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 FulbrightHays Travel
Grant, which allowed me to come to the University of Florida, and an
Educational Award from the Rotary International for the year 198182.
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 NavierStokes 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 TAGGEDPARTICLE FLUCTUATIONS IN SHEAR FLOW .......................91
1. Definition of the Problem ....................................91
2. Transformation to the Rest Frame .............................97
3. Twotime Velocity Autocorrelation Function ..................100
4. Equaltime Velocity Fluctuations ............................103
5. Results and Discussion ..................................... .105
VI DISCUSSION......................................................113
APPENDICES
A DERIVATION OF THE XDEPENDENT NONLINEAR NAVIERSTOKES
EQUATIONS............................................. .. ....... 118
B EVALUATION OF THE SOURCE TERM FOR
HYDRODYNAMIC FLUCTUATIONS......................................137
C DERIVATION OF THE XDEPENDENT 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 NavierStokes equations). From this equation the dynamics of
multispace 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 twotime 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 manybody
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 Nbody problem to a description in terms of
the oneparticle 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 oneparticle distribution function
is obtained by a low density closure of the BBGKY hierarchy.5,8 The
ChapmanEnskog 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 manyparticle
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 RayleighBrillouin
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 manybody 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',1924) 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 (GreenKubo
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,2633 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 OnsagerMachlup regression hypothesis to the nonequilibrium
case. The timedependent 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 authors264951 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
OnsagerMachlup 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 FokkerPlanck 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 OnsagerMachlup theory,
Graham25 has also analyzed in detail problems associated with the
stability and the breaking of symmetry in nonequilibrium stationary
states.
Several authors2932 have formulated studies of nonequilibrium
fluctuations based on a master equation in an appropriate stochastic
space. With the aim of generalizing LandauLifshitz'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
BoltzmannLangevin 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
used3438 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 twopoint 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.3942 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 fluctuationdissipation 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 Dorfman4346 developed a
hydrodynamic theory of nonequilibrium fluctuations in stationary states
based on the use of projection operator techniques and on the Kadanoff
Swift modecoupling 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 NavierStokes 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 NavierStokes equations. Additional
contributions appear as extra terms in the irreversible heat and
momentum fluxes. By functionally differentiating these generalized
NavierStokes 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 shortlived 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 velocityvelocity 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 6Ndimensional
space of the canonical coordinates.
In the language of statistical mechanics the state of the system at
the time t is described by the Nparticle 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
ith 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 nth 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
manybody 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 oneparticle
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
timecorrelation 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 timecorrelation
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 kth 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 twotime 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 (tlt)6A (t2t);0> (2.20)
Therefore the twotime 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 ktime 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, twoparticle 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(x1xi(t))
i=l
N N
f2(x1,x2,t) = I 6(x1xi(t))6(x2xi(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 (Ns)! 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 oneparticle 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 kth 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 twoparticle functions, etc., will
involve correlations of higher order phase space densities.
The equal time correlation functions of phase space fluctuations
introduced in Eqs.(2.2930) 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 oneparticle 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.(2.2931) 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 manybody problem to an effective one, two,
... sbody 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 (tX=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 manyparticle system (such as the Boltz
mann equation and the nonlinear NavierStokes 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 ChapmanEnskog 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 Xdependent 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.
(2.3536) 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 manyparticle 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 NavierStokes equations for the average densities and the non
linear Boltzmann equation for the oneparticle 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) appliesas long as the separation tlt2 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 twotime 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 NavierStokes 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 manyparticle 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(rq) ,
n=1 ,. .
N
g(r) = n 6(rqn
n=l
N 
g(P)= Pni6(rqn) (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 nth 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(rq) (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 onecomponent 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 onecomponent fluid is given by
pL(t) = exptqL(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 NavierStokes 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 NavierStokes 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 NavierStokes 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 NavierStokes
equations and preserves the properties of the generating functional.
In particular, a set of hdependent 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 Xdependent 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 Xdependent densities, { (r,t l)}.
The set of generalized NavierStokes equations for the five func
tionals {1 [X]} derived in Appendix A are formally identical to the
usual nonlinear NavierStokes 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
Xdependent 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 NavierStokes equations, appear in Eqs. (3.28). They originate from
the Xdependence 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 NavierStokes
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(tT) 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 NavierStokes 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 NavierStokes 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 Xdependent NavierStokes 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.(3.3839) are functionals of X
through the Xdependent 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,tx) = 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 irreversiblefluxes 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. (3.2325), such correlation functions can also be expressed
as the first functional derivatives of the Xdependent densities,
6
M ( tl ;r2 ,t2 ( =0} (3.43)
6s (r2,t2) a
A set of equations for the twotime correlation functions can then be
obtained by functionally differentiating the generalized NavierStokes
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 twentyfive 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 Xdependence 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 NavierStokes 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 NavierStokes 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 twotime 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 NavierStokes 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.26,3942,46 The
equations derived here represent therefore a generalization of these
results: they provide a hydrodynamic description of fluctuations, valid
to NavierStokes 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 NavierStokes 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 onecomponent 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(nm(q n))6(rq) (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. (3.5355) 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. (3.535), 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+
at 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 e0 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 ar)
(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 NavierStokes 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 twotime 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
fluctuationdissipation 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 fluctuationdissipation 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 NavierStokes 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 onecomponent fluid can be
obtained by using Eqs. (3.5859) 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 GreenKubo 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 authors26'27,3946 have recently formulated a description
of fluctuations in nonequilibrium hydrodynamic steady states which are
adequately described by the nonlinear NavierStokes 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
LandauLifschitz 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 semimicroscopic 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,3942 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 NavierStokes 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 Nparticle 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 oneparticle 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 oneparticle 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(tot) (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 eLt U[,]
... t ) = dx ..dx() .
s s' (Ns)! s+1 N N(O)
Nls
65
When evaluated at X=0, fs(xi,...,xs,tI) reduces to the sparticle
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 oneparticle distribution function5 can be
applied to the Xdependent 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
twoparticle distribution function for a dilute gas can be expressed in
terms of the oneparticle 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 sparticle 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 shortrange correlations (i.e. the sparticle 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 sparticle
s 1 s
functional analogous to the Nparticle 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 oneparticle 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,tX) 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
(sl)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 twotime correlation function, as long as tlt2 >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 twotime 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
oneparticle 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(xlx2) + 6(x3x2 ]
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 1t2 + 6(BX
1( )S (x3)[6(x1x2) + 5(x3x2)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 twotime 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 oneparticle distribution function in Eq.(4.37)
is evaluated at equilibrium (i.e. coincides with a MaxwellBoltzmann 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 threetime 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 stime 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 sth order
equation to all the lower order ones. As s increases, the equaicns
become therefore very complicated, but always conserve linearity.
75
Furthermore the solution of the equation for the sth 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.(4.2829). 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(x1X 2) + 6(x2x3)]
(4.41)
or, taking the limit,
P(x1,x2,t) (+P12) n f dx39(x1,x ){[S(:xx2) + 6(X3X2)] St(x1X3
s(x1,x2 )[6(x1x2) + 6(xx3)]}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(x2x3)F(x1,x3) = e(x1x2)F(x1,x2) (4.43)
(1 + P12) f dx3 e(x1,x3)8(x1x29F(X1,x3)
= 6(x1x2) 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(x1x2)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 1x 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(x1x2)7[f,f]1
n[A(x1,tlfI) + A(x2,tifl)]6(xlx2)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 Boltzmannlike
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(qq 2)K(p 2 f(xl,t)fl(x2,t)
+ n6(x1x )J[f 1,f i
n[I(x,tlfl) + I(x2,tlfl)]6(xx2)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 twotime 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(x1x2)(p1) (4.51)
where 4(p) is the MaxwellBoltzmann 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 BoltzmannLangevin 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
3437
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 oneparticle
distribution function. To close the first equation of the hierarchy
information on the twoparticle 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
sch 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 sth 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 oneparticle 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 oneparticle 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 Xdependent average densities can be defined as
((r,tlX) = f dp a^(p)nfl(x,th) (4.53)
where f1[X] is the oneparticle 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 Xdependent thermodynamic variables {y (r,tX)}
are defined by requiring
ca(r,tXl) = f dp a (p)fL(x,tlh) (4.54)
where fL is a oneparticle 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 Xdependent 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 dpv. a (p)A(x,tl\)
7t a br i a L Or. i a
i 1
= f dpa ()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(rq)] 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 oneparticle phase functions, a and b. Equation (4.70)
can be put in a more familiar form by observing that, to NavierStokes
(+ 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)](tT)} (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,tx);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,px) (p)[.q d nlL(r,p,t)]
x via (p)6(rq) (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 ChapmanEnskog
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)
tlt2>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
TAGGEDPARTICLE 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.19,20,2224
In particular the velocityvelocity 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. (5.14) 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