f2 in powers of no is therefore

appropriate. The cluster expansion in fact gives information on the

variation of the distribution functions over space and time scales of

the order of the force range a and the collision time T
c
respectively. In low density gases, however, second or higher order

correlations among particles vary appreciably also over distances and

times of the order of the mean free path Z and the mean free time tf,

both much larger than a and T A different and more systematic method

of expansion of the hierarchy can then be developed by better analyzing

the role of the various length and time scales in the problem. If the

s-ch equation of the hierarchy is scaled according to thn two character-

istic lengths over which a variation of the correlation functions is

expected, I and tf, the expansion parameter a = (n3 )1 naturally






81

appears in the equation. The terms of an expansion of the s-th order

distribution function in powers of a can then be systematically

evaluated. The condition a<<1 corresponds to a gas at low density in

the usual sense (i.e. no3 << 1, since a (no3) 2), but with a large

number of particles in a volume of the linear size of a mean free

path. This is exactly the condition under which the effect of

collisions will be important over the space and time scale considered

and will be responsible for higher order correlations. The scaling of

the BBGKY hierarchy and the evaluation of correlation functions to

zeroth and first order in a is carried out in Appendix D for a gas of

hard spheres. Again the results agree with Ref. 37 and with those

obtained by using the generating functional method.



IV.5 Hydrodynamic Limit

In the low density limit, the average conserved densities in a

fluid are defined as



(t(r,t) = f dp a(p)nf1(x,t) (4.52)



where {a (p)l are the one-particle conserved quantities, defined in Eq.

(3.5). In this section, in order to simplify the notation, x is used to

indicate also the set of variables (r,p), where r represents a point in

the fluid and p is the canonical momentum. The hydrodynamic equations

are then obtained by averaging the microscopic conservation laws, Eqs.

(3.6), over the nonequilibrium one-particle distribution function and by

supplementing the equations so obtained with constitutive equations for

the macroscopic fluxes as functionals of the average densities.




82

A set of X-dependent average densities can be defined as



((r,tlX) = f dp a^(p)nfl(x,t|h) (4.53)


where f1[X] is the one-particle functional defined in Eq. (4.7). When

evaluated at h=0 (and for to=0), Eqs. (4.53) reduce, at all times, to

the usual average densities, defined in Eq. (4.52). The functionals

[((r,tlh)} are the low density limit of the ones defined in Chapter

III.

Again, intensive X-dependent thermodynamic variables {y (r,t|X)}

are defined by requiring



ca(r,tXl) = f dp a (p)fL(x,tlh) (4.54)



where fL is a one-particle local equilibrium functional, given by



f (x,t I) = exp{-ya (,thX)a (p)} (4.55)


The local equilibrium distribution is normalized as follows,



f dx fL(x,tlh) = N,



where N is total number of particles in the system. The thermodynamic

variables {y (r,t)} are given explicitly in Eq. (3.16). The comments

made in Chapter III, when requiring the equality (3.13), apply her2.

Hydrodynamic equations for the functionals {( (r,tJl)} are simply

obtained by projecting Eq. (4.17) for f1[X] over the five conserved

densities {a( p)1 By writing
a Q('






83

fI(x,tjx) = fL(xtlx) + A(x,t Ix)


(4.56)


and substituting in Eq. (4.17), an equation for A is obtained



+ Lo(x) n AL(x,tlfL)}A(x,t|\)
bt AL(x'tifQ)IA(x't IX)


= [ + Lo(x )]fL(x,tlX) + W(x,tl) ,


(4.57)


where, for consistency with the purpose of deriving equations to Navier-

Stokes order, only terms linear in A have been retained. The operator

AL in Eq. (4.57) is the local equilibrium form of the operator defined

in Eq. (4.29), i.e.


AL(x',t fL) = f dx2e(x1,x2) 5(xl,x2)(1+Pl3)fL(x,t |)


The solution of Eq. (4.57) is given by



A(x,tlX) = T(x;t,0)A(x,OIX) + ft dt T(x;t,)W(x,T X)
0


-ft d- T(x;t~t)La + L (x)I]f(x,-rIx)


(4.58)


(4.59)


where the time evolution operator T is X-dependent and is defined as the

solution of the equation,


1 6 + L (x) n I 0
3t 0 AL'x nhXxjctlfL,>=


(4.60)


with initial condition T(x;t,t) = 1. Also, the initial deviation from






84

the local equilibrium distribution functional is given by



A(x,OIX) = fL(x,Ol) (4.61)



By projecting now the equation for fl[X] over the conserved

densities {a } and making use of Eq. (4.54), the following set of

equations for the { a[X]} is obtained



-- (r,tX) + f- dp v. a (p)f(x,tk) + -i f dp-v. a (p)A(x,tl\)
7t a br i a L Or. i a
i 1


= f dp-a ()W(x,tl ) -- (4.62)



The term on the right hand side of Eq. (4.62) can be neglected. In fact

it contains the potential part of the momentum and energy fluxes which

are of higher order in the density and therefore negligible in the Boltz-

mann limit. By inserting Eq. (4.59) for A, the set of Eqs. (4.62) beccmes



a( + r Ea(rt; [Al})


~ ft dT f dp via (p)T(x;t,T)[-O + v*V ]f (x,Tlx)
or. o ia U r L
1


= dp v.a (p)T(x;t,0)A(x,Olk)
1


t Jt dr J dp v. a (p)T(x;t,T)W(x,T X) (4.63)
1


where the Euler matrix Ei has been defined as
aX







85

E (r,t;([hX}) J dp v. a (p)f (x,tl) (4.64)



Furthermore, to lowest order in the gradients,



[- + VV fL(X,TIX) = fL(x',lX) (P) (.65)
J


where

j(p) = (i P )v a (p) (4.66"



Here P l) is the low density form of the local equilibrium projection

operator defined in Eq. (B.9) and it is given by



P()A(a) = a (p)g(r) f dp' fL(r,p',tI)a (p')A() (4.67)
t P U ap a


where

gg(~) = f dp fL(x,t |)a(p)a (p) (4.6 )


-1
and g is the (ap) element of the inverse matrix. The thermodynamic
ap
variables {y } and their gradients appearing in Eq. (4.65) can also be

expanded in a functional Taylor series around the point t=-. To Navier-

Stokes order only the first terms in these expansions have to be

retained. Substituting Eq. (4.65) into (4.63), the latter becomes


--Y((rrt)t I+)
tatIX) r a af
^ a | {-(rt;{}) +r, })


a (.69)






86

where L is the matrix of the transport coefficients f r a low density

gas, given by



L a(r,t;{}) = lim dt J dx fL(x,t X)[T (x;t,T)via(p)
t>>tf


6(r-q)] j(p) (4.70)



For convenience, the same notation that was adopted to indicate the

matrix of transport coefficient for a dense fluid is used here for the

case of a low density gas. The two differ because the coefficient of

bulk viscosity vanishes at low density. The operator T in Eq. (4.70)

is the adjoint of the operator T and is defined by



f dx a(x)T(x;t,T)b(x)f (x,T) = f dx fl(x,z)b(x)T (x;t,T)a(x) ,

(4.71)


A A
for any pair of one-particle phase functions, a and b. Equation (4.70)

can be put in a more familiar form by observing that, to Navier-Stokes
(+ t
order, the hydrodynamic variables {y (r,t)} in the operator T can be

evaluated at -=t. The time evolution operator is then given by



T (x;t,t) = exp{[v.V- nI (x,t)](t-T)} (4.72)



where
--I
nIL(x,t) = fL (x,tlX)nL(xt fL)fL(xt X) (4.73)



The time integral in Eq. (4.70) can then be performed with the result,






87

Li(r,t;()) = f dx fL(x,t|x);j (p)[v V niL(x,t)]-1



xv.a (p)6(rq-) (4.74)



The action of the derivative operator in Eq. (4.74) on the operator

IL(q,p,t) or on fL(x,tl ) generates gradients of the thermodynamic

variables {y }. To lowest order these terms can be neglected and Eq.

(4.74) can be written as



L (r,t;{}) = J dqdp fL(r,p|x) (p)[.q d- nlL(r,p,t)]



x via (p)6(r-q) (4.75)

or

L (r,t;{}) = fdp fL(r,p ) (p)[n L(r,p,t)] v ) (4.76)



which is the familiar expression for the transport coefficients in the

Boltzmann limit, derived, for example, by using the Chapman-Enskog

expansion.8 The inhomogeneous term on the right hand side of Eq. (4.69)

is



W (r,tX) = J- dp via (p)T(x;t,0)A(x,0jX)
1


ft dr f dp via (p)T(x;t,T)W(x,tI) (4.77)
1


When evaluated at X=0, the right hand side of Eq. (4.69) vanishes

identically and the left hand side reduces to the nonlinear hydrodyna'niic

equations for a low density gas.






88

The correlation functions of the fluctuations of the conserved

densities around their average value are given by


S 6c (r ,t 1x)
M(rltl;r2,t2) = f dp2 a(p2) [6( t2) (4.73)



Equations for the correlation functions are then obtained by

functionally differentiating Eq. (4.69) and projecting the resulting

equation over the conserved densities. The case of correlation of

fluctuations at different time is first considered, with the result



I Map (r tl;r2t2) + rlt a (r't;r2't2


1
= I (r tl ;r2,t2) (4.79)


where
6W (r ,tI |)
I(rltl;r 2,2 lim f dp2a (p2 x2,t2) \=0 (4.80)
tl>>t f


The operator LK is given in Eq. (3.45) and depends on the state of the

system through the solution of the nonlinear hydrodynamic equations. In

the low density limit considered here, the terms containing the bulk

viscosity and the derivative of the pressure or of the enthalpy density

with respect to the density vanish, and do not appear in L ap

The inhomogeneous term on the right hand side of Eq. (4.79) is

analyzed in Appendix E. The contribution from the second term on the

right hand side of Eq. (4.77) is shown to vanish on a time scale of the

order of the collision time, Tc. Therefore, it does not contribute to

the hydrodynamic equations for the correlation functions, which describe










89

variations over time scales large compared to the mean free time, tf.

The functional derivative of the first term in Eq. (4.77) vanishes in

the limit

lim a(ltl;r2,t2) --- 0
tl>tf (4.81)
tl-t2>t

and gives no contribution to the equation for the unequal time

correlation functions, which reduces to the same linear homogeneous

equation obtained in Section 111.2, Eq. (3.49).

An equation for the equal time correlation functions can be

obtained by using the same limiting procedure applied for example in

Section 111.3 (see Eq. (3.63)). The resulting equation is


34
M rl,t;rt) + (1 + P2P ) : (rl ,t; {4 )M (r ,t;r,t)
it UP1 2 12 ap a1 2


S (r,r ;t) (4.82)
ap 1 2


where

V (' 0 2P- ) (raiV ,t) (4.83)
i(r +,r2;t) = lim (1+P1 P )I ,t+e;r2 ,t) (4.83)



Only the contribution to I from the first term on the right hand side

of Eq. (4.77) survives on the time scale of interest, leading to the

result,
































90

The set of hydrodynamic equations derived here to describe the time

evolution of averages and fluctuations strictly applies only to a low

density gas. With this restriction in mind, they are identical to the

equations obtained in Chapter III for a dense fluid.









CHAPTER V
TAGGED-PARTICLE FLUCTUATIONS IN SHEAR FLOW

V.1 Definition of the Problem

To illustrate the application of the formalism developed in the

previous chapters, the problem of steady shearing flow is considered.

Substantial simplifications occur as the result of three conditions:

(1) low density, (2) Maxwell molecules, and (3) uniform shear flow. In

fact for Maxwell molecules the irreversible momentum flux can be

determined exactly from the nonlinear Boltzmann equation, as a nonlinear

function of the shear rate. The macroscopic conservation laws can then

be closed and solved exactly for the situation of uniform shear. All

the information on the macroscopic state of the system, which is needed

in the equations for the correlation functions, can be evaluated for

arbitrarily large shear rate, as has been shown elsewhere.19,20,22-24

In particular the velocity-velocity autocorrelation function of a tagged

particle in the fluid is evaluated here. For the particular system

chosen, this correlation function can be calculated without any

approximation. Due to the presence of the shear on the system, equal

time correlations which are zero in equilibrium are now nonvanishing,

and their amplitude increases with the shear rate.








*Haxwell (or Maxwellian-) molecules are monoatomic molecules inte acting
through a weakly repulsive potential of the form V(r) = eo(o/r) where
r is the interparticle distance.






92

The steady shear flow corresponds physically to a fluid between two

parallel plates at a fixed distance apart and in relative motion. The
4 + *
flow field, U(r), is expected to vary linearly between the plates

(except near the surfaces) and be of the form,


U.(r,t) = U + a..r. .
1 oi 13


(5.1)


The constant vector Uoi and the constant tensor aij are the velocity

the lower plate and the shear rate tensor, respectively. In the

following Uoi will be set equal to zero without loss of generality.

the geometry considered, the tensor aij has the following properties:


aii = 0,



where no summation is intended in Eq. (5.2), and


a.j.a = 0 .
1i Jk


(5.2)


(5.3)


It is also assumed

only inhomogeneity



p(r,t) = p(t)



u(r,t) = u(t)


that the externally imposed shear flow represents the

in the system, and



-q *
q (r4t) = qi(t) ,
1 i


Sij (r ,t) = t ij(t)


(5.4)


*In this chapter the macroscopic flow velocity of the fluid is indicated
with U(r,t), instead of v(r,t), as was done in Ch. III.


I







93

Therefore, the walls of the system are not maintained at a constant

temperature (the extraction of heat from the system would create an

inhomogeneity in the temperature field). Heat is produced in the system

through viscous friction and the temperature grows in time. The state

considered is therefore nonstationary.

By inserting Eqs. (5.1-4) into the hydrodynamic equations for a

simple fluid, Eqs. (3.20), these reduce to



ap(t)
t- 0 (5.5)
at


Bu(t) *
u(t) = a. .t..(t) (5.6)
at 1j ij



The mass density p is therefore constant,



p(t) = po (5.7)



The internal energy density u can be considered a function of p and of

the temperature T (or the pressure p). The hydrodynamic equations

reduce then to one single equation, given by


BT(t) *
( aij(t)t (t) (5.8)
at 13 ij


OT
where a(t) = -u) In the case of a low density gas, the ideal gas
ou p
2 -1
equation of state can be used, to obtain a = (nK where

n = Po/m is the constant number density. The temperature equation

becomes then




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