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\begin{document}
\title{Hydrodynamics and Fluctuations Outside of Local Equilibrium:\
Driven Diffusive Systems}
\author{Gregory L. Eyink\\{\em Department of Mathematics}\\ {\em Building No. 89}\\
{\em University of Arizona}\\{\em Tucson, AZ 85721}\\
{$\,$}\\
Joel L. Lebowitz\\{\em Departments of Mathematics and Physics }\\ {\em Rutgers University}\\
{\em Hill Center-Busch Campus}\\{\em New Brunswick, NJ 08903}\\
{$\,$}\\
Herbert Spohn\\{\em Theoretische Physik}\\ {\em Ludwig-Maximilians-Universit\"{a}t}\\
{\em Theresienstr. 37, 80333}\\ {\em M\"{u}nchen, Germany}}
\date{ }
\maketitle
\newpage
\begin{abstract}
We derive hydrodynamic equations for systems not in local thermodynamic
equilibrium, that is, where the local stationary measures are ``non-Gibbsian'' and do not
satisfy detailed balance with respect to the microscopic dynamics. As a main
example we consider the {\em driven diffusive systems}(DDS), such as electrical
conductors in an applied field with diffusion of charge carriers.
In such systems, the hydrodynamical description is provided by a nonlinear
drift-diffusion equation, which we derive by a microscopic method of {\em
nonequilibrium distributions}. The formal derivation yields a Green-Kubo formula for
the bulk diffusion matrix and microscopic prescriptions for the drift velocity
and ``nonequilibrium entropy'' as functions of charge density. Properties of
the hydrodynamical equations are established, including an ``H-theorem'' on
increase of the thermodynamic potential, or ``entropy,'' describing approach
to the homogeneous steady-state. The results are shown to be consistent with
the derivation of the linearized hydrodynamics for DDS by the Kadanoff-Martin
correlation-function method and with rigorous results for particular models.
We discuss also the internal noise in such systems, which we show to be
governed by a generalized {\em fluctuation-dissipation relation}(FDR), whose
validity is not restricted to thermal equilibrium or to time-reversible
systems. In the case of DDS, the FDR yields a version of a relation proposed
some time ago by Price between the covariance matrix of electrical current
noise and the bulk diffusion matrix of charge density. Our derivation of the
hydrodynamical laws is in a form---the so-called ``Onsager force-flux form''
---which allows us to exploit the FDR to construct the Langevin description of
the fluctuations. In particular, we show that the probability of large
fluctuations in the hydrodynamical histories is governed by a version of
the Onsager ``principle of least dissipation,'' which estimates the
probability of fluctuations in terms of the Ohmic dissipation required to
produce them and provides a variational characterization of the most probable
behavior as that associated to least (excess) dissipation.
Finally, we consider the relation of long-range spatial correlations in the
steady-state of the DDS and the validity of ordinary hydrodynamical laws. We
discuss also briefly the application of the general methods of this paper to
other cases, such as reaction-diffusion systems or magnetohydrodynamics of plasmas.
\end{abstract}
\newpage
{\bf TABLE OF CONTENTS}
\noindent{\bf 1. Introduction}
{\em (1.1) Description of the Program}......................................................................pp.5-7
{\em (1.2) Driven Diffusive Systems}.........................................................................pp.7-11
{\em (1.3) Outline of the Work}.................................................................................pp.11-13
\noindent{\bf 2. Nonequilibrium Distributions and Hydrodynamic Laws}
{\em (2.1) Nonequilibrium Thermodynamics of Driven Steady-States}.......................pp.13-20
{\em (2.2) The Nonequilibrium Distribution Formula}...............................................pp.20-26
{\em (2.3) Microscopic Derivation of Hydrodynamics}...............................................pp.26-38
{\em (2.4) Reversibility and Onsager Reciprocity}......................................................pp.33-38
{\em (2.5) Linear Response and the Einstein Relation}..............................................pp.38-40
\noindent{\bf 3. Current Noise and the Fluctuation-Dissipation Relation}
{\em (3.1) Review of Standard Theory}......................................................................pp.40-45
{\em (3.2) The Generalized FDR in Linear Theory}...................................................pp.45-50
{\em (3.3) The Linear Langevin Equation and Correlation Function Method}...........pp.50-56
{\em (3.4) Steady-State Correlations and Hydrodynamics}.........................................pp.56-60
\noindent{\bf 4. Large Hydrodynamic Fluctuations}
{\em (4.1) A Generalized FDR for the Nonlinear Fokker-Planck Equation}..............pp.60-67
{\em (4.2) Fluctuating Equations for the Driven Diffusion Model}............................pp.67-73
{\em (4.3) Onsager-Machlup Lagrangian and Least Excess Dissipation}....................pp.73-77
\noindent {\bf 5. Conclusions}
{\em (5.1) Outstanding Problems for DDS}................................................................pp.77-78
{\em (5.2) Remarks on the Nonequilibrium Distribution Method}..............................pp.78-79
{\em (5.3) Other Applications}...................................................................................pp.79-80
\newpage
\noindent {\bf 6. Appendices}
{\bf Appendix I. Are Stationary Measures of DLG Non-Gibbsian?}...................pp.80-82
{\bf Appendix II. Density, Chemical Potential, and Stationary Measures}......pp.82-85
{\bf Appendix III. The Asymmetric Simple Exclusion Process}.........................pp.40-45
{\bf Appendix IV. Steady-State Stability and the FDR of Kraichnan}...............pp.89-91
{\bf Appendix V. Microscopic Current Correlations of the DLG}......................pp.91-92
{\bf Appendix VI. Kadanoff-Martin Approach to Hydrodynamics in DLG}.....pp.92-95
\newpage
\section{Introduction}
\noindent {\em (1.1) Description of the Program}
This paper describes an approach to the derivation of hydrodynamical equations---
including also the fluctuations around the deterministic behavior due to internal
``molecular'' noise---for microscopic interacting particle systems. The method is quite general but
particular attention is given to cases where the system is not locally in thermodynamic equilibrium.
By ``local equilibrium'' we mean here that the system may be divided into cells small
on the macroscopic scale but each containing a large number of molecules whose statistics
are described by the equilibrium distributions (classical or quantum) of Gibbs. Many
nonequilibrium phenomena commonly encountered are in this class, including
ordinary hydrodynamics of fluids, heat conduction, Fickian diffusion of solutes, etc. Nonequilibrium situations of this
kind generally occur either through the relaxation of imposed initial inhomogeneities
of hydrodynamic densities or through the imposition of steady nonuniform thermodynamic
potentials on the boundaries. Therefore, even if very large overall departures of
the system from equilibrium occur, the small local regions are well-described
to lowest order in gradients by a Gibbs distribution with parameters slowly varying across the system.
Thermohydrodynamic fluid flow in B\'{e}nard cells is, for the convective regime with a large temperature
difference, a standard example which is globally far from equilibrium.
Nevertheless, there is another large class of nonequilibrium systems, e.g. those locally
driven by external fields, which can have statistics in small cells strongly
deviating from those of Gibbs. In such cases, there may still be a hydrodynamic behavior
associated to {\em locally stationary states} with slowly varying parameters, in which homogeneous,
stationary measures of the driven dynamics, generally non-Gibbsian, play the role of equilibrium
states. Examples of this sort include charge diffusion in semiconductors under applied electric
fields, chemical reaction-diffusion processes with a local supply of reagents, magnetohydrodynamics
of driven plasmas, etc. A crucial feature of these systems is that the local stationary measures
are {\em non-reversible}, i.e. they do not satisfy detailed balance with respect to the dynamics. Far less is
known for these systems than for the local equilibrium ones. For example, there is no generally
accepted prescription for the noise strength in a Langevin equation to describe the fluctuations
around such a state, such as the fluctuation-dissipation relation of equilibrium systems.
The method we use here to derive the hydrodynamic laws is a formalism based upon {\em nonequilibrium distributions}
which has been pioneered and applied by a number of authors, particularly H. Mori \cite{1}, D. N.
Zubarev \cite{2}, and J. A. McLennan \cite{3} among others. We may mention also the work of Ya. G. Sinai \cite{3a},
who made such techniques the basis of an attempt to rigorously derive the Euler hydrodynamic equations of
one-dimensional Hamiltonian particle systems. One of us (G.L.E.) has elsewhere
reviewed this method for local-equilibrium states of classical Hamiltonian
or quantum dynamics with conservation laws \cite{4}. The objective there was to discuss the mathematical foundations
of the method and to clarify the conditions required for a hydrodynamic description to be valid: the
separation of scales, dynamical properties of ergodic type, etc. Here let us just say that
the non-equilibrium distribution itself is a rigorous formula of Gibbs exponential type
for a time-evolved local equilibrium distribution, a kind of ``Girsanov formula'' for the density
with respect to a homogeneous reference measure. It is a suitable object for subsequent formal
expansion in the small ratio-of-scales parameter $\epsilon,$ which yields the constitutive laws
for fluxes and the hydrodynamic equations.
We shall discuss in this work the extension of this approach to systems without local equilibrium.
In particular, we will discuss the new features that appear with local stationary measures of non-Gibbsian
type and attempt to formulate explicitly the assumptions under which the derivations are obtained.
Our method is constructive, since we introduce basic quantities that appear in the hydrodynamical
laws by formally exact microscopic expressions which are calculable (in principle) analytically
and (in practice) by molecular dynamics. These quantities are: (i) a {\em nonequilibrium entropy},
or a ``fundamental equation'' for the entropy as function of the hydrodynamic densities, expressed
as a suitable large-volume limit; (ii) {\em drift} (or, {\em convection}){\em velocities} as
stationary averages of microscopic currents; and (iii) {\em Onsager relaxation coefficients}
or transport coefficients, given by Green-Kubo formulae in terms of the microscopic dynamics.
>From these microscopic quantities we construct also the statistical theories of
fluctuations about the deterministic hydrodynamic behavior. It is possible to develop
a nonequilibrium distribution method based upon the microscopic probability conservation
for the empirical distributions (a ``level-2'' approach) adequate to derive a hydrodynamic
Fokker-Planck equation for the fluctuations.
%* To clarify this remark, I add a foonote.
\footnote{The ``empirical distribution'' in this method
is just the delta-distribution $\hat{f}[\rho]=\prod_\br\delta(\rho(\br)-\hat{\rho}(\br)),$
supported on the density function of the $n$-body system, $\hat{\rho}(\br)=\sum_{i=1}^n
\delta(\br-\br_i).$ The latter is the ``empirical density,'' or level-1 object. It is easy to check
that the empirical distribution evolved under the microscopic dynamics satisfies a continuity equation
of the form $\partial_t \hat{f}_t[\rho]+\int\,d\br{{\delta}\over{\delta\rho(\br)}}\hat{J}_t[\br,\rho]=0.$
When this is averaged with respect to a suitable ensemble of initial conditions, the Fokker-Planck
equation results.} This has been done some time ago for local-equilibrium systems by Zubarev
and Morozov \cite{6}, applying a general method of Zwanzig \cite{7}. However, we shall
follow here a more economical procedure. In fact, it is an important principle of statistical
physics, emphasized by Einstein \cite{8} and Onsager \cite{9}, that fluctuations are determined
by hydrodynamics, so that a derivation of the latter also suffices to prescribe the
distribution law of fluctuations. In the present context this connection is given by a
{\em (generalized) fluctuation-dissipation relation} which we derive. For the small fluctuations
described by a linear Langevin equation, the stationary distribution must be calculated to
an approximation including correlations in the steady-state. For large deviations an ``Onsager-Machlup
action'' functional is constructed which gives the probability of spontaneous fluctuations of
the hydrodynamic histories, and which yields automatically a variational principle for the
most probable behavior, a version of Onsager's ``least-dissipation principle.''
\noindent {\em (1.2) Driven Diffusive Systems}
As a main example of the formalism we discuss in the text the {\em driven
diffusive systems} (DDS), and only consider other concrete areas of
application in the conclusion section. The DDS have a long history in statistical physics, going back
at least to the 1951 work of Wannier based upon a Boltzmann transport equation \cite{10,10a}.
Studying gaseous ionic conductors in a strong electric field, Wannier observed that inhomogeneities
of charge density imposed upon the current-carrying state slowly spread about their
drift displacement. He showed that the diffusion concept is still applicable in the nonequilibrium
system and was able to calculate the diffusion tensor for the driven state. There are a broad class of physical
situations in nature which exhibit a similar behavior. Certainly the technologically most important example
are the inhomogeneous semiconductor devices, such as the well-known P-N diode, which constitute
still an active field of research in condensed matter physics, electrical engineering and applied
mathematics : see \cite{11a} for a nice, up-to-date account.
Other physical examples include electrolyte solutions, certain solid electrolytes, and magnetohydrodynamics of plasmas.
Most of the past work on these systems has started from a kinetic equation
description. This should be adequate at low density of
charge carriers but will fail for higher concentrations. More
importantly, beginning with the kinetic Boltzmann level omits
consideration of fundamental issues of statistical physics for such systems.
Therefore, we study here instead a fully microscopic system, a lattice-gas with Kawasaki-type
stochastic dynamics, which was introduced earlier \cite{11} as a model of fast ionic conductors.
This so-called {\em driven lattice gas} or {\em DLG model}
of a driven-diffusive system has several advantages for our purposes. First, its simple dynamical
rules are ideal for computer realization and have allowed large-scale
simulations to be performed, revealing a wealth of unexpected phenomena.
For a review of these and a survey of present theoretical understanding,
see the recent article of Schmittmann and Zia \cite{12}. In addition,
there are for particular simple cases of DLG models rigorous results with
which to compare our theory, e.g. see \cite{13} and our Appendix III. Such checks are
important because the methods we use are formal and the physics is
largely unexplored, so that corroboration by rigorous results is very
reassuring when it is available. There are many deficiencies of the DLG models as a realistic
portrayal of nature \cite{11a} and one may expect technical problems in extending
our treatment to real systems with long-range Coulombic interactions. We believe
that the difficulties already present at this stage justify their removal
until a later time.
The DLG models were defined in the paper of Katz et al. (KLS) \cite{11} and
discussed systematically in Chapter 4 of \cite{14a}. Here we just recall
that particles in the model hop on a simple hypercubic lattice $\bZ^d$ in
$d$ dimensions with a hard-core exclusion condition (at most one particle per site). The particles are subjected to driving by an external electric field
$\bE$ and confined to a box $\Lambda\subset \bZ^d$ with periodic boundary
conditions. The stochastic hopping is ``thermally activated'' by a heat
reservoir at inverse temperature $\beta=1/k_BT,$ which absorbs the Ohmic power
produced by the field. More mathematically, the state space of the system
is the set of all possible particle configurations $\eta=\{\eta_\bx:
\bx\in\Lambda\}$, where $\eta_\bx$ is the site occupation variable
\be \eta_\bx=\left\{ \begin{array}{ll}
1 & \mbox{if $\bx\in\Lambda$ is occupied,} \cr
0 & \mbox{if $\bx\in\Lambda$ is empty.}
\end{array}
\right. \lb{1} \ee
The dynamics is prescribed by the exchange rates $c_E(\bx,\by,\eta)\geq 0$ between pairs of sites $\bx,\by\in\Lambda.$
Letting $\eta^{\bx\by}$ denote the configuration with occupations at sites $\bx$ and $\by$ interchanged, the
time evolution of a probability distribution $P_t(\eta)$ is given by the master equation
\begin{eqnarray}
{{d}\over{dt}}P_t(\eta) & = & {{1}\over{2}}\sum_{\bx,\by\in\Lambda}\left\{ c_E(\bx,\by,\eta^{\bx\by})P_t(\eta^{\bx\by})
-c_E(\bx,\by,\eta)P_t(\eta)\right\} \cr
\, & \equiv & L^*_{E,\Lambda} P_t(\eta). \lb{2}
\end{eqnarray}
The exchange rates are assumed, for simplicity, to be only nearest neighbor, i.e. $c_E(\bx,\by,\eta)=0$ for $|\bx-\by|>1.$
It is also useful to assume that $c_E(\bx,\by,\eta)>0$ for $|\bx-\by|=1$ and $\eta_\bx\neq\eta_\by$ to avoid degeneracies.
Another natural condition is space homogeneity, guaranteed by invariance of the rates under translation, i.e.
$c_E(\bx+\ba,\by+\ba,\tau_\ba\eta)=c_E(\bx,\by,\eta)$ where $(\tau_\ba \eta)_\bx=\eta_{\bx-\ba({\rm mod}\,\Lambda)}$ is
the periodically shifted configuration. Finally, and most importantly, we impose {\em local detailed balance} with respect to a
lattice-gas Hamiltonian $H(\eta)$ including the work done by the electric field $\bE$ in the jump. This has the form
\be c_E(\bx,\by,\eta)=c_E(\bx,\by,\eta^{\bx\by})e^{-\beta[H(\eta^{\bx\by})-H(\eta)]}e^{-q\beta \bE\cdot(\bx-\by)(\eta_\bx-\eta_\by)}, \lb{3} \ee
where $q$ is the particle charge and $H(\eta)$ is the Hamiltonian. For example, $H$ could be chosen as the Ising
model Hamiltonian $H(\eta)={{-J}\over{2}}\sum_{\bx,\by\in\Lambda,|\bx-\by|=1}\eta_\bx\eta_\by.$ If $E=0,$ then this
is the usual condition of detailed balance, which implies stationarity and reversibility of the Gibbs measure $P\propto e^{-\beta H}$
for the dynamics. For any value of field $\bE$ and number of particles $N$ the dynamics has a unique stationary measure $P_{E,N,\Lambda}$ in the
periodic domain $\Lambda.$ However, if $E\neq 0,$ then the measure $P_{E,N,\Lambda}$ supports a finite mean current and is non-reversible.
In this work we shall consider the hydrodynamical description of the relaxation of an inhomogeneous but slowly-varying
distribution of charge $\rho(\epsilon \bx)$ in the large domain $\epsilon^{-1}\Lambda,$ for $\epsilon\rightarrow 0.$
This is equivalent to the situation with a smooth distribution of charge $\rho(\bq)$ in the fixed domain $\Lambda$
as the lattice spacing $\epsilon\rightarrow 0.$ We shall derive a diffusion equation with drift to describe the relaxation
of this charge density on a macroscopic time scale $\tau\sim \epsilon^{-1}t$, of the form
\be \partial_\tau\rho(\bq,\tau)= -\grad\cdot[\hat{\jj}(\rho(\bq,\tau))-\en \bD(\rho(\bq,\tau))\cdot
\grad\rho(\bq,\tau)], \lb{3a} \ee
in which the drift current is given at low fields by Ohm's law as $\hat{\jj}(\rho)=\bsigma(\rho)\cdot\bE.$
It differs primarily from the ``drift-diffusion equation'' of semiconductor physics \cite{11a} in containing just one
sign of charge and in lacking the terms describing recombination and generation of electron-hole pairs. Microscopic
expressions for the quantities appearing in the equation will be one output of our derivation, as well as results
for its solutions such as an ``H-theorem,'' etc. It will be a chief concern of this paper to stress the
statistical and dynamical conditions underlying the validity of the equation. Such properties are commonly
assumed in the distribution function derivations of hydrodynamics but ordinarily only implicitly. This is particularly
important for DDS since the conditions may even be violated there!
A second main problem addressed in our work is the theory of fluctuations for these same systems.
It might be expected from analogy with other hydrodynamic systems that the fluctuations in the
charge density will be described by adding to the conservation law a ``noisy current,'' as
\be \partial_\tau\rho(\bq,\tau)= -\grad\cdot[\hat{\jj}(\rho(\bq,\tau))-\en \bD(\rho(\bq,\tau))\cdot
\grad\rho(\bq,\tau)+\en^{d/2}\bj^\prime(\bq,\tau)]. \lb{3b} \ee
In local equilibrium systems such fluctuating hydrodynamics has been long used, with the stochastic current
$\bj^\prime$ distributed according to a ``fluctuation-dissipation relation.'' We shall show that even for irreversible
systems such as the DDS there is a generalized fluctuation-dissipation relation. Furthermore, our derivation of the
hydrodynamics is in a form (so-called {\em force-flux form}) which allows us to exploit the FDR to write down
directly the Langevin equation for fluctuations.
Although the DLG models are our primary example, our {\em methods} are not at all restricted to that context
and there is little difficulty
in extending them to microscopic particle systems governed by classical Hamiltonian
or quantum dynamics. In fact, Zubarev and his collaborators have for some years employed
similar methods to derive results for physical nonequilibrium systems, including those
without local equilibrium. For example, see the early work of Kalashnikov on hot electron
kinetics in semiconductors \cite{13a}. At least in one simple {\em deterministic} model
of DDS type---the Lorentz gas with applied field and ``Gaussian'' thermostat---the program
we outline can be carried through in a mathematically rigorous way to derive Ohm's law
of charge transport \cite{14}.
\noindent {\em (1.3) Outline of the Work}
\noindent The remainder of this paper is organized as follows:
In Section 2 we give our derivation of the hydrodynamical law. We start with some preliminaries on the thermodynamics
of the stationary non-Gibbsian states of the dynamics. Thereafter, we derive the nonequilibrium distribution formula,
which is an exact and rigorously valid expression for a time-evolved local-equilibrium measure. Using this formula
as the starting point, the constitutive laws for fluxes and the hydrodynamical equations are derived by formal expansions
in the separation of scales parameter $\epsilon.$ The perturbation expansions are based upon an asymptotic method
\`{a} la Chapman-Enskog, exploiting the ``normal form'' of the nonequilibrium distribution. Mathematical rigor
is lost at this stage and various approximations, e.g. neglect of memory effects, are made which lack justification
and may even prove false for certain applications. However, we find the results to agree with the recent rigorous
theorems of Esposito et al. \cite{13} and Landim et al. \cite{76} for a simple example, the {\em asymmetric simple
exclusion process} (ASEP) \cite{13}. We conclude this section by discussing linear transport near $E=0$ including
consequences of reversibility, the celebrated {\em Onsager reciprocal relations}.
In Section 3 we turn our attention to the theory of internal noise for these systems at the linear level, i.e. for small
fluctuations about the steady-state. We derive the fluctuation-dissipation relation and emphasize the important fact
that it is a {\em macroscopic relation} independent of many details of the particle system, such as microscopic reversibility.
In particular, we obtain in this way a form of the {\em generalized Einstein
relation} for DDS proposed in 1965 by Price \cite{15}, which relates the covariance of microscopic current fluctuations and the bulk diffusion
tensor of charge inhomogeneities. Calculating the steady-state accurately to a Gaussian approximation, correctly incorporating the
steady-state two-point spatial correlations, we can use it to write down the linear Langevin equation for small fluctuations.
An alternative approach to deriving the linearized hydrodynamics can be based upon this theory---the ``correlation function
method'' of Kadanoff and Martin \cite{16},\cite{5}. This method was already employed in the 1984 paper of KLS
and we show that the results are consistent with those of the present approach.
However, we also show that the phenomenon of long-range power-law decay of
static correlations in the DLG models poses some severe difficulties for
the derivation of hydrodynamical laws in these systems.
Section 4 extends some of these results to the case of large fluctuations described by a nonlinear Fokker-Planck equation.
In this context there is an FDR, due in a general form to Graham \cite{17}, which allows us likewise to write down directly
the Fokker-Planck equation for the distribution of the fluctuating density field. It will hold under the same assumptions
which justify the hydrodynamic law itself. In the limit $\epsilon\rightarrow 0$ the probability of large fluctuations are
exponentially small and a precise asymptotic formula for the decay exponent can be derived by steepest descent from
a path-integral solution of the Fokker-Planck equation. This is also a procedure pioneered by Graham \cite{18}.
The result has the form of a nonlinear Onsager-Machlup action. Its physical interpretation is as an {\em excess
dissipation function} and it leads to a variational characterization of the most probable behavior as the state
of least excess dissipation.
Our concluding remarks are in Section 5 where we consider generalizations and limitations of the formalism. Some
material, subsidiary to the main discussion, is collected in various Appendices in Section 6.
\section{Nonequilibrium Distributions and Hydrodynamical Laws}
\noindent {\em (2.1) Nonequilibrium Thermodynamics of Driven Steady-States}
As a necessary preliminary to the discussion of hydrodynamics and fluctuations, we must
first address a different issue, the nonequilibrium thermodynamics of the DLG models, which
involves the large-volume limit $\Lambda\rightarrow \infty$ of its stationary, translation-invariant measures.
Thermodynamics plays a crucial role in the distribution function method, since generally in this method
the hydrodynamic density $n$ is spatially-modulated by the imposition of a local chemical potential $\lambda.$
It is therefore required to specify a functional relationship, $\lambda=\lambda(n)$ of the chemical
potential to produce a given density $n.$ The traditional nonequilibrium thermodynamic description
of electrical conducting systems postulates a local ``free energy function'' $f(\beta,n),$ in terms of
which an ``electrochemical potential'' may be defined by the isothermal derivative:
\be \mu=\left({{\partial f}\over{\partial n}}\right)_\beta. \lb{23b} \ee
See Landau and Lifshitz \cite{19a}, Sections 25-26. Such a description is appropriate for a
system in contact with a heat bath at inverse temperature $\beta.$ Alternatively, a ``nonequilibrium
entropy'' density $s(u,n)$ may be employed which is a function of internal energy $u,$ as in DeGroot and
Mazur \cite{19b}, Ch.XIII,3d, and Callen \cite{19c}, Ch. 17. These functions are presumed to have also
the relation to microscopic fluctuations proposed by Einstein for thermal equilibrium \cite{8}. Namely,
the probability to observe the density $n$ in a large volume $\Lambda$ should be proportional to
$\sim \exp[|\Lambda|\cdot s(u,n)/k_B],$ with $k_B$ the Boltzmann constant; see \cite{19b,19c}.
This aspect of the thermodynamic functions will play an important role in our discussion of fluctuations in the DDS.
However, except for $E=0$ or for special choices of the rates in $d=1,$ the stationary measures of the DLG models are not
Gibbs measures $\propto e^{-\beta H}$ with respect to the {\em a priori} Hamiltonian $H$ appearing in the definition
of the dynamics. (It is an open question, discussed in Appendix I, whether they are also not Gibbs for any
other Hamiltonian, or, more technically, not Gibbs probability measures for any summable potential \cite{19}.)
Therefore, standard proofs of the infinite-volume limit and of the validity of thermodynamic formalism for Gibbs measures
do not obviously apply. Furthermore, although the stationary measures of the DLG for fixed density are expected to be
space-ergodic, they have been observed numerically to have a very slow power-law decay of correlations for all temperatures
and densities when $E\neq 0,$ rather than the exponential clustering exhibited in Gibbs measures away from critical points
\cite{20}. This means that much of the standard lore of equilibrium thermodynamics has no sound mathematical foundation
for driven steady-states. The present section will review what little is known concerning the stationary
measures of the DLG in infinite volume, and, in particular, will attempt to systematize the standard hypotheses
on thermoelectric systems \cite{19a,19b,19c} in the context of the DLG models. All of our discussion will
be confined to the high-temperature region of those models. As for the Ising-type lattice gas at $E=0,$ a second-order
transition appears to occur at some critical $T_c,$ below which there is a two-phase coexistence region for
``liquid'' and ``gas'' phases with different densities. In this work, we stay well away from the critical point and
coexistence regions.
For finite $\Lambda,$ the basic facts about stationary measures
are quite straightforward to establish. Because the number of particles
$N$ is conserved, there is a stationary distribution satisfying
\be L^*_{E,\Lambda} P_{E,N,\Lambda}=0, \lb{4} \ee
for each $N=0,1,...,|\Lambda|.$ Since the jump rates are non-degenerate, these are unique,
and the theory of finite-state Markov chains ensures an exponential approach to stationarity.
$P_{E,N,\Lambda}$ must inherit the symmetries of the dynamics; for example, it is invariant under
(periodic) translations:
\be P_{E,N,\Lambda}(\tau_\ba\eta)=P_{E,N,\Lambda}(\eta). \lb{5} \ee
The zero-field measure $P_{0,N,\Lambda}$ is the canonical equilibrium state but no explicit expression is
generally available for $P_{E,N,\Lambda},$ which is defined only implicitly by Eq.(\ref{4}). The states with $E\neq 0,$
will support a nonvanishing number current of charge carriers
\be \bj(\bE)=\langle \bj_E(\bx)\rangle_{E,N,\Lambda}\neq 0, \lb{9a} \ee
where the $m$th vector component of the current,
\be j_{E,m}(\bx,\eta)=c_E(\bx,\bx+\ve_m,\eta)(\eta_\bx-\eta_{\bx+\ve_m}), \lb{9b} \ee
is the expected jump rate from $\bx$ to $\bx+\ve_m$ conditioned on being in the configuration $\eta.$
However, very little is rigorously known about the stationary states of the DLG dynamics for the limit
$\Lambda\uparrow\bZ^d.$ Therefore, we shall proceed by making plausible hypotheses. It seems reasonable to believe
that a (weak) limit exists,
\be P_{E,n}= w-\lim_{\Lambda\uparrow\bZ^d}P_{E,N,\Lambda}, \lb{6} \ee
when the density $n_\Lambda=N/|\Lambda|$ is selected so that
\be \lim_{\Lambda\uparrow\bZ^d}n_\Lambda=n. \lb{7} \ee
Existence of limits along suitable subsequences follows from a compactness argument, but one believes
that the limit is, in fact, independent of the choice of subsequence. The limit point $P_{E,n}$ is a
translation-invariant probability measure on the infinite-volume configuration space $\Omega=\{0,1\}^{\bZ^d}.$
It is also stationary under the infinite-volume version of the DLG dynamics, i.e. it satisfies
\be \langle L_E f\rangle_{E,n}=0, \lb{8} \ee
for any local function $f(\eta)$ depending upon a finite number of occupation numbers with
\be L_Ef(\eta)\equiv {{1}\over{2}}\sum_{\bx,\by\in\bZ^d}c_E
(\bx,\by,\eta)[f(\eta^{\bx\by})-f(\eta)]. \lb{9} \ee
Furthermore, the measures $P_{E,n}$ will presumably be the unique stationary, homogeneous measures of the dynamics
for each fixed density $n.$
The fundamental hypothesis we will need in our later development is the following: for any subset
$\Lambda\subset\bZ^d,$ define the {\em empirical density} $n_\Lambda(\eta)$ as
\begin{eqnarray}
n_\Lambda(\eta) & = & {{N_\Lambda(\eta)}\over{|\Lambda|}} \cr
& = & {{1}\over{|\Lambda|}}\sum_{\bx\in\Lambda}\eta_\bx. \lb{10}
\end{eqnarray}
Our basic assumption is then that
\begin{Hyp}
There exists, for a chosen ``reference density''
$n^*\in [0,1]$ (say, $n^*={{1}\over{2}}$), a concave function $s_E: [0,1]\rightarrow [-\infty,0]$ such that, for any of the intervals
$J=[a,b],(a,b],[a,b),(a,b)\subset [0,1],$
\be \lim_{\Lambda\uparrow\bZ^d}{{1}\over{|\Lambda|}}\log P_{E,n^*}\{n_\Lambda\in J\}= \sup_{n\in J} s_E(n). \lb{11} \ee
\end{Hyp}
Mathematically, this is an assumption of {\em large deviations} type \cite{22}. Our fundamental hypothesis
has not been rigorously established up until now for the stationary measures $P_{E,n^*}$ of the DLG dynamics
with $E\neq 0.$ Such a result has been proved for invariant measures of non-conservative, so-called ``attractive'' interacting
particle systems on $\Omega,$ for which an FKG property is satisfied \cite{21}.
The result means, heuristically, that
\be P_{E,n^*}\{n_\Lambda\approx n\}\sim \exp[ |\Lambda|\cdot s_E(n)], \lb{12}
\ee
and the function $s_E$ plays the same role as the entropy function in the Einstein formula for equilibrium fluctuations \cite{8}.
For this reason, we refer to $s_E(n)$ as the {\em nonequilibrium entropy}, or to $s=s_E(n)$ as the
{\em fundamental equation} in the language of Callen \cite{19c}. It would be more appropriate
to use the notation $s_E(n,\beta),$ but we shall not generally make the temperature-dependence explicit.
The physical interpretation will be discussed further below.
It is a consequence of the Hypothesis 1 that the limit
\be \lim_{\Lambda\uparrow\bZ^D}{{1}\over{|\Lambda|}}
\log\langle\exp(\lambda N_\Lambda)\rangle_{E,n^*}= p_E(\lambda),
\lb{13} \ee
exists for all real $\lambda,$ and that
\be p_E(\lambda)=\sup_{n\in [0,1]}(\lambda n+s_E(n)). \lb{14} \ee
This follows from Varadhan's theorem on asymptotics of integrals (e.g. see Section II.7 of \cite{22}).
In particular, the ``pressure function'' $p_E(\lambda)$ is convex in $\lambda.$Because $s_E(n)$ was assumed to
be concave, it follows that it is conjugate to $p_E$ in the sense that
\be s_E(n)=\inf_{\lambda\in\bR}(p_E(\lambda)-\lambda n), \lb{15} \ee
and, furthermore, the derivative
\be \lambda_E(n)= -s_E'(n) \lb{16} \ee
exists except possibly at a countable set of points in $[0,1].$ For $\beta$ small enough (or, for temperature high
enough), the function $s_E$ is expected to be strictly concave as a function of density. Therefore, the derivative $n_E(\lambda)=p_E'(\lambda)$
will exist for every $\lambda\in\bR.$
The latter fact has an important consequence. Let $P^*_{E,\Lambda}$ be the marginal measure induced by $P_{E,n^*}$
on $\Omega_\Lambda=\{0,1\}^\Lambda.$ Then, we may define an ``exponential family'' of measures $P^*_{E,\Lambda,\lambda}$ on $\Omega_\Lambda$ by
\be P^*_{E,\Lambda,\lambda}(\eta)={{1}\over{Z_{E,\Lambda}(\lambda)}}\exp[\lambda N_\Lambda(\eta)]P^*_{E,\Lambda}(\eta), \lb{17} \ee
with
\be Z_{E,\Lambda}(\lambda)=\langle\exp(\lambda N_\Lambda)\rangle_{E,n^*}. \lb{18} \ee
Because of the differentiability of $p_E,$ it can be shown that, for every $\epsilon>0,$
\be \lim_{\Lambda\uparrow\bZ^d}P^*_{E,\Lambda,\lambda}\{|n_\Lambda-n_E(\lambda)|<\epsilon\}=1. \lb{19} \ee
(For example, see Lemma VII.4.2 in \cite{22}.) In other words, the empirical density $n_\Lambda(\eta)$ distributed
with respect to the measure $P^*_{E,\Lambda,\lambda}$ takes on the value nearly equal to $n_E(\lambda)$ with
overwhelming probability in a large region $\Lambda.$ Under our assumptions it is possible to show even a little more,
namely, that the large-deviations hypothesis made for the single reference density $n^*$ in fact implies the same property
for the exponential modifications:
\be \lim_{\Lambda\uparrow\bZ^d}{{1}\over{|\Lambda|}}\log P^*_{E,\Lambda,\lambda}\{n_\Lambda\in J\}= \sup_{n\in J} s_E(n|\lambda), \lb{21} \ee
with
\be s_E(n|\lambda)= s_E(n)+\lambda n-p_E(\lambda). \lb{22} \ee
For example, see Theorem 3 in \cite{21}. This implies that for any choice of reference density $n,$ the Hypothesis 1
should hold for the measure $P_{E,n}$ also with ``entropy function'' $s_E(\cdot|\lambda_E(n)).$
The main point we wish to stress here is that, assuming only existence and concavity of $s_E,$ it is possible to change the
density to any desired value $n$ by making the exponential modification with the chemical potential $\lambda=-s_E'(n)$
of the reference measure. In particular, there is no need to assume the stationary measure to be Gibbsian for this
procedure to be valid. Unfortunately, while the use of the chemical potential as in Eq.(\ref{17}) to change the density
from $n^*$ to $n$ appears reasonable, it is not necessarily the ``correct'' method. One can argue that, since the number
$N_\Lambda(\eta)$ of particles in a large volume $\Lambda$ is conserved up to flux across the boundary, the members of
the exponential family in Eq.(\ref{17}) will be nearly stationary for a long time interval. In fact, it is to be expected
that
\be P_{E,n_E(\lambda)}=w-\lim_{\Lambda\uparrow\bZ^d}P^*_{E,\Lambda,\lambda}. \lb{20} \ee
so that the (presumably unique) stationary measures $P_{E,n}$ with other densities $n\neq n^*$ ought to be recoverable
from the infinite-volume limits of $P^*_{E,\Lambda,\lambda}$ with $\lambda=-s_E'(n).$ In the general context there
is little we can say on this point, but for the specific case of the DLG models we can go a little further. It is
shown in Appendix II that, under the assumption that {\em one} of the stationary measures of the DLG model
is canonical Gibbs with a suitable Hamiltonian, then any of its ergodic, invariant measures is Gibbs for the
same Hamiltonian and for a certain choice of chemical potential. The proof depends upon the extension of a
result of K\"{u}nsch \cite{23} for stochastic spin-flip dynamics to our conservative models (due to A. Asselah),
which states that if any translation-invariant, stationary measure of the DLG is canonical Gibbs with some Hamiltonian
then all of them are canonical Gibbs with the same Hamiltonian. Of course, this does not settle the issue even
for the DLG, since the invariant measures may well not be Gibbs. However, it does give some firmer dynamical
grounds to our procedure. In general, our method must be judged solely on the basis of its results.
The concavity part of our hypothesis may be regarded as a kind of ``stability assumption'' on the stationary measures
$P_{E,n}.$ For example, it implies that $\chi_E^{-1}= \partial\lambda_E/\partial n= -d^2s_E/dn^2 >0.$ Therefore, an
increase in the ``potential'' $\lambda$ is required to raise the density $n.$ Also, if the fluctuation formula analogous
to Eq.(\ref{12}) is considered for $P_{E,n}$---which follows from the analogue of Eq.(\ref{21})---then in a quadratic
approximation
\be P_{E,n}\{n_\Lambda\approx n'\}\sim \exp[-|\Lambda|\cdot \chi_E(n)(n'-n)^2/2], \lb{23} \ee
and the probabilities of spontaneous fluctuations of the density away from $n$ are damped out. Concavity of the entropy also implies heuristically
a stability property in the coupling of systems in two large regions $\Lambda_1,\Lambda_2$ both distributed according to the measure $P_{E,n}.$
The two systems are imagined to be coupled at a common boundary in an arbitrary way permitting free exchange of particles. If
the volume-fractions $\lambda_1=|\Lambda_1|/|\Lambda|,\lambda_2=|\Lambda_2|/|\Lambda|$ are fixed and it is assumed that correlations
of the subsystems are an ignorable boundary effect in the limit $|\Lambda|\rightarrow \infty,$ then
\begin{eqnarray}
P(n_\Lambda\approx n) & \sim & \sup_{\{n_1,n_2:\lambda_1n_1+\lambda_2n_2=n\}}\left\{P(n_{\Lambda_1}\approx n_1)P(n_{\Lambda_2}\approx n_2)\right\} \cr
\, & \sim & \exp\left\{|\Lambda|\sup_{\{n_1,n_2:\lambda_1n_1+\lambda_2n_2=n\}}[\lambda_1s_E(n_1)+\lambda_2s_E(n_2)]\right\} \cr
\, & \sim & \exp[|\Lambda|s_E(n)], \lb{23a} \end{eqnarray}
where the assumption of concavity was used in the last line. In other words, the distribution of density fluctuations is unchanged for
the compound system and is something like an asymptotically ``stable law'' in the sense of limit theorems of probability.
We believe the concavity is likely to hold for the systems we consider, or in general for steady-states with such stability.
To conclude this section, let us make a few explanatory remarks concerning the physical interpretation of the various
quantities introduced above. Since the lattice-gas dynamics is at a fixed temperature, the appropriate thermodynamic
potential is actually the {\em free-energy function} $f_E(n,\beta),$ with units of energy per volume, related to
our ``entropy'' as
\be s_E(n,\beta)= -\beta f_E(n,\beta). \lb{12a} \ee
For notational convenience, however, we have preferred to work with the ``entropy'' $s_E$ with units of inverse volume.
Our parameter $\lambda$ is likewise related to the usual electrochemical potential $\mu$ as $\lambda=\beta\mu.$
In Section 4.3 of this paper we shall show that $f$ has a simple operational significance as the minimum energy dissipated
by external fields required to change the density from its reference value $n^*$ to a new value. We have chosen to work
with number density $n,$ but we could just as well have used the charge density $\rho=q\cdot n.$ In that case total
charge $Q_\Lambda$ would replace particle number $N_\Lambda$ in the preceding arguments. The quantity $\varphi=\mu/q$
can then be interpreted as the ``electric potential'' of the system. If the need arises, we may denote the
``number current'' previously introduced as $\bj^N,$ and the ``electric current'' $q\cdot \bj^N$ as $\bj^Q.$
\noindent {\em (2.2) The Nonequilibrium Distribution Formula}
The problem we wish to study is the relaxation of an inhomogeneous density distribution $n_0(\bq)$ imposed upon the
homogeneous measure $P_{E,n^*}.$ A basic condition necessary for validity of a hydrodynamical description is the
{\em separation of scales} between length- and time-scales associated to the microscopic conservative dynamics
and the length- and time-scales over which the macroscopic density variations occur. We always denote by $\en$ the small
ratio of micro/macro length-scales, which is here just $\en=$(lattice-spacing)/(gradient-length). Thus, we wish to consider
states in which the density variation in
lattice units is $n_0(\en\bx).$ From the discussion in the previous section, one guesses that an appropriate way to modify
the reference measure $P_{E,N^*,\Lambda}$ is by adding a local chemical potential term. More precisely, we define a
{\em (canonical) local stationary measure} in the scaled domain $\Lambda_\en=\left(\en^{-1}\Lambda\right)\cap \bZ^d$ as
\be P_\en^{ls}(\eta)={{1}\over{Z_\en^{ls}}}\exp\left({\sum_{\bx\in\Lambda_\en}\lambda_0(\en\bx)\eta_\bx}\right)P_{E,N^*_\en,\Lambda_\en}(\eta), \lb{24} \ee
with
\be Z_\en^{ls}=\sum_{\eta\in\Omega_{\Lambda_\en}}\exp\left({\sum_{\bx\in\Lambda_\en}\lambda_0(\en\bx)\eta_\bx}\right)
P_{E,N^*_\en,\Lambda_\en}(\eta). \lb{25} \ee
Here $\lambda_0(\bq)$ is the smooth chemical potential field given by the ``local prescription''
\be \lambda_0(\bq)= -s'(n_0(\bq)). \lb{26} \ee
(We drop $E$-subscripts in this section when no confusion will result.) Also, $N^*_\en=[\![n^*\cdot\Lambda_\en]\!],$ where $[\![\cdot]\!]$ denotes
the integer part. For simplicity in the following expressions, we shall often abbreviate $P_{E,N^*_\en,\Lambda_\en}=P^*_\en.$ We consider always periodic
b.c. on $\Lambda.$
The ``nonequilibrium distribution formula'' is an exact expression for the time-evolution of the initial measure $P_\en^{ls}$
under the DLG dynamics. It can be written in the ``weak'' form for averages, as:
\begin{eqnarray}
\,& &\langle e^{\en^{-1}\tau L}f\rangle^{ls}_\en=\langle f\rangle^{ls,\tau}_\en \cr
\,& &\,\,\,\,\,\,\,\,\,\,-\en\int_0^{\en^{-1}\tau}ds\sum_{\bx\in\Lambda_\en}\left(\partial_m^+\lambda(\en\bx,\tau-\en s)
\langle(\Delta_{\bx,\bx+\ve_m}e^{sL}f)I_{\en,m}^{\tau-\en s}(\bx)\rangle^{ls,\tau-\en s}_\en \right. \cr
\,& &\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+\left. \partial_\tau\lambda(\en\bx,\tau-\en s)
\langle(e^{sL}f)(\eta_\bx-\langle\eta_\bx\rangle^{ls,\tau-\en s}_\en)\rangle^{ls,\tau-\en s}_\en\right).
\lb{27}
\end{eqnarray}
We must explain our various notations. The parameter $\tau$ is a ``macroscopic time,'' associated to a microscopic time $\en^{-1}\tau$
which goes to infinity as $\en\rightarrow 0.$. The formula itself is valid without any need for $\en$ to be small
(e.g. it is true with $\en=1$), but we wish to consider the evolution of the density distribution slowly varying on the spatial scale
$\sim \en^{-1}.$ A change of order 1 in the density occurs in a time $\sim \en^{-1}$ through the transport of particles over
distances $\sim \en^{-1}$ by the local drift velocity. The formula depends upon a space-time chemical potential field $\lambda(\bq,\tau)$
in the macroscopic variables which---we emphasize---is at this stage
{\em arbitrary} except for the initial condition
$\lambda(\bq,0)=\lambda_0(\bq)$ and some reasonable smoothness.
The expectation $\langle\cdot\rangle_\en^{ls,\tau}$ is with respect to the
local stationary distribution
for the profile $\lambda(\cdot,\tau).$ For any choice of a chemical potential profile, $\bI_\en $ is a ``modified current'' defined by
\begin{eqnarray}
I_{\en,m}(\bx,\eta) & \equiv & c(\bx,\bx+\ve_m,\eta)
\left[{{{\exp\left((\lambda(\en(\bx+\ve_m))-\lambda(\en\bx))(\eta_\bx-\eta_{\bx+\ve_m})\right)-1}}
\over{\lambda(\en(\bx+\ve_m))-\lambda(\en\bx)}}\right] \cr
\, & = & j_m(\bx,\eta)+O(\en). \lb{28}
\end{eqnarray}
The symbol $\partial^+_m$ denotes the operation of forward difference, $(\partial^+_m\lambda)(\bq)\equiv{{\lambda(\bq+\en\ve_m)-\lambda(\bq)}\over{\en}},$
and $\Delta_{\bx\by}$ is the ``exchange operator,'' $(\Delta_{\bx\by})f(\eta)\equiv f(\eta^{\bx\by}).$ We use the summation convention for the
repeated index $m$ (except in Eq.(\ref{28})) and also below for other repeated Latin indices. The basic advantage of the ``nonequilibrium
distribution formula'' is that it separates the true time-evolved distribution into a leading part of order $\sim 1,$ which is given by
a canonical local stationary distribution with a reference profile $n(\cdot,\tau),$ and a correction term of order $\sim \en$
involving microscopic space-time correlations.
The proof of Eq.(\ref{27}) is straightforward, based upon a simple trick using the fundamental theorem of calculus. In fact, it is easy to see that
\begin{eqnarray}
\langle e^{\en^{-1}\tau L}f\rangle^{ls}_\en
& = & {{1}\over{Z_{\en}^{ls,\tau}}}\sum_\eta P^*_\en(\eta)\exp\left(\sum_{\bx\in\Lambda_\en}\lambda(\en\bx,\tau)\eta_\bx\right)f(\eta) \cr
\,& &\,\,\,\,\,\,\,\,\,+\sum_\eta\int_0^{\en^{-1}\tau}ds\,\,{{d}\over{ds}}
\left[{{\exp\left(\sum_{\bx\in\Lambda_\en}\lambda(\en\bx,\tau-\en s)\eta_\bx\right)}\over{{Z_{\en}^{ls,\tau-\en s}}}}(e^{sL}f)(\eta)\right]
P^*_\en(\eta), \lb{29}
\end{eqnarray}
where $Z^{ls,\tau}_\epsilon$ is as defined in Eq.(\ref{25}) but with $\lambda_0(\en \bx)$ replaced by
$\lambda(\en\bx,\tau).$ Carrying out the differentiation inside the integral then gives
\begin{eqnarray}
\langle e^{\en^{-1}\tau L}f\rangle^{ls}_\en
& = & \langle f\rangle^{ls,\tau}_\en \cr
\,& &\,\,\,\,+\sum_\eta\int_0^{\en^{-1}\tau}ds\,\,{{1}\over{{Z_{\en}^{ls,\tau-\en s}}}}\left[(Le^{sL}f)(\eta)\right. \cr
\,& &\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left. -\en\sum_{\bx\in\Lambda_\en}\partial_\tau\lambda(\en\bx,\tau-\en s)
(\eta_\bx-\langle\eta_\bx\rangle^{ls,\tau-\en s}_\en)\times (e^{sL}f)(\eta)\right] \cr
\,& &\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
\times\exp\left(\sum_{\bx\in\Lambda_\en}\lambda(\en\bx,\tau-\en s)\eta_\bx\right)P^*_\en(\eta). \lb{30}
\end{eqnarray}
%*I add a sentence of explanation about the adjoint.
In the first term in the square brackets one obtains the factor $(L^*P_\en^{ls,\tau-\en s})(\eta)$ by taking
the adjoint $L^*$ of the operator $L$ with respect to counting measure, which is given by Eq.(\ref{2}).
For any local stationary distribution this factor is easily calculated to be
\be (L^*P_\en^{ls})(\eta)=
-\en \sum_{\bx\in\Lambda_\en}\partial^+_m\lambda(\en\bx)\Delta_{\bx,\bx+\ve_m}\left[I_{\en,m}(\bx,\eta)P^{ls}_\en(\eta)\right], \lb{31} \ee
when the stationarity condition $(L^*P^*_\en)(\eta)=0$ is used. If this result is now substituted into Eq.(\ref{30}), the originally
claimed formula is obtained.
Another useful form of the expression can be established by noting that, for any local stationary distribution,
\be \sum_{\bx\in\Lambda_\en}\partial^+_m\lambda(\en\bx)\langle I_{\en,m}(\bx)\rangle^{ls}_\en=0 \lb{32} \ee
as an exact identity. This will also turn out to be very important in the discussion of hydrodynamics in the next subsection.
It is easily proved by summing equation Eq.(\ref{31}) over $\eta\in\Omega_{\Lambda_\en}.$ As a consequence of this identity, the
entire distribution formula can be expressed in terms of truncated expectations $\langle fg\rangle^\top=\langle fg\rangle-
\langle f\rangle\langle g \rangle,$ as
% \newpage?
\begin{eqnarray}
\,& &\langle e^{\en^{-1}\tau L}f\rangle^{ls}_\en=\langle f\rangle^{ls,\tau}_\en \cr
\,& &\,\,\,\,\,\,\,\,\,\,-\en\int_0^{\en^{-1}\tau}ds\sum_{\bx\in\Lambda_\en}\left(\partial_m^+\lambda(\en\bx,\tau-\en s)
\langle(\Delta_{\bx,\bx+\ve_m}e^{sL}f)I_{\en,m}^{\tau-\en s}(\bx)\rangle^{ls,\tau-\en s,\top}_\en \right. \cr
\,& &\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+\left. \partial_\tau\lambda(\en\bx,\tau-\en s)
\langle(e^{sL}f)\eta_\bx\rangle^{ls,\tau-\en s,\top}_\en\right)+o(\en).
\lb{33}
\end{eqnarray}
The $o(\en)$ error term arises from the fact that $\langle\Delta_{\bx,
\bx+\ve_m}e^{sL}f\rangle^{ls,\tau-\en s}_\en$ becomes independent of
$\bx$ only in the limit $\en\rightarrow 0.$ This new form is very
crucial in taking that limit.
Until this last point we have not made any simplifications based upon small $\en$ and the results are formally exact. We
have also not made any particular choice of reference profile $\lambda(\cdot,\tau),$ except for the required initial
condition. However, we wish now to make an evaluation of the formula which should be asymptotically exact for
$\en\rightarrow 0.$ In particular, we wish to approximate the expectation $\langle e^{\en^{-1}\tau L}
\tau_{[\![\en^{-1}\bq]\!]}f\rangle^{ls}_\en,$ when $f$ is a local function supported
in a neighborhood of the origin shifted to the lattice point
$[\![\en^{-1}\bq]\!].$ The first term in the desired approximation ought
to be the ``local steady-state'' part, in which the average is with
respect to the homogeneous stationary measure at density $n(\bq,\tau)$
and the second term is a correction. Of course,
to have a chance of getting a small correction of order $\sim \en$ it is necessary to
choose the correct hydrodynamic profile in the leading term of order $\sim 1,$
which should be determined by the solution of the ``Euler-level''
hydrodynamic equation. The resulting formula, which characterizes the
behavior of the time-evolved measure in the neighborhood
of the space-time point $(\bq,\tau)$ is:
\begin{eqnarray}
\,& &\langle e^{\en^{-1}\tau L}\tau_{[\![\en^{-1}\bq]\!]}f\rangle^{ls}_\en=\langle f\rangle_{n(\bq,\tau)} \cr
\,& &\,\,\,\,\,\,\,\,\,\,+\en\partial_m\lambda(\bq,\tau)\sum_{\bx}x_m\langle \eta_\bx f\rangle^\top_{n(\bq,\tau)} \cr
\,& &\,\,\,\,\,\,\,\,\,\,-\en\partial_m\lambda(\bq,\tau)\int_0^{\infty}dt\,\,\sum_{\bx\in\bZ^d}
\left[\langle(\Delta_{\bx,\bx+\ve_m}e^{tL}f)j_m(\bx)\rangle^\top_{n(\bq,\tau)} \right. \cr
\,& &\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
-\left. \chi((n(\bq,\tau))\hat{\j}_m'\left(n(\bq,\tau)\right)\hat{f}'\left(n(\bq,\tau)\right)\right]+o(\en).
\lb{34}
\end{eqnarray}
We have introduced a new notation
\be \hat{f}(n)\equiv \langle f\rangle_n, \lb{35} \ee
where the expectation is with respect to the infinite-volume stationary measure at density $n$ and $\hat{f}'$ denotes derivative with respect to $n.$
(We also use the same notation $\hat{f}(\lambda)$ for $\hat{f}(n(\lambda)),$ hopefully without any confusion.)
In Eq.(\ref{34}) we have now made a special selection of the reference profile, as the solution of the initial-value problem for the first-order hyperbolic equation
\be \partial_\tau\lambda(\bq,\tau)=-\hat{\Bj}'(n(\lambda(\bq,\tau)))\cdot\nabla\lambda(\bq,\tau). \lb{36} \ee
As we discuss in the following subsection, this is indeed the ``Euler'' hydrodynamic equation of the system, correct to
describe the evolution of the density for microscopic times $\sim \en^{-1}.$ We shall see below that it is this choice of
the profile which produces the term proportional to $\hat{\Bj}'(n)$ in Eq.(\ref{34}). A formal argument, given in
Appendix 3 of \cite{11}, shows that this is precisely the subtraction necessary for the convergence of the time-integral.
The derivation of the asymptotic formula Eq.(\ref{34}) is no longer rigorous, but is based upon plausible formal reasoning. The first two terms come
from an evaluation of the local stationary expectation in Eq.(\ref{33}). In fact, if one approximates
\be \lambda(\en\bx,\tau)=\lambda(\bq,\tau)+\en\grad\lambda(\bq,\tau)\cdot(\bx-\en^{-1}\bq)+O(\en^2) \lb{37} \ee
for $\en\bx\approx\bq$ and expands out the exponent in the canonical form, one obtains
\be \langle \tau_{[\![\en^{-1}\bq]\!]}f\rangle^{ls,\tau}_\en=\langle f\rangle_{n(\bq,\tau)}
+\en\partial_m\lambda(\bq,\tau)\sum_\bx x_m\langle\eta_\bx f\rangle^\top_{n(\bq,\tau)}+o(\en). \lb{38} \ee
To evaluate the time-integral term, one first substitutes Eqs.(\ref{28}) and (\ref{36}), to obtain for it
\begin{eqnarray}
\,& &\en\int_0^{\en^{-1}\tau}ds\sum_{\bx\in\Lambda_\en}\partial_m\lambda(\en\bx,\tau-\en s)
\left[\langle \Delta_{\bx,\bx+\ve_m}e^{sL}\tau_{[\![\en^{-1}\bq]\!]}f\cdot j_m(\bx)\rangle^{ls,\tau-\en s,\top}_\en\right. \cr
\,& &\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
-\left. \hat{\j}_m'(n(\en\bx,\tau-\en s))\langle e^{sL}\tau_{[\![\en^{-1}\bq]\!]}f\cdot \eta_\bx\rangle^{ls,\tau-\en s,\top}_\en\right]+o(\en).
\lb{39}
\end{eqnarray}
If the truncated correlations decay rapidly enough on the microscopic scale, then the time-integral and space-sum in the above expression
get their main contributions from the regions $s\approx 0$ and $\en\bx\approx\bq.$ In that case, the reference profile may be evaluated
everywhere to leading order by its value at space-time point $(\bq,\tau)$ and the integral and sum may be extended to infinity. In this way, the
previous expression is estimated as
% \newpage?
\begin{eqnarray}
\,& &\en\partial_m\lambda(\bq,\tau)\int_0^\infty ds
\left[\sum_{\bx\in\bZ^d}\langle \Delta_{\bx,\bx+\ve_m}e^{sL}f\cdot j_m(\bx)\rangle^\top_{n(\bq,\tau)}\right. \cr
\,& &\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
-\left. \hat{\j}_m'(n(\bq,\tau))\sum_{\bx\in\bZ^d}\langle e^{sL}f\cdot \eta_\bx\rangle^{\top}_{n(\bq,\tau)}\right]+o(\en).
\lb{40}
\end{eqnarray}
However, appealing to the fact that the state $\langle\cdot\rangle_n$ may be obtained by adding the chemical potential
term $\exp[\lambda\sum_\bx \eta_\bx]$ to the reference measure in finite volume, one finds by a formal exchange of limits that
\begin{eqnarray}
\hat{f}'(n) & = & {{1}\over{\chi(n)}}\hat{f}'(\lambda) \cr
\,& = & {{1}\over{\chi(n)}}\sum_\bx\langle f\eta_\bx\rangle^\top_n. \lb{41}
\end{eqnarray}
Substituting this expression into the final term of (\ref{40}), one checks that the claimed asymptotic formula Eq.(\ref{34})
is the result. Although no careful justification was provided for the various steps in the previous argument, it is clear
that the main requirement is a rapid decay of the truncated space-time correlation functions. This certainly restricts us to dimensions $d>2,$ since
there are non-integrable long-time tails in dimensions $d=1,2$
(see Appendix III.) If the assumption is violated---for any reason
---then a similar evaluation could still be made, but with a result
nonlocal in the space-time profile $\lambda(\bq',\tau')$ and involving an integral kernel defined in terms of truncated correlations.
The assumption of fast decay has allowed us to ``Markovianize'' the expression.
To conclude this subsection, we will observe that the same types of formulae, like both the exact expression Eq.(\ref{27}) and the
approximate asymptotic expression Eq.(\ref{34}), can be derived as well for classical or quantum microscopic dynamics. The exact expression
can also be written as a formula for the density (or, Radon-Nikod\'{y}m derivative) of the true time-evolved measure with respect
to a reference local stationary measure. In the case of classical or quantum dynamics there is another version of this formula
which can be proved by exploiting the Liouville theorem in which the space-time integral appears in an exponent \cite{1,2,3,5}. This density
of exponential type is closer to the ``Girsanov formula'' for change of measure in stochastic analysis. In all cases, the exact expression
for the time-evolved measure involves an integral over the past history of the system.
\noindent {\em (2.3) Microscopic Derivation of Hydrodynamics}
Hydrodynamic behavior in the systems we consider is due to the presence of microscopic conservation. In the DLG model
there is {\em local conservation} of particle number, expressed as
\be \partial_tN(\bx,t)+\nabla_m^{-} J_m(\bx,t)=0. \lb{42} \ee
Here $N(\bx,t)$ is the occupation number at point $\bx$ at time $t$ in a particular realization of the dynamics,
$J_m(\bx,t)$ is the instantaneous flux of particles from site $\bx$ to site $\bx+\ve_m$ at time $t$ for the same stochastic
trajectory, and $(\nabla_m^{-} b)(\bx)=b(\bx)-b(\bx-\ve_m)$ is the backward difference operator.
The expectation of this relation at $t=0,$ conditional on being in configuration $\eta,$ is
\be L\eta_\bx+\nabla_m^{-}j_m(\bx,\eta)=0 \lb{43} \ee
(which is quite easy to verify also by direct calculation). These two relations are the expression of the fact that
particles are neither created nor destroyed in the DLG dynamics but simply hop to neighboring sites.
The derivation of the hydrodynamic law, at least for ensemble averaged densities, is considerably simplified by the
existence of the microscopic conservation laws. In fact, let us define the mean number-density for the time-evolved
measure, as
\be \bar{n}_\en(\bq,\tau)\equiv \langle e^{\en^{-1}\tau L}\eta_{[\![\en^{-1}\bq]\!]}\rangle^{ls}_\en \lb{44} \ee
and the mean current density as
\be \bar{\Bj}_\en(\bq,\tau)\equiv \langle e^{\en^{-1}\tau L}\bj([\![\en^{-1}\bq]\!])\rangle^{ls}_\en. \lb{45} \ee
Then, using the conservation relation Eq.(\ref{43}), it follows that
\be \partial_\tau \bar{n}_\en(\bq,\tau)+\partial^-_m\bar{\j}_{\en,m}(\bq,\tau)=0. \lb{46} \ee
Therefore, the ``balance equation'' for the mean density follows straightforwardly. At this point we have not even made any use of the fact that the averages
are with respect to a ``local stationary'' measure, and the above equation
holds quite generally.
However, we have seen also in the preceding subsection that the time-evolved measure can be separated into
two contributions, the leading-order term having the canonical local stationary form and a correction term
of order $\sim\en.$ The local stationary part must, for consistency, have a reference profile which is the
same as $\bar{n}_\en(\cdot,\tau),$ up to possible deviations of order $\sim\en.$ This means that an equation for the
mean density can be derived by evaluating the mean current correctly up to that same order and substituting
into the balance Eq.(\ref{46}). The required result for the mean current is
\be \bar{\j}_{\en,m}(\bq,\tau)=\hat{\j}_m(\bar{n}_\en(\bq,\tau))+O(\en). \lb{47} \ee
This ``constitutive law'' closes the balance equation in terms of the density at that order, yielding the
following first-order hyperbolic equation
\be \partial_\tau \bar{n}(\bq,\tau)+\partial_m\hat{\j}_m(\bar{n}(\bq,\tau))=0. \lb{48} \ee
It is easy to see that this Eq.(\ref{48}) is the same as Eq.(\ref{36}), and is the ``Euler hydrodynamic equation''
for the DLG model. It is expected to give a correct description of the evolution in the initial density distribution
over microscopic times of order $\en^{-1}$ up to corrections $\sim \en$:
\be \bar{n}_\en(\bq,\tau)=n(\bq,\tau)+O(\en). \lb{48a} \ee
The equation does not involve $\en,$ so that its solution is likewise independent of $\en$ and has been denoted simply as $n(\bq,\tau).$
An important fact concerning the Euler hydrodynamics can be deduced from the exact relation Eq.(\ref{32})
in the previous section. Consider that equation for any chosen reference profile $\lambda(\cdot).$
Recalling that $I_{\en,m}(\bx,\eta)=j_m(\bx,\eta)+O(\en),$ substituting into Eq.(\ref{32}), multiplying
by $\en^d$ and taking the limit $\en\rightarrow 0$ gives
\be \int_\Lambda d\bq\,\,\partial_m\lambda(\bq)\hat{\j}_m(n(\bq))=0, \lb{49} \ee
whenever the reference profile $\lambda$ has at least one continuous derivative. This result can be
given a direct physical interpretation by defining the following ``local entropy functional''
\be S(n)\equiv \int_\Lambda d\bq\,\,s(n(\bq)), \lb{50} \ee
in terms of density profiles $n(\cdot)$ over $\Lambda.$ Then, noting that
\be \lambda(\bq)= -{{\delta S}\over{\delta n(\bq)}}, \lb{51} \ee
a simple integration by parts gives
\be \int_\Lambda d\bq\,\,{{\delta S}\over{\delta n(\bq)}}\partial_m\hat{\j}_m(n(\bq))=0. \lb{52} \ee
Therefore, this relation just states that $dS/dt=0$ instantaneously for the Euler evolution, which is
therefore {\em conservative} of the entropy $S.$ \footnote{For the situation we are considering here with
a single conserved quantity, this is actually a trivial result. In fact, for {\em any} differentiable function
$h(n),$ one can define the associated flux $\bj^h(n)=\int_{n^*}^n d\bar{n}\,
h'(\bar{n})\hat{\Bj}'(\bar{n}),$ so that a smooth
solution of the Euler equation satisfies $\partial_\tau h(n(\bq,\tau))+\partial_m j^h_m(n(\bq,\tau))=0.$
Therefore any ``local functional'' $H(n)=\int_\Lambda d\bq \,\,h(n(\bq))$ is conserved. However, our result holds
also with an arbitrary number of conserved densities $\brho=(\rho_1,...,\rho_r)$ and then the existence of
a conserved, concave entropy function $s(\brho)$ is a nontrivial and important fact. Indeed, it is not hard to
show that there is an associated current function $\bj^s(\brho)$ so that a local conservation holds as above,
i.e. $(s,\bj^s)$ are a convex Lax entropy pair for the drift-diffusion equation Eq.(\ref{58}). Existence of such
an entropy pair is generally true of the PDE's which arise in statistical mechanics, as discussed elsewhere
\cite{23a}.} However, it should be noted that this result assumed the differentiability of the chemical potential
profile $\lambda(\cdot)$ (or of the density profile $n(\cdot)$). Even if this were true for the initial density
$n_0(\cdot),$ it need not be true for the solution $n(\cdot,\tau)$ of the Euler equations. Since the function
$\hat{\j}(n)$ is nonlinear in general, the characteristic velocities ${\bf c}(n)=\hat{\Bj}'(n)$ depend upon the
density and shock-type singularities may appear after a finite time if characteristics intersect. Thereafter,
the equations would have to be interpreted in a suitable ``weak'' sense and the entropy conservation
law for the Euler dynamics would break down.
The Euler equations, while adequate in certain regimes, will not be sufficient for long times. For example,
they cannot describe the relaxation of the initial density $n_0(\bq)$ to a final homogeneous state $n_\infty(\bq)$ of
constant density $\bar{n}_0={{1}\over{|\Lambda|}}\int_\Lambda d\bq\,\,n_0(\bq)$: if the solution stays smooth, then
$S$ will keep its initial value $S(n_0)$ forever and never even begin to approach the expected final value
$S(n_\infty)=|\Lambda|\cdot s(\bar{n}_0)>S(n_0).$ (The inequality follows by strict concavity of $s$.) In fact,
production of entropy and the relaxation to the final stationary state will occur on a longer time scale $\sim \en^{-2}$
in microscopic units due to an additional {\em diffusive} contribution to the current of order $\sim \en.$
To derive the corresponding ``Navier-Stokes hydrodynamic equation,'' therefore, the mean current in Eq.(\ref{46})
must be evaluated to that order.
The evaluation may be accomplished using the general formula Eq.(\ref{34}) for the
order $\sim\en$ correction to the local stationary state. Observe that this formula is valid if the correct
mean density $\bar{n}_\en$ is used for the reference density in the local stationary state rather than the
solution of the Euler equation, because $\partial_\tau\lambda(\bar{n}_\en(\bq,\tau))=-\hat{\Bj}'(\bar{n}_\en(\bq,\tau))
\cdot\nabla\lambda(\bar{n}_\en(\bq,\tau))+O(\en)$ and the changes to the formula Eq.(\ref{34}) are $O(\en^2).$
Therefore, it is easy to calculate the ``constitutive law'' at the next to leading order, as
\be \bar{\j}_{\en,m}(\bq,\tau)=\hat{\j}_m(\bar{n}_\en(\bq,\tau))-\en L_{ml}(\bar{n}_\en(\bq,\tau))
\partial_l\lambda(\bar{n}_\en(\bq,\tau))+o(\en), \lb{53} \ee
with
\be L_{ml}(n)= -\sum_{\bx}x_l\langle \eta_\bx j_m\rangle^\top_n+\int_0^{\infty}dt\left[ \sum_{\bx\in\bZ^d}
\langle \Delta_{\bx,\bx+\ve_l}e^{tL}j_m(\bo) \cdot j_l(\bx)\rangle^\top_n-\chi(n)\hat{\j}_m'(n)\hat{\j}_l'(n)\right], \lb{54} \ee
or, equivalently,
\be \bar{\j}_{\en,m}(\bq,\tau)=\hat{\j}_m(\bar{n}_\en(\bq,\tau))-\en D_{ml}(\bar{n}_\en(\bq,\tau))\partial_l \bar{n}_\en(\bq,\tau)+o(\en), \lb{55} \ee
with
\be L_{ml}(n)=D_{ml}(n)\chi(n). \lb{56} \ee
Since $\bD$ is the coefficient of the contribution to the particle flux linearly proportional to the density gradient, it
may be identified as the {\em (bulk) diffusion matrix}, whereas $\bL$ is the so-called {\em Onsager coefficient}. The
first version of the constitutive law is said to be in the {\em (Onsager) force-flux form}, because $\bX= -\nabla\lambda$
is the ``thermodynamic force'' conjugate to the ``flux'' $\bj.$ It is interesting to note that if we discuss charge density
$\rho$ rather than number density $n,$ then the constitutive law for the electric current is obtained as
\be \bar{\j}_{\en,m}^Q(\bq,\tau)=\hat{\j}_m^Q(\bar{\rho}_\en(\bq,\tau))-\en \Sigma_{ml}
(\bar{\rho}_\en(\bq,\tau))\partial_l\bar{\varphi}_\en(\bq,\tau)+o(\en), \lb{57} \ee
with $\bSigma=\beta q^2 \bL.$ The latter is a kind of ``conductivity matrix''
for the electric field $\bE=-\nabla\bar{\varphi}$
due to the inhomogeneous ``electrochemical potential.'' (See Section 2.5 for
further discussion.)
Employing the order $\sim \en$ constitutive law in the local balance Eq.(\ref{46}) then yields the ``Navier-Stokes equation,''
in the form
\be \partial_\tau{n}_\en(\bq,\tau)=-c_m({n}_\en(\bq,\tau))\partial_m{n}_\en(\bq,\tau)+
\en \partial_m\left[D_{ml}({n}_\en(\bq,\tau))\cdot \partial_l{n}_\en(\bq,\tau)\right]. \lb{58} \ee
This is precisely a {\em driven diffusion equation}, where ${\bf c}$ represents the drift and $\bD$ the diffusion.
\footnote{ A peculiarity of the single-component case is that only the
{\em symmetric part} of the diffusion matrix $\bD$ enters into the hydrodynamic
equations. Indeed, the diffusive term can be written as, $D_{ml}(n)
\partial_m\partial_ln+ D'_{ml}(n)(\partial_m n)(\partial_l n),$ which makes explicit
that only the symmetric part enters. This is responsible for a number of
special features that appear here. In particular, if the drift current happens to vanish---
as it does in some known cases (Appendix III)---then time-reversibility is restored at
the macroscopic level, although it is broken microscopically. This will become
clear from our discussions in Sections 2.4 and 4.1, which imply that the
Graham-Haken ``potential conditions'' are satisfied here. It is not
true for multi-component DDS, or more generally, and we shall make no
use of this fact in our discussions. Note that even for the single-component
case an unambiguous definition of the full diffusion tensor, antisymmetric as
well as symmetric part, is provided by the current response to a local
chemical potential gradient: see Eq.(\ref{55}) and Section 2.5}
It is also easy to write down the equation in ``Onsager form'' using the Onsager matrix rather than the diffusion
matrix. This equation contains the small parameter $\en$ and its solution, which depends now upon $\en,$ may be denoted as
$n_\en(\bq,\tau).$ (Physicists may not be accustomed to writing the small parameter explicitly in this equation, but elementary
kinetic theory estimates show that transport coefficients like $D$ are proportional to the mean-free-path in physical
systems and, therefore, $\sim\en$ when lengths are measured in macroscopic units.) It is expected that the solution
$n_\en$ will be accurate to describe the mean density $\bar{n}_\en$ not only for times $\sim \en^{-2}$ but even uniformly in time,
in the sense that
\be \sup_{\bq\in\Lambda,0<\tau<\infty}|n_\en(\bq,\tau)-\bar{n}_\en(\bq,\tau)|=o(\en), \lb{59} \ee
as long as the solution stays smooth (i.e. no density gradients of order $\en^{-1}$ develop) and the basic assumption
of a separation of scales remains valid. However, if the Euler equations develop shocks, then the shock-width for the
Navier-Stokes equation will be of order $\sim\en$ and the scale-separation assumption will break down. In that case, the
approximation of the Navier-Stokes density to the true density in the region of the shock may be poor and an alternative
description may be necessary. On the other hand, the Navier-Stokes solution should undergo a smooth approach
to the final homogeneous stationary state. Entropy is no longer conserved, but instead
\be {{d}\over{d\tau}}S(n_\en(\tau))=
\en \int_\Lambda d\bq\,\, L_{ml}(n_\en(\bq,\tau))\cdot\partial_m\lambda_\en(\bq,\tau)\partial_l\lambda_\en(\bq,\tau). \lb{60} \ee
Observe that only the symmetric part $\bL^s={{1}\over{2}}(\bL+\bL^\top)$ contributes in this expression. We shall, in fact, show in the
next section that $\bL^s$ is a {\em positive-definite matrix}, so that the entropy must {\em increase}, $dS/d\tau\geq 0,$
and $dS/d\tau=0$ only for a homogeneous state with $\partial_m\lambda\equiv 0.$ Among all of the density profiles
with the same mean density ${{1}\over{|\Lambda|}}\int_\Lambda d\bq\,\,n(\bq)=\bar{n}_0,$ the homogeneous state
$n_\infty(\bq)=\bar{n}_0$ is the unique, absolute maximum of $S$ when $s(n)$ is strictly concave. Therefore, $S(n)$
is a {\em Lyapunov functional} for the Navier-Stokes dynamics, which guarantees the approach to the final
stationary state $n_\infty.$
The derivation we have given of the hydrodynamical equations is essentially an analogue for microscopic systems
of the {\em Chapman-Enskog expansion} in the Boltzmann equation context, a fact which was previously observed by Zubarev \cite{2}
for local-equilibrium systems. Chapman-Enskog expansion was also the method employed by Wannier in his original discussion
of DDS at the Boltzmann level \cite{10a}. The ``nonequilibrium distribution'' may be regarded as an explicit solution of the master
equation, Eq.(\ref{2}), in so-called {\em normal form}. That is, it is a solution which is a functional of the conserved
density field $n_\en(\bq,\tau)$ through the chemical potential histories $\lambda_\en(\bq,\tau)$ used in its construction.
The basic perturbation method employed for the constitutive relations is one in which the fluxes are assumed to have an asymptotic
expansion in the {\em explicit} $\en$-dependence (at least when truncated at order $\en$), but the {\em implicit} $\en$-dependence
via the solutions $n_\en$ themselves is not expanded. This is exactly the procedure used in the Chapman-Enskog method for
the Boltzmann equation, which yields at zeroeth order the Euler equations, at first order the Navier-Stokes equations, and
at second order the so-called ``Burnett equations.'' We have not attempted to carry out the expansion in the DLG model to second-order,
since, for most microscopic systems in $d=3,$ the resulting expressions for the Burnett-order transport coefficients will
diverge as a consequence of ``long-time tails'' \cite{3}, \cite{24a}.
%*I made part of the change but also restored most of my original wording.
We note at this point that using Hypothesis 1 we have not only succeeded to derive the hydrodynamic laws, but also we have
developed microscopic expressions for all of the quantities involved: the entropy function or ``fundamental equation,''
the Eulerian fluxes, and the Onsager relaxation coefficients ( see Eqs.(\ref{11}),(\ref{47}),(\ref{54})).
In a more realistic context, therefore, the methods developed here should be
useful in molecular dynamics calculation of nonequilibrium transport behavior.
The exact expressions may also be useful for theoretical evaluation by
approximation methods, such as low-density expansions. We should emphasize
that, although our methods are formal, our results agree with available
rigorous results for the infinite-temperature ($\beta=0$) case of the DLG,
the so called Assymmetric Simple Exclusion Process, or ASEP (Appendix III). We shall
show furthermore in Section 3.3 that the results are consistent with those
of another formal method, the so-called ``correlation-function approach.''
A final comment here concerns the probability sense of our derivation. We have only given a (heuristic)
derivation of the hydrodynamical laws for the ensemble-averages. In fact, the equations should hold in a much stronger
sense, i.e. they should be true for empirical densities in individual
realizations with probability going to one as $\en\rightarrow 0$
(hydrodynamic law of large numbers). We shall discuss a refinement of this statement, the hydrodynamic large-deviations principle,
in Section 4. No mathematical derivation of these probabilistic statements will be attempted, even at a formal
level. (However, this could be done by the ``level-2'' method of Zwanzig \cite{7}). When we speak of ``derivation'' we
mean it in the sense of Physics as ``guessing the correct answer.''
\noindent {\em (2.4) Reversibility and Onsager Reciprocity}
At this point we shall consider the subject of {\em time-reversal} for the stochastic DLG dynamics, and, in particular,
the consequences for the hydrodynamics of time-reversal invariance of the stationary measures when $E=0.$ The
meaning of time-reversal for stochastic dynamics is essentially the same as for classical or quantum dynamics but
may be somewhat less well-known to physicists. Therefore, we shall start by briefly reviewing the subject.
Basically it means that a history of the system which is {\em typical} of the stationary state will appear
the same when run forward or backward in time. Another way of saying it is that transitions from a set $A\subset \Omega$
to a set $B\subset\Omega$ occur, in the stationary state, with the same frequency as transitions from $B$ to $A.$
On the more formal level, let $\{N(\bx,t)\}$
as before denote the ensemble of particle histories composed of realizations of
the stationary Markov process obtained by using the stationary distribution $P_n$ as the initial distribution on $\Omega$ and
$P_t(B|\eta)=(e^{tL}\chi_B)(\eta)$ as the Markov transition probability from configuration $\eta$ into subset
$B\subset\Omega$ ($\chi_B$ is the characteristic function of the set $B.$) Then, the ``time-reversed process'' $N^r$ is,
very naturally, defined by \be N^r(\cdot,t)=N(\cdot,-t). \lb{61} \ee
This is also a stationary Markov process, with single-time distribution $P_n$. The generator $L^r$ can be obtained from
the definition $\langle f\cdot e^{tL}g\rangle_n =\sE_n[f(N(0))g(N(t))]$ and $\sE_n[f(N^r(0))g(N^r(t))]=\sE_n[f(N(0))g(N(-t))]=\sE_n[f(N(t))g(N(0))].$
We use the symbol $\sE_n(\cdot)$ to denote the average in path-space over
ensembles of histories, as opposed to the single-time average
$\langle\cdot\rangle_n$ with distribution $P_n.$ Therefore,
\be \langle f\cdot e^{L^r t}g\rangle_n=\langle e^{Lt}f\cdot g\rangle_n, \lb{62} \ee
and we conclude that $L^r$ is the adjoint $L^\dagger$ of $L$ with respect to $P_n.$ A point worth emphasizing is that---unlike the adjoint $L^*$ with respect
to counting measure---the adjoint $L^\dagger$ requires knowledge of the
stationary measure $P_n$ for its calculation.
It follows from these general results that the Kawasaki dynamics we consider have a time-reversal of the
same type. For example, it is also conservative
\be \partial_tN^r(\bx,t)+\nabla_m^{-} J_m^r(\bx,t)=0, \lb{62a} \ee
with local current $J_m^r(\bx,t)=-J_m(\bx,-t).$ In a finite volume $\Lambda$ it is easy to show also that
the time-reversed dynamics is a Markov jump process with rates given by
\be {{c^r(\bx,\by,\eta^{\bx\by})}\over{c(\bx,\by,\eta)}}={{P_n(\eta)}\over{P_n(\eta^{\bx\by})}}. \lb{63} \ee
Indeed, it is direct to check, with Eq.(\ref{63}) as a definition, that $\langle f\cdot Lg\rangle_n-\langle L^r f\cdot g\rangle_n
=\langle L(fg)\rangle_n=0.$ It is helpful here to note the identity
\be \langle f\cdot c^r(\bx,\by)\rangle_n =\langle \Delta_{\bx\by}f\cdot c(\bx,\by)\rangle_n. \lb{64} \ee
The result Eq.(\ref{63}) has also an extension to infinite-volume, if the righthand side is expressed in terms
of local conditional distributions \cite{23}.
The process is said to be {\em reversible} if $N^r=N$ in distribution. This definition has a number of mathematically
equivalent formulations. Obviously, one equivalent statement is that $L^\dagger=L,$ i.e. the generator of the process
should be self-adjoint with respect to measure $P_n.$ A more intuitively appealing formulation follows from Eq.(\ref{63})
when $c^r=c,$ namely:
\be c(\bx,\by,\eta^{\bx\by})P_n(\eta^{\bx\by})=c(\bx,\by,\eta)P_n(\eta), \lb{65} \ee
which is the usual condition for {\em detailed balance} of the rates with respect to $P_n.$ The DLG dynamics
are not reversible if $E\neq 0.$ In fact, applying the relation Eq.(\ref{64}) for $f=\eta_\bx-\eta_\by$ gives
\be \langle j^r_m(\bx)\rangle_n= -\langle j_m(\bx)\rangle_n \lb{66} \ee
as a general relation and $\langle j_m\rangle_n=0$ in a reversible case.
(Of course, it may be true even in a non-reversible case that $\langle j_m\rangle_n=0$: see Appendix III for rigorous results on such examples.) However, the DLG measures at finite $E$ support a nonvanishing mean current. Of
course, for $E=0$ the DLG dynamics is reversible with respect to
the Gibbs measures with Hamiltonian $H$ by construction (see Eq.(\ref{3})).
%*This paragraph is revised
An important consequence of micro-reversibility for hydrodynamics was derived by Onsager in 1931,
the famous {\em Onsager reciprocity} (OR) for the transport coefficients: see \cite{9}, usually referred to as RRIP I,II,
and also the later elaboration with his student Machlup \cite{23a}. To explain the
reciprocal relations in a wide context (e.g. see \cite{24}), we consider a general hydrodynamic law for conserved densities
$\rho_\alpha,\alpha=1,...,k$ in the ``Onsager form'':
\be \partial_t\rho_\alpha(\bq,t)=-\partial_m\hat{\j}_{m \alpha}(\rho(\bq,t))+
\partial_m\left[L_{m\alpha,l\beta}(\rho(\bq,t))\cdot \partial_l\lambda_\beta(\bq,t)\right]. \lb{67} \ee
(For simplicity of notation we set $\en=1$ here.) As with our particular example, the ``chemical potentials''
$\lambda_\alpha$ are conjugately related to the conserved densities through a ``local entropy functional'' $S$ as,
$\lambda_\alpha(\bq)=-\delta S(\rho)/\delta\rho_\alpha(\bq).$ The entropy $S$ is conserved by the Eulerian
flux $\hat{\j}_{m\alpha}(\rho).$ If the time-reversal parity of the density $\rho_\alpha$ is $\en_\alpha=\pm,$
then Onsager reciprocity states that
\be \en_\alpha\en_\beta L_{m\alpha,l\beta}(\en\cdot \rho)=L_{l\beta, m\alpha}(\rho). \lb{68} \ee
One can also add to this the relation
\be \en_\alpha\hat{\j}_{m \alpha}(\en\cdot\rho)= -\hat{\j}_{m\alpha}(\rho) \lb{69} \ee
for time-reversal of the Eulerian flux. Notice that this relation implies that the Euler part must
vanish if all time-parities are even, $\en_\alpha\equiv 1.$ Onsager's explanation of these very general phenomenological
relations as a consequence of time-reversibility of the microscopic dynamics is still considered one of the great
achievements of nonequilibrium statistical physics. We shall demonstrate here that these relations
are valid for the expressions we have derived from the microscopic DLG dynamics when $E=0$ and micro-reversibility holds.
%* The next paragraph has been set off and rewritten
It should be pointed out that Onsager's original derivation, unlike ours below, did not use any microscopic expressions
for the transport coefficients. In fact, the key hypothesis---Eq.(1.2) in RRIP II---can be reinterpreted in the
Langevin formulation of Onsager and Machlup as stochastic reversibility at the level of a macroscopic thermodynamic
description. In other words, with a suitable interpretation, Onsager's derivation did not even use micro-reversibility
in any essential way except to motivate the assumption of {\em macro-reversibility} of the Langevin fluctuating hydrodynamic
description. This means that it is actually somewhat more general than our derivation, because it applies
also to systems without micro-reversibility but where detailed balance is restored on the macroscopic level (e.g.
Rayleigh-Ben\'{a}rd cells near onset of convection and lasers near the critical inversion \cite{25}). Therefore,
Onsager reciprocity ought to be considered a result of macroscopic thermodynamics proper.
In our derivation of the OR for the DLG model transport coefficients, it is useful to consider
a generalization of those relations which applies even when $E\neq 0.$ Such a {\em generalized Onsager
reciprocity} (GOR) was considered recently by Dufty and Rub\'{\i} \cite{26} and gives a formal extension
of OR to systems without detailed balance (see also \cite{39}). GOR relates quantities
for the forward and reverse dynamics, as:
\be \en_\alpha\en_\beta L_{m\alpha,l\beta}^r(\en\cdot \rho)=L_{l\beta, m\alpha}(\rho), \lb{70} \ee
and
\be \en_\alpha\hat{\j}_{m \alpha}^r(\en\cdot\rho)= -\hat{\j}_{m\alpha}(\rho). \lb{71} \ee
Unlike the original OR relation, it does not relate physically measurable quantities, since
it is impossible in practice to realize the time-reversed dynamics in the laboratory. Nevertheless,
they will turn out to be formally useful to us in our discussion of the DLG model in order to establish the equivalence of the Kadanoff-Martin and nonequilibrium distribution function methods for deriving hydrodynamics. Here the GOR
relations take the simple form
\be L_{ml}^r(\rho)=L_{lm}(\rho), \lb{72} \ee
and
\be \hat{\j}_{m}^r(\rho)= -\hat{\j}_{m}(\rho), \lb{73} \ee
since there is only one component ($k=1$) and the time-parity of the number density is even. We verify now that these
relations hold for our microscopic expressions. The second relation, in fact, follows directly from the definitions and
Eq.(\ref{66}). For the main GOR, Eq.(\ref{72}), it is helpful to distinguish two contributions to $\bL,$ a ``static'' term
\be L^{{\rm stat}}_{ml}= -\sum_{\bx}x_l\langle \eta_\bx j_m(\bo)\rangle^\top_n, \lb{73a} \ee
and a ``dynamic'' one
\be L^{{\rm dyn}}_{ml}= \int_0^{\infty}dt\left[ \sum_{\bx\in\bZ^d}
\langle \Delta_{\bx,\bx+\ve_l}e^{tL}j_m(\bo) \cdot j_l(\bx)\rangle^\top_n-\chi(n)\hat{\j}_m'(n)\hat{\j}_l'(n)\right]. \lb{80} \ee
We verify the GOR for each separately.
For the static contribution we use the following alternative form:
\be L^{{\rm stat}}_{ml}= -\lim_{\Lambda\uparrow\bZ^d}{{1}\over{|\Lambda|}}\sum_{\bx,\by\in\Lambda}
x_m y_l\langle L\eta_\bx \cdot\eta_\by\rangle^\top_{\Lambda,n}. \lb{73b} \ee
In fact,
\begin{eqnarray}
L^{{\rm stat}}_{ml} & = & -\lim_{\Lambda\uparrow\bZ^d}\sum_{\by\in\Lambda}y_l\langle j_m(\bo)\eta_\by \rangle^\top_{\Lambda,n} \cr
\,& = & -\lim_{\Lambda\uparrow\bZ^d}{{1}\over{|\Lambda|}}\sum_{\bx,\by\in\Lambda}(y_l-x_l)
\langle j_m(\bx)\eta_\by\rangle^\top_{\Lambda,n} \cr
\,& = & \lim_{\Lambda\uparrow\bZ^d}{{1}\over{|\Lambda|}}\sum_{\bx,\by\in\Lambda}
x_m y_l \langle (\nabla^-_kj_k)(\bx)\eta_\by\rangle^\top_{\Lambda,n}. \lb{73c}
\end{eqnarray}
In the last line, a contribution from the $x_l$-term in the previous line was neglected since it may be shown by using
particle conservation, Eq.(\ref{43}), that it is a surface term, which vanishes in the limit if correlations decay rapidly
enough. Finally, Eq.(\ref{73b}) follows by applying again particle conservation. However, the new expression for $\bL^{{\rm stat}}$
makes GOR manifest, since
\be L^{{\rm stat},r}_{ml}= -\lim_{\Lambda\uparrow\bZ^d}{{1}\over{|\Lambda|}}\sum_{\bx,\by\in\Lambda}
x_m y_l\langle L^r\eta_\bx \cdot\eta_\by\rangle^\top_{\Lambda,n}. \lb{73d} \ee
Therefore, $L^{{\rm stat},r}_{ml}=L^{{\rm stat}}_{lm}$ follows using $L^r=L^\dagger.$
For the dynamic contribution we also utilise an alternative form, using the identity Eq.(\ref{64}) with $f=e^{tL}j_m(\bo)
\cdot(\eta_\bx-\eta_{\bx+\ve_l}),$ to rewrite it as:
\be L^{{\rm dyn}}_{ml}= -\int_0^{\infty}dt\left[ \sum_{\bx\in\bZ^d}\langle e^{tL}j_m(\bo) \cdot
j_l^r(\bx)\rangle^\top_n-\chi(n)\hat{\j}_m^{\prime}(n)\hat{\j}_l^{r,\prime}(n)\right]. \lb{81} \ee
The relation Eq.(\ref{73}) was also used. However, this expression can be directly verified to satisfy GOR, since
\begin{eqnarray}
L^{{\rm dyn},r}_{ml} & = & -\int_0^{\infty}dt\left[ \sum_{\bx\in\bZ^d}
\langle e^{tL^r}j_m^r(\bo) \cdot j_l(\bx)\rangle^\top_n-\chi(n)\hat{\j}_m^{r,\prime}(n)\hat{\j}_l^{\prime}(n)\right] \cr
\, & = & -\int_0^{\infty}dt\left[ \sum_{\bx\in\bZ^d}
\langle j_m^r(-\bx) \cdot e^{tL}j_l(\bo)\rangle^\top_n-\chi(n)\hat{\j}_m^{r,\prime}(n)\hat{\j}_l^{\prime}(n)\right] \cr
\, & = & -\int_0^{\infty}dt\left[ \sum_{\bx\in\bZ^d}
\langle e^{tL}j_l(\bo) \cdot j_m^r(\bx) \rangle^\top_n-\chi(n)\hat{\j}_l^{\prime}(n)\hat{\j}_m^{r,\prime}(n)\right] \cr
\, & = & L^{{\rm dyn}}_{lm}, \lb{82}
\end{eqnarray}
as required. As for the static part, the main property used here was $L^r=L^\dagger,$ in the second line. This concludes
the verification that GOR is satisfied.
If we make a decomposition of $\bL$ into even and odd parts, as:
\be \bL^e={{1}\over{2}}(\bL+\bL^r)\,\,\,\,\,\,\,\,\bL^o={{1}\over{2}}(\bL-\bL^r), \lb{75} \ee
then the GOR states that the even and symmetric parts of $\bL$ coincide and, likewise, so do the odd and antisymmetric parts:
\be \bL^e=\bL^s\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\bL^o=\bL^a. \lb{77} \ee
Specializing to the case when $E=0,$ we recover the traditional OR in the DLG model. In that case $\hat{\j}_m(n)=0,$
so the ``Euler term'' of the dynamics disappears. For reversible lattice gases, the entire density change occurs on the ``diffusive''
time scale $\sim\en^{-2}$ in microscopic units, and, by making the time scaling of this type, the factor of $\en$ in front of the
diffusion term is removed. Notice also that $\bL^o=0,$ and only the even terms remain. OR here states simply that these
remaining terms are symmetric,
\be \bL=\bL^s. \lb{83} \ee
\noindent { \em (2.5) Linear Response and the Einstein Relation}
Locally the lattice gas sets up a diffusive flux in response to a small
density gradient. As a consistency check, in linear response we should
find a coefficient of proportionality which is identical to the one from
the nonequilibrium distribution method. The formal manipulations in
this calculation are essentially identical but the intuitive content
is perhaps clearer. Let us prepare then, at $t=0,$ a state with a small negative density gradient by setting
\begin{equation}
\mu_0(\eta)=Z(\lambda')^{-1}P_n(\eta)\exp[-\sum_{\bf x}\lambda' x_l\eta_{\bf x}] \lb{83a} \end{equation}
The response in the average excess current at the origin at time $t$ is given by
\begin{equation}
Z(\lambda')^{-1}\la(e^{tL}(j_m(\bo) - \la j_m(\bo)\ra_n)) \exp[-\sum_{\bf x}\lambda' x_l \eta_{\bf x}]\ra_n, \lb{83b}
\end{equation}
with $\la\cdot\ra_n$ the stationary state at density $n$. To first order in $\lambda',$ Eq.(\ref{83b}) equals
\begin{eqnarray}
\, & &\Delta_lj_m(t) = \lambda' \{ - \sum_{{\bf x}} x_l\la j_m(\bo)\eta_{{\bf x}}\ra_n^{\top} \cr
\, & &\,\,\,\,\,\,\,\,\,\,\,
-\int_0^t ds [ \sum_{{\bf x}}\la(e^{sL}j_m(\bo))j_l^r({\bf x})\ra_n^{\top} - \chi(n) \hat{\j}_m'(n)
\hat{\j}_l'(n) ] - t \chi(n) \hat{\j}_m'(n)
\hat{\j}_l'(n) \}. \lb{83c}
\end{eqnarray}
Here we differentiated the exponential and, as before, once iterated the time evolution operator
$e^{tL}=1+\int_0^tds\,\,e^{L(t-s)}L.$
Initially the average density equals $n - \lambda' \chi(n) x_l$ for small
$\lambda'$. According to the Euler equations Eq.(\ref{48}), linearized
at $n$ since
$\lambda'$ is small, the density at the origin at time $t$ is $n + \lambda'
t \chi(n)\hat{\j}_l'(n)$. This induces a change in the current at the origin
as $-\hat{\j}_m'(n)\lambda't\chi(n)\hat{\j}_l'(n)$. Subtracting
out this term yields
\begin{equation}
\lim_{t \rightarrow \infty} \frac{1}{\lambda'} \Delta_lj_m(t) - t \chi(n)
\hat{\j}_m'(n)\hat{\j}_l'(n) = L_{ml}(n) \lb{83d}
\end{equation}
in agreement with Eq.(\ref{54}). Note that we have obtained the full,
in general nonsymmetric, Onsager matrix.
Besides the response to a density gradient, it is of interest physically
to consider also the response to ``mechanical'' perturbations, i.e. to
small changes in the dynamics. In our case the driving field is singled
out. The response of the average current to it defines the conductivity
matrix, $\sigma_{ml}(n,{\bf E})$, at field strength ${\bf E}$. At
$E=0$ the local
detailed balance condition (3) implies the Einstein relation
\begin{equation}
\bsigma(n,E=0) = \frac{q^2}{k_BT}\bL(n,E=0). \lb{83e}
\end{equation}
In fact, this relation does not depend on how the driving field is
incorporated into the rates as long as (3) is satisfied: see \cite{14a}, Section II. 2.5.
For $E\neq 0$, this will no longer be the case. One fairly natural choice is
\begin{equation}
c_E(\bx,\by,\eta) = c_0(\bx,\by,\eta)\exp[- \frac{1}{2} q \beta {\bf E}\cdot
({\bf x} - {\bf y})(\eta_{{\bf x}} - \eta_{{\bf y}}]) \lb{83f}
\end{equation}
with $c_0$ the equlibrium rates satisfying
\begin{equation}
c_0(\bx,\by,\eta) = c_0(\bx,\by,\eta ^{\bx\by})\exp[-\beta(H(\eta^{\bx\by}) - H(\eta))]. \lb{83g}
\end{equation}
Then (3) holds and, taking into account that $j_m(\bo)$ also depends on ${\bf
E}$,
\begin{eqnarray}
\sigma_{ml}(n,E)& = &\frac{\partial}{\partial E_l} \la j_m(\bo)\ra_n \cr
\,& = &q^2\beta \left(\frac{1}{2} \delta_{ml} \la c(\bo,\ve_m\ra_n
- \int_{0}^{\infty} dt \sum_{{\bf x}}\la\frac{1}{2}(j_l({\bf x}) +j_l^r({\bf x}))e^{tL}j_m(\bo)\ra_n\right). \lb{83h}
\end{eqnarray}
We note that $\la j_l({\bf x}) + j_l^r({\bf x})\ra_n=0$ and no subtraction in
(\ref{83h}) is needed. At $E=0$, since $j_l = j_l^r$, (\ref{83h}) reduces to the Einstein
relation (\ref{83e}), but such an identity will not hold, in general.
If the electric field is represented by the gradient of an electrostatic
potential $\varphi,$ as $\bE= -\grad\varphi,$ then as long as the field is
weak enough for linear response to be valid, there is a useful
combination of electrostatic and chemical potentials. (Note that the
gradient representation by a single-valued potential is not possible for a constant
$E$ with our periodic b.c., but we have in mind more physically realistic
situations.) According to the Einstein relation Eq.(\ref{83e}),
the constants of proportionality in the linear response to $-\grad\varphi$
and $-{{1}\over{q}}\grad\mu$ are identical. In that case, it is useful to
consider a combined {\em electrochemical potential}
$\varphi+{{1}\over{q}}\mu;$ cf. Landau and Lifshitz, \cite{19a}, Section 25.
However, it should be clear that this is only possible in the
weak-field regime, and that the response to chemical potential gradient
and the differential response to the electric field in the strong
field case are generally distinct.
\newpage
\section{Current Noise and the Fluctuation-Dissipation Relation}
\noindent {\em (3.1) A Review of Standard Theory}
We now turn to a somewhat different subject, the theory of fluctuations about the
hydrodynamic behavior whose description was derived in the previous section. While
the predictions of the hydrodynamic equations, like Eq.(\ref{58}), are observed
with a probability going to one as $\en\rightarrow 0,$ there will
also be small, random corrections $\sim \en^{d/2}$. That is, if the ``fluctuation variable''
\be X^\en(A,\tau)={{1}\over{\en^{d/2}}}\left[\en^d\sum_{\en\bx\in A} e^{\en^{-1}\tau L}\eta_\bx-\int_A d\bq\,\,\bar{n}_\en(\bq,\tau)\right], \lb{84} \ee
is considered for any macroscopic time $\tau$ and region $A\subset\Lambda,$ then it is expected that its
distribution will converge to a Gaussian for $\en\rightarrow 0.$ Observe that the normalization $\sim\en^{-d/2}$ of
the sum corresponds to a standard central limit theorem scaling, since the size of the region $A$ in microscopic units
grows as $\sim \en^{-d}.$ The basic problem of fluctuation theory is to derive the form of the limiting distribution
laws. A physical theory for this in the case of the Navier-Stokes system was proposed by Landau and Lifshitz in 1959 \cite{27}, the
so-called ``fluctuating hydrodynamics,'' which was an application to simple
fluids of general principles set out in the work of Onsager and
Machlup \cite{23a}. Extended to the more general context of hydrodynamics in ``Onsager form,'' their hypothesis was that
$X^\en_\alpha(A,\tau)$ is approximated in the limit $\en\rightarrow 0$ by $\int_A d\bq\,\,\xi_\alpha^\en(\bq,\tau),$
where $\xi_\alpha^\en(\bq,\tau)$ is the solution of the stochastic PDE (Langevin equation)
\be \partial_\tau\xi^\en_\alpha(\bq,\tau)= -{\cal H}_{\alpha\beta}(\bar{\rho}(\bq,t))\xi_\beta^\en(\bq,\tau)
-\grad\cdot\bj_{\alpha}^\prime(\bq,\tau), \lb{85} \ee
with ``linearized hydrodynamic operator'' given by ${\cal H}^\en_{\alpha\beta}(\bar{\rho}(\bq,t))f(\bq)=\partial_m[H^\en_{m,\alpha\beta}(\bar{\rho}(\bq,t))
f(\bq)]$ and
\be H^\en_{m,\alpha\beta}(\rho)= {{\partial\hat{\j}_{m \alpha}}\over{\partial\rho_\beta}}(\rho)
-\en L_{m\alpha,l\gamma}(\rho){{\partial\lambda_\gamma}\over{\partial\rho_\beta}}(\rho)\partial_l. \lb{86} \ee
The $j_{m\alpha}^\prime(\bq,\tau)$ are ``random fluxes,'' which were taken to be Gaussian fields with zero-mean.
In other words, the fluctuations were assumed to be governed by the hydrodynamic equations linearized about
the deterministic solution $\bar{\rho}_\alpha(\bq,\tau),$ to which were added additional stochastic currents representing
effects of molecular noise. To complete the theory, a prescription was given also for the covariance of the random fluxes, as
\be \langle j_{m\alpha}^\prime(\bq,\tau)j_{l\beta}^\prime(\bq',\tau')\rangle=
2k_B\cdot\en L^s_{m\alpha,l\beta}(\bar{\rho}(\bq,\tau))\delta(\bq-\bq')\delta(\tau-\tau'), \lb{87} \ee
where $k_B$ is Boltzmann's constant and $\bL^s$ is the symmetric part of the Onsager matrix. This is a particular
example of a {\em fluctuation-dissipation relation}, since it relates the covariance of fluctuating currents to the
dissipative Onsager coefficients.
There are actually a number of distinct---but related---general types of FDR's.
A quite recent monograph of Stratonovich \cite{32} gives a detailed discussion
of linear and nonlinear fluctuation-dissipation theorems with a wide variety of applications. This work makes a classification of FDR's into 1st, 2nd, and 3rd types. We shall briefly discuss here the various types
following Stratonovich, although his terminology for FDR's of 1st and 2nd kind is exactly the {\em opposite} of that in
the Western literature. Note: {\em we follow here Stratonovich' classification!} Although there is some
weight of tradition against this usage, we find it supported by the internal logic of the subject.
In Stratonovich' classification, the FDR of 1st type is a relation between noise characteristics and dissipation
elements, e.g. the noise covariance and the (Onsager) relaxation coefficient at linear level such as is given in
Eq.(\ref{87}). The Einstein relation (ER) \cite{33}
can be considered a prototype of such a relation.
In fact, if the conductivity $\bsigma$
is considered to be the linear coefficient in the Langevin dynamics
\be \dot{x}_i(t)= -\sigma_{ij} {{\partial U}\over{\partial x_j}}(x(t))+\eta_i(t) \lb{88} \ee
and ${\bf D}$ is taken to be the noise-strength parameter
\be \la\eta_i(t)\eta_j(0)\ra=2D_{ij}\delta(t) \lb{89} \ee
then
\be P(\bx)\sim \exp[-\beta U(\bx)] \lb{90} \ee
is the stationary measure if and only if
\be \bsigma=\beta{\bf D}. \lb{91} \ee
This is a typical FDR of 1st type, and it is the type which is most important for this work.
The FDR of 2nd type according to Stratonovich is a relation between time-correlation functions and (dissipative) response functions
to an external perturbation. The Callen-Welton relation \cite{34}was a first case of 2nd-type FDR for quantum systems. In our previous example,
the linear FDR of 2nd type is
%*Note the change below: expectations are after the functional derivative. Add something on Lebowitz-Rost?
\be \beta \la\dot{x}_i(t) x_j(0)\ra= -\langle{{\delta x_i(t)}\over{\delta E_j(0)}}\rangle,\,\,\,\,\,\,t>0 \lb{92} \ee
for expectations in the stationary state, when the external field ${\bf E}(t)$ is coupled into the Langevin equation as:
\be \dot{x}_i(t)=\sigma_{ij}\left[-{{\partial U}\over{\partial x_j}}(x(t))+E_j(t)\right]+\eta_i(t). \lb{93} \ee
The ER is also closely related to the FDR of 2nd type, if one now considers $\bsigma$ as a response to an external field and ${\bf D}$
as defined in terms of time-correlations. The 2nd-type FDR is actually slightly more general than the ER. In fact,
one can define a conductivity response kernel $\sigma_{ij}(t)=\langle \delta v_i(t)/\delta E_j(0)\rangle,$ giving
the delayed current response, with $\bv(t)\equiv\dot{\bx}(t).$ Taking a time-derivative of the FDR, one obtains the
relation
\be \beta \la v_i(t) v_j(0)\ra = \sigma_{ij}(t). \lb{94} \ee
In the Fourier representation this becomes
\be \sigma_{ij}(\omega)= \beta \int_0^\infty dt e^{-i\omega t} \la v_i(t)v_j(0)\ra, \lb{95} \ee
which gives the full frequency response. It is not hard to show that Eq.(\ref{95}) yields $\sigma_{ij}(\omega=0)
=\beta D_{ij}.$
It is a general feature that the FDR of 2nd type implies the FDR of 1st type (e.g. \cite{32}, Section 5.5.3), as one sees here
by taking $\omega=0.$ To obtain the FDR of second type requires an appropriate special coupling of the external field, which can
always be done (\cite{32}, Section 5.5.1, also \cite{17} and the following section). The relation between FDR's of 1st and 2nd types
is intuitive since the roles of fluctuational and (appropriate) external forces are essentially interchangeable. Stratonovich also
defines another set of relations, FDR's of 3rd type, as those existing between noise characteristics and response functions,
e.g. random force covariance and linear response function. The Nyquist relation \cite{35} in electrical circuits between the power spectrum
of the noisy EMF (electromotive force) and the frequency-dependent impedance is a prototypical example. We will not
consider this subject here.
It is our purpose here to consider the extension of the Landau-Lifshitz method
to the systems without local equilibrium
and to develop corresponding prescriptions for the random fluxes, such as
current noise in electrical conductors.
It seems to be still not generally appreciated that there is a {\em generalized fluctuation-dissipation relation}
which applies to such systems. In fact, the theory is rather scattered throughout the literature, and we shall
develop the subject {\em ab initio} here. The presentation we give below is therefore not precisely the same as
that we have found anywhere in the literature of the subject, but it borrows from a number of others.
An important early work was that of Price \cite{15}, who proposed a charge diffusion-current noise relation in
semiconductors and plasmas, which we shall discuss in the next subsection. The classic work of Fox
and Uhlenbeck \cite{28} also derived the generalized FDR in the Langevin equation context in a way that extends
directly to systems without local equilibrium (although they did not consider that extension). A paper of
Tomita and Tomita \cite{29} made an important decomposition of the systematic flux in the linear Langevin equation
which is relevant to the FDR, since it corresponds to a separation into ``conservative'' and ``dissipative'' terms.
Our perspective on this has been strongly influenced by the work of Graham in 1977 \cite{17}, who generalized
this decomposition to general nonlinear Langevin equations. In fact, we believe that Graham's paper is an important
contribution to the subject which has been unduly neglected, perhaps because of the abstract style of presentation.
The last section of this paper will discuss that work extensively and hopefully will make its physical relevance more clear.
A number of review articles and books have also been useful to us. The paper of Gantsevich et al. \cite{30},
contains a wealth of information on the theory of fluctuations in nonequilibrium electron gases, developed
from the point of view of Boltzmann transport equations and slanted toward the Soviet work on semiconductors.
A more general review article is that of Tremblay \cite{31}, which discusses these developments and parallel work in
the West up to a decade ago. The general philosophy of Stratonovich
in \cite{32} is also very close to and has influenced ours.
\noindent {\em (3.2) The Generalized FDR in Linear Theory}
The FDR's of 1st and 2nd type are established quite easily in the context of a general linear Langevin equation, as we now
discuss. In particular, we wish to make clear in our derivation that the FDR is independent of any near-equilibrium
assumption. We consider, as in the original paper of Onsager-Machlup, a discrete set of variables $\balpha=
\{\alpha_i|i=1,...,p\}.$ It is quite easy to extend the theory to spatially extended systems as above by
considering the index $i$ to stand for $(\bq,m,\alpha).$ The variables evolve according to
\be \dot{\alpha}_i= -H_{ij}\alpha_j +\eta_i, \lb{96} \ee
a linear relaxational equation, $\bH^s\geq 0,$ with Gaussian white-noise force
\be \la\eta_i(t)\eta_j(t')\ra= 2k_B Q_{ij} \delta(t-t'). \lb{97} \ee
The stationary probability distribution of $\balpha$ is then also Gaussian:
\be P(\balpha)\propto \exp\left[-{{1}\over{2k_B}}(\bg^{-1})_{ij}\alpha_i\alpha_j\right], \lb{97a} \ee
where $k_B\cdot\bg$ is the steady-state covariance
\be \la\alpha_i\alpha_j\ra = k_B g_{ij}. \lb{98} \ee
To obtain $\bg$ ones solves the Langevin equation as
\be \balpha(t)=e^{-\bH t}\balpha_0+\int_0^tds\,\,e^{-\bH(t-s)}\boeta(s), \lb{102a} \ee
and then calculates $\langle \alpha_i(t)\alpha_j(t)\rangle$ in the limit $t\rightarrow +\infty.$ This yields
\be {\bf g}=2\int_0^\infty dt\,\,e^{-\bH t}\bQ e^{-\bH^\top t}. \lb{102b} \ee
As a direct consequence, we obtain
\be {\bf Hg}+{\bf gH}^\top=2{\bf Q}. \lb{102bb} \ee
It is this relation which is the FDR of first type and arises as a simple ``stationarity'' or ``balance'' condition.
The above general derivation goes back to \cite{28}.
However, the Eq.(\ref{102bb}) does not give the FDR in its most familiar form, which involves ``dissipative Onsager
coefficients'' appearing in a ``force-flux form'' of the equations. Even without the near-equilibrium assumption, we may
{\em define} an ``entropy'' associated to the steady-state distribution $P(\balpha)$ as
$s(\balpha)= k_B\cdot\log P(\balpha),$ or
\be s(\balpha)= s_0-{{1}\over{2}}(\bg^{-1})_{ij}\alpha_i\alpha_j. \lb{99} \ee
We can also introduce a ``force''
\be X_{i}= -{{\partial s}\over{\partial \alpha_i}}. \lb{100} \ee
The Langevin equation is then expressed in ``force-flux form,'' as:
\be \dot{\alpha}_i= -L_{ij}X_j +\eta_i, \lb{101} \ee
with ${\bf L}={\bf Hg},$ which are the ``Onsager coefficients.'' In terms of these the Eq.(\ref{102bb}) becomes simply
\be 2{\bf Q} = {\bf L}+{\bf L}^\top. \lb{102} \ee
It just means that the noise strength $Q_{ij}$ is the symmetric
part of $L_{ij},$
\be {\bf Q}=\bL^s, \lb{103} \ee
which is the standard linear FDR of 1st type. Its derivation is completely straightforward. The distinction of the
general case from the near-equilibrium one is that, for equilibrium systems, the ``entropy'' is the ordinary thermodynamic
equilibrium entropy which has been much studied and is known for many systems by a theoretical or empirical determination
of the fundamental equation. For nonequilibrium systems, such as the DDS, the quantity $s(\balpha),$ or $\bg,$ is not
readily available. Nevertheless, the relationship between the variables $\bQ,\bH,$ and $\bg$ remains valid.
The terminology ``fluctuation-dissipation'' is explained in the present context, by considering the decomposition of
$\bL$ into its symmetric part $\bL^s$ and it anti-symmetric part denoted $\bL^a,$
\be L_{ij}=L^a_{ij}+L^s_{ij}. \lb{104} \ee
Correspondingly, one can define a ``conservative flux''
\be r_i= -L^a_{ij}X_j, \lb{105} \ee
and a ``dissipative flux''
\be d_i= -L^s_{ij}X_j. \lb{105a} \ee
Since $\partial s/\partial \alpha_i= -X_i$ and $\partial X_j/\partial\alpha_i= \left(
\bg^{-1}\right)_{ij},$ the $r_i$ automatically satisfy
\be {{\partial s}\over{\partial \alpha_i}}r_i(\alpha)=0 \lb{106} \ee
from symmetry of $X_iX_j,$ and
\be {{\partial r_i}\over{\partial \alpha_i}}=0 \lb{107} \ee
from symmetry of $g_{ij}.$ The first of these implies that the evolution
under $r_i$ indeed conserves the ``entropy'' $s,$ while the second is
a Liouville theorem. Hence, the
entire evolution of the ``entropy'' under the systematic (non-random) evolution arises from $d_i,$ and is calculated as
\be {{d}\over{dt}}s(\alpha)= L^s_{ij}X_iX_j\geq 0. \lb{108} \ee
The positivity follows because $\bL^s$ is, by the FDR, a covariance matrix, which is necessarily positive-definite.
Therefore, the general decomposition of the linear Langevin equation
\be \dot{\alpha}_i= r_i+d_i+\eta_i \lb{109} \ee
is achieved into parts ``conservative'' and ``dissipative'' for the ``entropy'' of the stationary state.
The same decomposition was made by Tomita and Tomita \cite{29} from a different motivation, discussed below.
%* I've added the Langevin forces back into the equation
Note that one can also show easily in this context the FDR of 2nd type. The principle is to add the external force
${\bf F}$ in the equation so that the stationary measure is changed by the addition of a linear term ${\bf F}\cdot \balpha$
in the exponent. It is not hard to show that for this the driven equation must be taken to be
\begin{eqnarray}
\dot{\alpha}_i& = & -L_{ij}(X_j -F_j)+\eta_i \cr
\,& = & -H_{ij}\alpha_j +L_{ij}F_j+\eta_i. \lb{110}
\end{eqnarray}
Then the response function $G_{ij}(t)=\delta\alpha_i(t)/\delta F_j(0)$ (which is now deterministic) satisfies
\be \dot{G}_{ij}(t)=-H_{ik}G_{kj}(t)+L_{ij}\delta(t), \lb{111} \ee
whose solution is
\be G_{ij}(t)= \left[ e^{-{\bf H}t}{\bf L}\right]_{ij}\theta(t). \lb{112} \ee
On the other hand, the linear regression solution to the Langevin equation is
\be \la\alpha_i(t)\alpha_j(0)\ra
= k_B\left[ e^{-{\bf H}|t|}{\bf g}\right]_{ij}\theta(t)
+k_B\left[ {\bf g}e^{-{\bf H^\top}|t|}\right]_{ij}\theta(-t). \lb{113} \ee
The FDR of 2nd type
\be \la\dot{\alpha}_i(t)\alpha_j(0)\ra
= -k_B\left[ G_{ij}(t)-G_{ji}(-t)\right], \lb{114} \ee
follows automatically.
It is important to note that the FDR's all follow (rather simply!) from the assumption of a general
description in terms of a random process described by a Langevin stochastic equation. No condition of time-reversibility
or detailed balance is required. In fact, they are results of pure theory of random processes and involve ``no physics.''
Actually, the physics enters in from the validity of the description by a Gaussian random process with some stable
stationary state. This imposes some nontrivial requirement on the microscopic dynamics (Appendix IV).
Although the FDR as such does not require detailed balance, an important simplification occurs in that case.
In fact, suppose that the variables $\alpha_i$ have time parities $\en_i,$ so that the Onsager coefficient transforms
under time-reversal as
\be {L}^r_{ij}=\en_i\en_j L_{ij}. \lb{118} \ee
We are considering a generalized time-reversal transformation in which the histories $\{\alpha_i(t)\}\rightarrow
\{\alpha^r_i(t)\equiv\en_i\alpha_i(-t)\},$ with $\en_i=\pm.$ If detailed balance holds and the OR relations are
therefore satisfied, then
\be {L}^r_{ij}=L_{ji}, \lb{119} \ee
so that the decomposition of $\bL$ into parts even and odd under time-reversal coincides with the decomposition
into a symmetric, or ``dissipative,'' part and an anti-symmetric, or ``conservative,'' part: $\bL^o=\bL^a,\bL^e=\bL^s.$ Equivalently, the ``conservative''
and ``dissipative'' fluxes are, respectively, odd and even under time-reversal,
\be r_i(\en\cdot\balpha)=-\en_ir_i(\balpha),\,\,\,\,\,\,\,\,
d_i(\en\cdot\balpha)=+\en_id_i(\balpha), \lb{120} \ee
and they can be distinguished {\em a priori} on that basis. Without the
reversibility condition, this decomposition into ``conservative'' and ``dissipative'' terms requires knowledge of the stationary measure, as above.
In this connection, we would like to make a few remarks on the interesting work of Tomita \& Tomita (TT) \cite{29}.
They made as well the decomposition of the Onsager matrix $\bL$ into its symmetric part and its antisymmetric part. They referred to the latter as the ``irreversible circulation of
fluctuation'' and connected its nonvanishing with the violation of OR, interpreted as the symmetry of $\bL.$
However, it seems to us that TT's interpretation of their results is somewhat misleading. The presence or absence of the anti-symmetric
part $\bL^a$ is not related to microscopic reversibility in general, as they suggest. In fact, the
interpretation of symmetry of ${\bf L}$ as Onsager symmetry is also wrong in general. It seems to be the source of a lot of
confusion that the ``dissipative'' Onsager coefficient is {\em always} the symmetric part $\bL^s,$ but this has nothing to do
with time-reversal or Onsager reciprocity. Second, it is not in general implied by time-reversibility that $\bL^a=\bo$, i.e.
that $\bL=\bL^s.$ This is only the case if all the parities $\epsilon_i=1.$ While this was, in fact, true for the systems
TT considered, it need not always be the case. An obvious example, already pointed out in this context in the classic work
of Fox and Uhlenbeck \cite{28}, is the simple fluid described by the compressible Navier-Stokes equations which has a non-vanishing Euler part.
The possibility to have such a term here arises from the -1 parity of the velocity field, or momentum density.
It is only for the cases with all parities $+1,$ as for the DDS example, that the presence of the Euler term, or ``irreversible
circulation,'' can be attributed to violation of detailed balance.
\noindent {\em (3.3) The Correlation Function Approach to Hydrodynamics}
One of the main assumptions in the theory of fluctuations, reviewed in the previous section, is that the
small spontaneous fluctuations from the stationary mean will decay according to the linearized hydrodynamic law
governing macroscopic disturbances. This idea goes back to the original work of Onsager in 1931 and is known as
the {\em Onsager regression hypothesis}. This assumption can be made the basis of a microscopic method
for deriving hydrodynamics and calculating transport coefficients, the so-called ``correlation-function method''
which was pioneered in work of Kadanoff and Martin \cite{16} (see also \cite{4}). In fact, if
$\Delta\alpha_i(t)\equiv \alpha_i(t)-\alpha_i(0),$ then it follows from the linear regression solution Eq.(\ref{113}) that
\be \langle\Delta\alpha_i(t)\alpha_j(0)\rangle= k_B\left[e^{-{\bf H}t}{\bf g}\right]_{ij}-k_B\cdot g_{ij} \lb{121} \ee
for $t>0.$ Hence
\begin{eqnarray}
\lim_{t\rightarrow 0}{{1}\over{t}}\langle\Delta\alpha_i(t)\alpha_j(0)\rangle
\,& = & -k_B\left[{\bf Hg}\right]_{ij} \cr
\,& = & -k_B L_{ij}. \lb{122}
\end{eqnarray}
The Onsager matrix $L_{ij}$ can thus be calculated from the correlation function $\langle\Delta\alpha_i(t)\alpha_j(0)\rangle,$
determined by another method, e.g. a microscopic calculation. When using the latter, then the limit $t\rightarrow \infty$
should be considered instead, since the time in the Langevin equation must be considered a ``macroscopic time.'' The limit
of ``macroscopic time'' $\tau\rightarrow 0$ must match onto the limit of ``microscopic time'' $t\rightarrow \infty.$
Although this method was originally applied by Kadanoff and Martin to hydrodynamics for near-equilibrium fluids,
it should be quite general since its basis is just the regression hypothesis. In fact, the method was used
already in the original paper of KLS to derive the linearized hydrodynamics of their lattice DDS model.
We shall reconsider the calculation here because it provides an important check on the ``nonequilibrium distribution
method'' developed in the first section. Also, the original discussion of KLS contained a few minor mistakes
which this work corrects.
To begin, let us write down the form of the driven diffusion equation derived previously, Eq.(\ref{58}),
linearized about the homogeneous state of density ${n}:$
\be \partial_t\xi(\bq,t)=-c_m({n})\partial_m\xi(\bq,t)+
D_{ml}^s({n})\partial_m\partial_l\xi(\bq,t). \lb{125} \ee
(Again we set $\en=1$ for simplicity.) Note that the linearized equation contains only the
{\em symmetric part} of the diffusion matrix (a feature special to the 1-component case.)
The equation can also be Fourier transformed in space, to give
\be \partial_t\hat{\xi}(\bk,t)= i\bc\cdot\bk\hat{\xi}(\bk,t)
-(\bk\cdot\bD\cdot\bk)\hat{\xi}(\bk,t), \lb{126} \ee
where $\bc=\bc({n}),\bD=\bD^s({n}).$ This is often more convenient since the different wavenumber
components are uncoupled. If the fluctuation theory of the previous subsection is applied, then we should
add to this equation a noisy current $\hat{\bJ}^\prime,$ as
\be \partial_t\hat{\xi}(\bk,t)= i\bc\cdot\bk\hat{\xi}(\bk,t)
-(\bk\cdot\bD\cdot\bk)\hat{\xi}(\bk,t)-i\bk\cdot\hat{\bJ}^\prime(\bk,t), \lb{127} \ee
with
\be \langle\hat{J}^\prime_i(\bk,t)\hat{J}^\prime_j(\bk',t')\rangle=
2 R_{ij}(\bk)\delta(\bk-\bk')\delta(t-t'). \lb{128} \ee
Then, from the regression solution, Eq.(\ref{113}), the ``structure function''
\be S(\bk,t)=\int d\br\,\,e^{i\bk\cdot\br}\langle\xi(\br,t)\xi(\bo,0)\rangle, \lb{129} \ee
takes the value for $t>0$
\be S(\bk,t)=S(\bk)\exp[i(\bc\cdot\bk)t-(\bk\cdot\bD\cdot\bk)t]. \lb{130} \ee
Here, $S(\bk)$ is the static value of the structure function, or stationary covariance, which from the FDR Eq.(\ref{103}), is given as
\be S(\bk)={{\bk\cdot\bR(\bk)\cdot\bk}\over{\bk\cdot\bD\cdot\bk}}. \lb{131} \ee
It is then easy to calculate that
\be \left.{{\partial}\over{\partial k_m}}S(\bk,t)\right|_{\bk=\bo}
= \left({{\partial}\over{\partial k_m}}S\right)(\bo)-ic_mtS(\bo). \lb{132} \ee
In fact, it follows from the evenness in $\br$ of the density-density correlation that $S(\bk)$ is even in $\bk,$ so that
\be \left({{\partial}\over{\partial k_m}}S\right)(\bo)=0. \lb{132a} \ee
In the same way, it is easy to check that
\be \left.{{\partial^2}\over{\partial k_m\partial k_l}}S(\bk,t)\right|_{\bk=\bo}
= \left({{\partial^2}\over{\partial k_m\partial k_l}}S\right)(\bo)-2D_{ml}tS(\bo)-c_mc_lt^2S(\bo). \lb{133} \ee
Here we have assumed that $S(\bk)$ is twice-differentiable at $\bk=\bo,$ an assumption that will be critically
re-examined in the following subsection. If we now go back to the microscopic model and identify $S(\bo)=\chi(n),$
then these results suggest that $\bc,\bD^s$ should be determined from the microscopic dynamics as
\be c_m(n)=\lim_{t\rightarrow\infty}{{1}\over{t}}\left({{1}\over{\chi(n)}}\sum_\bx x_m\langle e^{tL}\eta_\bx\eta_\bo\rangle^\top_n\right), \lb{134} \ee
and
\be D^s_{ml}(n)=\lim_{t\rightarrow\infty}{{1}\over{2t}}\left({{1}\over{\chi(n)}}\sum_\bx x_mx_l\langle e^{tL}\eta_\bx\eta_\bo\rangle^\top_n
-c_m(n)c_l(n)t^2\right). \lb{134a} \ee
The righthand sides of these equalities were calculated in a formal (non-rigorous) way by KLS in Appendix 3 of \cite{11}. The result for $\bc$ was
\be c_m(n)=\hat{\j}_m^\prime(n), \lb{135} \ee
and the result for $\bD^s$ was
\be
D_{ml}^s(n)={{1}\over{2}}\int_{-\infty}^{+\infty}dt\,\,\left[{{1}\over{\chi(n)}}\sum_\bx\sE_n(J_m(\bx,t)J_l(\bo,0))^\top_n
-\hat{\j}_m^\prime(n)\hat{\j}_l^\prime(n)\right], \lb{136} \ee
where $\bJ(\bx,t)$ is the instantaneous particle current introduced in Eq.(\ref{42}).
Note that the result for $\bc$ is exactly that found by the distribution function method in the previous section. To compare
the results for $\bD$ the current-current correlation must be evaluated. This was done in Appendix 3 of \cite{11}, but the
result there contained an error (it does not give a symmetric result for $D^s_{lm}.$) A correct calculation
(Appendix V) gives
\begin{eqnarray}
\sE_n(J_m(\bx,t)J_l(\by,s)) & = & \langle c(\bo,\ve_m)\rangle_n\delta_{ml}\delta_{\bx\by}\delta(t-s) \cr
\, & & \,\,\,\,\,\,\,\,\,\,-\theta(t-s)\langle e^{L(t-s)}j_m(\bx)\cdot j_l^r(\by)\rangle_n \cr
\, & & \,\,\,\,\,\,\,\,\,\,-\theta(s-t)\langle e^{L(s-t)}j_l(\by)\cdot j_m^r(\bx)\rangle_n. \lb{137}
\end{eqnarray}
Substitution into Eq.(\ref{136}) gives
\begin{eqnarray}
\, & & D_{ml}^s(n)={{1}\over{2\chi(n)}}\langle c(\bo,\ve_m)\rangle_n\delta_{ml} \cr
\, & & \,\,\,\,\,\,\,-\int_0^{\infty}dt\,\,\left[{{1}\over{2\chi(n)}}\sum_\bx
\left(\langle e^{tL}j_m(\bx)\cdot j_l^r(\bo)\rangle^\top_n+\langle e^{tL}j_l(\bx)\cdot j_m^r(\bo)\rangle^\top_n\right)
-\hat{\j}_m^\prime(n)\hat{\j}_l^{r,\prime}(n)\right]. \lb{138}
\end{eqnarray}
Comparison with Eq. (\ref{81}) shows that the ``dynamic term'' above agrees with the result of the distribution function calculation for the
symmetric part of the diffusion matrix. To verify that the ``static terms'' are also the same we note that, by GOR, the symmetric and
even parts must coincide. Using this fact, the symmetric part from the distribution method can be written as
\be D^{{\rm stat},s}_{ml}= -{{1}\over{2\chi(n)}}\sum_{\bx}x_l\langle \eta_\bx (j_m+j_m^r)\rangle^\top_n. \lb{78a} \ee
However, this can be explicitly calculated using the identity Eq.(\ref{64}), with $f(\eta)=\eta_\bz(\eta_\bx-\eta_\by),$
and the result is
\begin{eqnarray}
D^{{\rm stat},s}_{ml} & = & -{{1}\over{2\chi(n)}}\sum_{\bx}x_l\langle (\eta_\bx-\eta_\bx^{\bo,\ve_m})j_m)\rangle^\top_n \cr
\, & = & -{{1}\over{2\chi(n)}}\delta_{lm}\langle (\eta_{\ve_m}-\eta_\bo)j_m)\rangle_n \cr
\, & = & {{1}\over{2\chi(n)}}\delta_{lm}\langle c(\bo,\ve_m)\rangle_n. \lb{79}
\end{eqnarray}
(Observe that the result is indeed symmetric, as required by GOR.) Comparing with Eq.(\ref{138}), we see that the ``static''
term there also coincides with the symmetric part from the distribution function calculation. This is an important
consistency check on both methods. However, the correlation function method in the 1-component case is unable to determine
the anti-symmetric part of the diffusion matrix (as defined by the current response).
To complete this subsection, we shall now show that the matrix $\bD^s(n)$ is positive-definite and (since $\chi(n)>0$) so also is
$\bL^s(n),$ as claimed in the previous section. This is important since it guarantees that the ``entropy'' $S(n)$ is a Lyapounov functional for
the nonlinear dynamics, monotonically increasing in time to its maximum value. The positivity is a consequence of the fact that the
expression Eq.(\ref{136}) for the symmetric diffusion is, in fact, a standard {\em Green-Kubo formula}, as we now demonstrate. For
this we need to make some definitions. For local random variables $A(\bx)$ in the probability space of the microscopic process $N(\bx,\cdot),$
we define an inner-product, as
\be \langle A|B\rangle_n=\sum_\bx\,\sE_n(A(\bx)B(\bo))^\top. \lb{139} \ee
The elements $A(\bx)$ subject to the finite-norm condition $\|A\|_n^2=
\langle A|A\rangle_n<\infty$ (defined modulo total gradients
$A(\bx)\sim A(\bx)+\nabla \Phi(\bx),$ which have zero norm) form a Hilbert space. In the physics literature, this is
known as the {\em Zwanzig-Mori space} (e.g. see \cite{4}). With our normalization of the inner-product
\be \|N\|^2_n=\chi(n), \lb{140} \ee
by definition. Notice that
\begin{eqnarray}
\langle N|J_m\rangle_n & = & \sum_\bx\,\sE_n(N(\bx)J_m(\bo))^\top \cr
\, & = & \sum_\bx\,\langle\eta_\bx j_m(\bo)\rangle_n^\top \cr
\, & = & \hat{\j}^\prime_m(n)\chi(n), \lb{141}
\end{eqnarray}
where the last line follows from Eq.(\ref{41}) in the previous section. Thus, the Eulerian velocity $c_m(n)=\langle N|J_m\rangle_n/\|N\|_n$
is the projection of the microscopic flux onto the space of conserved variables spanned by $\{N\}.$ In fact, observe that with
these definitions the formula Eq.(\ref{136}) may be rewritten as
\be D_{ml}^s(n)={{1}\over{2\chi(n)}}\int_{-\infty}^{+\infty}dt
\,\,\left[\langle J_m(t) | J_l\rangle_n-{{\langle J_m|N\rangle_n\langle N| J_l\rangle_n}\over{\langle N|N\rangle_n}}\right].
\lb{142} \ee
This is the standard form of the Green-Kubo formula in which the time-integrand is a current autocorrelation function in the Zwanzig-Mori
space with the projection onto the conserved subspace subtracted. It is usual to define projection operators,
\be P={{|N\rangle\langle N|}\over{\langle N|N\rangle}},
\,\,\,\,\,\,Q=\bw-{{|N\rangle\langle N|}\over{\langle N|N\rangle}},
\lb{143} \ee
the so-called {\em Zwanzig-Mori projectors}, so that Eq.(\ref{142}) can also be written as
\be D_{ml}^s(n)={{1}\over{2\chi(n)}}\int_{-\infty}^{+\infty}dt\,\,
\langle J_m(t) | Q\cdot J_l\rangle_n. \lb{144} \ee
This exhibits $\bD^s$ in an explicitly positive-definite form, since it is the integrated autocorrelation of the
``fast'' component of the current, $I_m\equiv Q\cdot J_m.$
It is quite generally true that the Green-Kubo formulae for the dissipative
Onsager coefficients can be regarded as {\em microscopic versions} of the
FDR of 1st type, where now the relaxation coefficients are related to the
covariance of microscopic phase space functions, the ``fast'' components
of currents, and the averages are taken with respect to the steady-state
measures on phase space. In the case of the DDS, this relation is essentially
the same as one proposed some time ago by Price \cite{15} which relates
the covariance of electric current noise in semiconductors and plasmas to
the symmetric part of the bulk charge diffusion matrix. (See also
\cite{30}, Sections 3.2 and 3.5, for discussions of the experimental
validity and usefulness of this relation in semiconductors.) The KLS
derivation with the correlation function method has given this relation
an exact microscopic basis.
>From the discussion it is clear that the covariance of the ``fast''
component of microscopic currents should be identified with the
covariance of noisy currents in the Langevin equation, Eq.(\ref{127}).
Therefore, we have now a complete set of prescriptions to calculate,
in principle, the parameters of that equation from the microscopic
dynamics. The drift velocity $\bc$ and the (full) diffusion matrix $\bD$
are given by the already presented microscopic expressions, $\bc=\hat{\j}'$
and the Green-Kubo formula Eq.(\ref{54}). The noise covariance $\bR$ is then
calculated from the FDR of Price's form:
\be \bR=\bD^s\chi. \lb{144a} \ee
The use of this relation requires that the non-equilibrium entropy
$s(n)$ be known, in order to calculate $\chi(n)=-1/s''(n).$
Of course, it is quite generally true that the fluctuation-dissipation
relation, Eq.(\ref{102}), allows one to determine the Langevin noise
covariance $\bQ$ from the steady-state covariance $\bg,$ assuming that the
macroscopic hydrodynamics (and the linearized operator $\bH$) is known.
When that is the strategy, some other means must be employed to find $\bg,$
such as the high-temperature series expansion of \cite{20}.
Of course, if one's only interest were the steady-state correlations
themselves, then it would be an empty exercise to calculate $\bQ.$ However,
knowledge of $\bQ$ allows arbitrary multi-time correlations to be obtained.
These may be of interest, e.g. to describe inelastic scattering of light by
the steady-state charge density fluctuations.
\noindent {\em (3.3) Steady-State Correlations and Hydrodynamics}
If we compare the Price-type FDR Eq.(\ref{144a}) we derived from the
correlation function method with the FDR at the Langevin level, Eq.(\ref{131}),
\be \bk\cdot\bR\cdot\bk=(\bk\cdot\bD^s\bk) S(\bk), \lb{145} \ee
then we see that they are consistent if we require that the static covariance
in the Langevin formalism be wavenumber independent, as
\be S(\bk)=\chi. \lb{146} \ee
{\em This is known to be generally false!} In fact, there is a problem even with
the existence of the limit as $k\rightarrow 0.$ It was observed
in \cite{36a} from computer simulation of the DLG model that the decay of
correlations are actually a slow power-law, like $\sim r^{-d}.$
This type of behavior has been corroborated in stochastic lattice dynamics, like the DLG models,
by independent calculations using high-temperature series expansions \cite{20}. Also, the existence
of the power-laws has been confirmed in more phenomenological Ginsburg-Landau models of DDS by
field-theoretic RG calculations \cite{36b}. The generic behavior of the correlation in space seems to be
\be C(\br)\sim {{\sum_i b_i r_i^2}\over{r^{d+2}}}, \lb{148} \ee
corresponding to a behavior in wavenumber space
\be S(\bk)\sim {{\sum_i b_i k_i^2}\over{k^2}}. \lb{149} \ee
Hence, the wavenumber independence predicted by the naive form
of the Price FDR, Eq.(\ref{144a}), is contradicted by both
simulations and microscopic calculations. This may be the proper point
to remark that the simple proportionality proposed by Price is known
also to be violated at the kinetic level of description due to correlations
created by collisions: see Section 2.2 of \cite{30}. It should be emphasized
that there is no inconsistency of the long-range correlations and the
Langevin FDR Eq.(\ref{145}). In fact, as was pointed out in \cite{20},
the correlation behavior exactly like that in Eq.(\ref{149})
is recovered from the Langevin FDR, Eq.(\ref{145}), when the simple
proportionality of $\bR$ and $\bD^s$ is abandoned. The crucial difference
with the naive form of the Price FDR is the appearance of the dot products
with $\bk,$ which allows $S(\bk)$ to be wavenumber dependent even with
$\bD,\bR$ constant. This appearance of $\bk$ is due to the fact that really
only the {\em divergence} $\nabla\cdot\bJ^\prime$ of the noisy current appears
in the fluctuating hydrodynamic equations.
Since the conclusion of the correlation function argument in the previous
section seems to be generally false, the question is raised which of
the assumptions it employed might be erroneous. In addition to rather
plausible assumptions---such as the Onsager regression hypothesis---there
was also, as we have already emphasized, a rather strong analyticity
assumption on the structure function $S(\bk,t)$ at small wavenumber.
This assumption is in contradiction with the observed long-range correlations,
or their wavenumber version Eq.(\ref{149}), which implies that the limit
\be \chi(n,\hat{\bk})\equiv \lim_{k\rightarrow 0}\sum_\bx
e^{i\bk\cdot\bx}\langle \eta_\bx\eta_\bo\rangle_n \lb{150} \ee
will exist but depend upon the direction vector $\hat{\bk}.$ In other words,
the structure function cannot be expected to be even continuous at $\bk=\bo.$
Nevertheless, it is possible to repeat the KLS correlation function argument
under these weaker assumptions and, in that case, exactly the weaker FDR
Eq.(\ref{145}) is re-obtained, fully consistent with the Langevin formalism
and the microscopic results on long-range correlations. This is done
in Appendix VI of this paper. In consequence, the correlation
function derivation of the {\em linearized} hydrodynamics seems to be
consistent with all other known results.
However, the derivation of the nonlinear hydrodynamics in Section 2
by the method of nonequilibrium distributions is in worse shape.
One of its main assumptions was that there is a {\em unique} relation
between the density $n$ and chemical potential $\lambda,$ as $\lambda
=\lambda(n).$ Integrating the function $\chi^{-1}(n,\hat{\bk})$
from Eq.(\ref{150}) with respect to $n,$ one would instead expect to
have independent relations $\lambda(n,\hat{\bk})$ for each direction
vector $\hat{\bk}.$ The problem shows up already at the level of our
fundamental large deviations hypothesis in Section 2.1. Indeed, simple
examples show that when the static structure function is discontinuous at
zero wavenumber, the large deviations hypothesis must undergo some
substantial revision. Even if it remains true, the resulting function
$s(n)$ may depend upon the exact way in which the sequences of volumes
$\Lambda$ are taken to infinity! This can already be seen to occur
for a Gaussian random field of spins $\{\sigma_\bx:\bx\in\bZ^d\}$ when
the static spin structure function
\be S(\bk)=\sum_\bx e^{i\bk\cdot\bx}\langle\sigma_\bx\sigma_\bo\rangle
\lb{151} \ee
has the form of Eq.(\ref{149}) (or is otherwise discontinuous at $\bk=\bo.$)
In fact, in that case, the ``smooth'' version of the volume-average
magnetization
\be m_\en(\varphi)
\equiv \en^d\sum_\bx\,\,\varphi(\en\bx)\sigma_\bx \lb{152} \ee
satisfies the LD hypothesis for every square-integrable test-function
$\varphi,$ \footnote{This can be proved most easily by calculating the
free energy
\be f_{\varphi,\en}(h)\equiv \en^d\log\langle\exp\left(
h\sum_\bx\varphi(\en\bx)\sigma_\bx\right)\rangle,
\lb{153} \ee
to be
\be f_{\varphi,\en}(h)={{h^2}\over{2}}\cdot {{1}\over{(2\pi)^d}}
\int_{\left[-{{\pi}\over{\en}},{{\pi}\over{\en}}\right]} d^d\bk\,\,
|\hat{\varphi}(\bk)|^2 S(\en\bk). \lb{154x} \ee
Since its limit exists
\begin{eqnarray}
f_\varphi(h) & \equiv & \lim_{\en\rightarrow 0}f_{\varphi,\en}(h) \cr
& = & {{1}\over{2}}K_\varphi h^2, \lb{155x}
\end{eqnarray}
and is differentiable for all real $h,$ Theorem II.6.1 of Ellis
\cite{22} applies.} with the quadratic entropy function
\be s_\varphi(m)={{m^2}\over{2K_{\varphi}}}, \lb{156x} \ee
in which
\be K_\varphi={{1}\over{(2\pi)^d}}\int d^d\bk\,\,
|\hat{\varphi}(\bk)|^2 \chi(\hat{\bk}). \lb{157x} \ee
Because of the $\hat{\bk}$ dependence of $\chi,$ the result clearly
depends upon the precise test function adopted. For instance, if a
function $\varphi(\bx)=\prod_{i=1}^d{{1}\over{2a_i}}\chi_{[-a_i,a_i]}(x_i)$
were chosen, corresponding to the rectangular parallelopiped
$[-a_1,a_1]\times\cdots\times[-a_d,a_d]$ scaled to infinity, then the
function $s_\varphi(m)$ obtained would depend upon the aspect ratios
of the box sides. This may seem an artificial mathematical example,
but it is just the Gaussian model of a uniaxial ferromagnet/ferroelectric with
dipolar interactions, which has appeared in studies of critical behavior
of those systems \cite{36c}. It is rather well-known for dipolar systems
that the thermodynamics may be shape-dependent (see \cite{5}) and the
structure-function singularity in that case has even been experimentally
observed \cite{36d}. However, it is perhaps less surprising to encounter
such phenomena in systems with long-range forces, rather than in the DLG
models where all particle-interactions are short-ranged.
Since the nonequilibrium distribution function method used rather
crucially the relation $\lambda(n)$ between chemical potential and
density to guess the correct modification of the reference measure
to produce a given smooth density profile, it is unclear to us how
to proceed when that relation breaks down. Therefore, it is not
clear to us that the nonlinear drift-diffusion equation of the
form of Eq.(\ref{58}) even still holds in the general case! It should
be stressed that the derivation of hydrodynamics in Section 2 was based
upon the explicit assumption of fast decay of correlations, and its
validity is not questioned in such cases. Situations with such
``fast decay'' do exist, such as the ASEP discussed in Appendix III,
whose stationary measures are completely uncorrelated (product measures),
and, more generally, DLG models of ``gradient type'' (see \cite{11},
Appendix 4). These even include cases where the measures are not Gibbsian \cite{45b,75}.
However, it is rather disconcerting that such situations
seem, according to present evidence, to be {\em non-generic} and,
in general, the long-range correlations and discontinuous susceptibility
at zero-wavenumber must be expected. On the other hand, we know of
no experimental evidence for these phenomena in physical examples
of DDS, although they should, in principle, show up clearly in
electromagnetic scattering from the homogeneous steady state.
%* The following footnote is added
\footnote{Note that the $k^{-4}$ Rayleigh peaks which have been observed \cite{99} in the
steady-states with temperature gradients have a different character. These latter systems
are local equilibrium states and the physical space correlation is there $\sim r^{-(d-2)}$
but proportional to $\en^2$ (or the square of the temperature gradient.) Thus,
while longer-range than those considered above, the correlations in those states
are {\em weaker,} locally vanishing as $\en\rightarrow 0.$ It seems to be harder
to realize the periodic geometry of the DDS than the steady states driven by boundary
conditions, making experimental study difficult.} We regard the reconciliation of the
long-range correlations and the nonlinear hydrodynamics to be one of the main outstanding
theoretical puzzles in the driven-diffusive systems.
\section{Large Hydrodynamic Fluctuations}
\noindent {\em (4.1) A Generalized FDR for the Nonlinear Fokker-Planck Equation}
In the previous section we considered the problem of constructing the
proper Langevin model of the small ($O(\en^{d/2})$) fluctuations about the
hydrodynamic behavior governed by the driven diffusion equation
Eq.(\ref{58}). However, there are also very rare large fluctuations
of order O(1). For the description of these there are a number of
formalisms in the literature: for example, nonlinear Fokker-Planck and Langevin
equations, or nonlinear Onsager-Machlup action functionals. For a
discussion of each of these and their interrelations see \cite{6,24,37}
and references therein. What we shall investigate here is the possibility
of applying such methods also to the systems without local equilibrium,
focusing on our main example of the DDS models. In general, the same types
of dynamical and statistical assumptions enter into the derivation of
the fluctuation theory as into the derivation of the hydrodynamical
equations themselves. Therefore, we believe that the ``level-2''
nonequilibrium distribution method used by Zubarev and Morozov in \cite{6}
could be also successfully applied to such systems (at least in the
``fast-decay case'' discussed in the last section).
However, rather than carry through such a microscopic derivation here, we would
like to point out a quicker route to the final result, which may be more generally
useful. This method exploits the fact that hydrodynamical equations derived by
the previous procedure are automatically in the ``Onsager force-flux form.'' In
that case it is possible to invoke a general fluctuation-dissipation relation
of first type for nonlinear Fokker-Planck equations, which allows one to write
down directly the fluctuation theory by inspection of the hydrodynamic equations
without further calculation. It was R.Graham in his paper \cite{17} who formulated
the FDR's we find useful in this context and who, particularly, emphasized their
validity without any restriction of time-reversibility, even at the
macroscopic level. (We should point out that other nonlinear FDR's of a
different character and with their own specific applications are discussed by
Stratonovich in his monograph \cite{32}.) It may be that this work has not
received sufficient attention because of its abstract style and its formal
differential-geometric language. Therefore, we would like here to
illustrate its usefulness in the very concrete setting of electrical
current-carrying systems, where it will be shown to give a nonlinear
generalization of Price's noise-diffusion relation. Furthermore, we shall present
a simplified version of Graham's relation, which turns out to avoid the
geometric complexities of his original work but to be perfectly adapted
for application to spatially-extended, macroscopic systems. The rest
of this subsection will be devoted to the first task of explaining the generalized
nonlinear FDR's of Graham in our simplified version and then comparing them
with his more general formulation.
The most arbitrary Fokker-Planck equation in the $p$ variables
$\{\alpha^i|i=1,...,p\}$ may be written in the form
\be {{\partial P}\over{\partial t}}(\balpha,t)=-{{\partial}\over{\partial \alpha^i}}
\left[ J^i(\balpha)P(\balpha,t)\right]
+{{\partial^2}\over{\partial \alpha^i\partial \alpha^j}}
\left[Q^{ij}(\balpha)P(\balpha,t)\right], \lb{154} \ee
where the sum over repeated indices always goes from $1$ to $p$ and
we have chosen notations in close accord with our discussion of
linear Langevin equations in section 3.1. The Fokker-Planck equation itself
corresponds to the Ito stochastic differential equation
\be \dot{\alpha}^i=J^i(\balpha)+g^i_\mu(\alpha)\eta^\mu, \lb{155} \ee
in which $\eta^\mu(t)$ are white-noise forces with zero mean and covariance
\be \langle\eta^\mu(t)\eta^\nu(t')\rangle=2\delta^{\mu\nu}\delta(t-t'),
\lb{156} \ee
and the functions $g^i_\mu(\balpha)$ are a set of multiplicative
noise-strengths such that
\be Q^{ij}(\balpha)=g^i_\mu(\balpha)g^j_\mu(\balpha). \lb{157} \ee
(Throughout this section we take $k_B\equiv 1$ for simplicity).
All of the analysis we shall make here depends upon a crucial assumption
on the Fokker-Planck diffusion matrix $\bQ(\balpha)$
\footnote{ Note that the Fokker-Planck equation is itself a driven
diffusion equation, but describing flow of probability in the state space of
the $\balpha$'s rather than of particle number in physical space! No confusion
should result when we use the standard terminology of ``diffusion'' for $\bQ$
and ``drift'' for $\bJ,$ qualified, if necessary, by the adjective
``Fokker-Planck.''}, namely, that it is ``divergence-free'' in the sense that
\be {{\partial Q^{ij}(\balpha)}\over{\partial \alpha^j}}=0. \lb{158} \ee
As shown in the original paper of Graham \cite{17}, and as we shall discuss
further below, results similar to those obtained here may be obtained
without this assumption. However, although it is a rather special
restriction in the class of general Fokker-Planck equations, we shall
show that the spatially-extended, conservative systems always satisfy this assumption
automatically and that it provides a suitable basis for our applications.
Under this single assumption, Eq.(\ref{158}), we now show that the systematic part of the Langevin
dynamics, the Fokker-Planck drift vector $J^i,$ may be decomposed
into a ``conservative'' part $r^i$ and a ``dissipative'' part $d^i,$
\be J^i(\balpha)=r^i(\balpha)+d^i(\balpha), \lb{159} \ee
in a way quite analogous to our decomposition for the linear Langevin
dynamics. (Of course, corresponding to different choices of $\bQ$ there will be
different decompositions of $\bJ$!) As in the linear case, making the decomposition
requires knowledge of the stationary measure $P_0(\balpha),$ or of the corresponding
``entropy'' defined by
\be S(\balpha)\equiv \log P_0(\balpha). \lb{160} \ee
The key point of the argument is to use
the ``divergence-free'' assumption Eq.(\ref{158}) to rewrite the
Fokker-Planck equation as
\be {{\partial P}\over{\partial t}}(\balpha,t)=-{{\partial}\over{\partial \alpha^i}}
\left[ J^i(\balpha)P(\balpha,t)-Q^{ij}(\balpha)
{{\partial}\over{\partial \alpha^j}}P(\balpha,t)\right]. \lb{161} \ee
We can then {\em define}
\be d^i(\balpha)\equiv Q^{ij}(\balpha)
{{\partial S(\alpha)}\over{\partial \alpha^j}}, \lb{162} \ee
and
\be r^i(\alpha)\equiv J^i(\balpha)-d^i(\balpha). \lb{163} \ee
Substituting $P=e^S$ into the Eq.(\ref{161}), we find that the
$\bQ$ diffusion term cancels with the $\bd$ part of the drift, giving
\be {{\partial}\over{\partial \alpha^i}}
\left[ r^i(\balpha)e^{S(\balpha)}\right]=0. \lb{164} \ee
This expresses the stationarity of $P_0=e^S$ under the deterministic
evolution by $\br$ alone.
We can further verify that the above decomposition corresponds to one into
``conservative'' and ``dissipative'' parts, and, indeed,
puts $\bJ$ into the ``Onsager form'' with respect to the potential S.
To see this, we identify $L^{ij}_s\equiv Q^{ij},$ which gives
\be d^i(\balpha)= -L^{ij}_s(\balpha)X_j(\balpha), \lb{166} \ee
in terms of the ``force''
\be X_i(\balpha)\equiv -{{\partial S(\balpha)}\over{\partial \alpha^i}}.\lb{167} \ee
Furthermore, following an idea of Graham \cite{17}, we represent
the divergence-free vector $r^ie^S$ as
\be r^i(\balpha)e^{S(\balpha)}\equiv {{\partial
F^{ij}(\balpha)}\over{\partial \alpha^j}}, \lb{168} \ee
with $F^{ij}$ antisymmetric, so that
\be r^i(\balpha)=L^{ij}_a(\balpha)
{{\partial S(\balpha)}\over{\partial \alpha^j}}
+{{\partial L^{ij}_a(\balpha)}\over{\partial \alpha^j}},
\lb{169} \ee
in terms of a new antisymmetric matrix $L^{ij}_a\equiv F^{ij}e^{-S}.$
Together Eq.(\ref{166}) and Eq.(\ref{169}) give the representation of the
full drift
\be J^i(\balpha)=-L^{ij}(\balpha)X_j(\balpha)
+{{\partial L^{ij}_a}\over {\partial \alpha^j}},
\lb{170} \ee
with
\be L^{ij}(\balpha)\equiv L^{ij}_s(\balpha)+L^{ij}_a(\balpha). \lb{171} \ee
The latter is now naturally interpreted as the ``Onsager coefficient'' and
Eq.(\ref{170}) is in the usual ``force-flux'' form except for the additional
divergence term $\partial L^{ij}_a/\partial \alpha^j.$ In fact, for the
applications we shall consider it will be generally true also that
\be {{\partial L^{ij}_a(\balpha)}\over{\partial \alpha^j}}=0, \lb{172} \ee
for the same reasons which imply the divergencelessness of $L^{ij}_s$
(see below). In that case, the standard Onsager form is obtained.
Furthermore, it follows under the condition Eq.(\ref{172}) that
the $\br$ part of the drift enjoys the Liouville property, or divergence-free condition
\be {{\partial r^i(\balpha)}\over{\partial \alpha^i}}=0, \lb{172a} \ee
by a direct calculation. Therefore, $\br$ is indeed ``conservative''
in the sense that
\be {{\partial S(\balpha)}\over{\partial \alpha^i}}r^i(\balpha)=0,
\lb{173} \ee
which follows from the combination of Eqs.(\ref{164}) and (\ref{172a}). In that
case, the entire change of $S$ under the systematic evolution is due to $\bd,$
which gives
\begin{eqnarray}
{{d}\over{dt}}S(\balpha) & = & {{\partial S(\balpha)}\over{\partial \alpha^i}}
d^i(\balpha) \cr
\,& = & L^{ij}_s(\balpha)X_i(\balpha)X_j(\balpha) \cr
\,& \geq & 0. \lb{174}
\end{eqnarray}
Therefore, $\bd$ is truly the ``dissipative'' part.
>From these considerations we can see that the identification
\be L^{ij}_s(\balpha)\equiv Q^{ij}(\balpha), \lb{165} \ee
is in fact a nonlinear extension of the {\em FDR of first type}, of the sort
established in the linear Langevin case in Section 3.1. It was first obtained
(in a more general setting) by Graham in \cite{17}. Before discussing
his results, let us reformulate the relation in a way directly
useful to us below. The starting point above was the Fokker-Planck equation
itself. However, it is instead our goal to deduce that equation, having in hand
the microscopic derivation of the hydrodynamic law in the Onsager form.
Since the systematic part of the Langevin dynamics must be the
deterministic hydrodynamics itself, we would like to have a way to guess
the proper choice of the noise to add to it. For that purpose, we may reverse
the previous considerations to derive a result expressed formally in
the following
\begin{Prop}
Consider the Fokker-Planck Eq.(\ref{154}) with the divergence-free
condition $\partial Q^{ij}/\partial\alpha^j$ \newline
$=0.$ If $J^i=r^i+d^i$ with ${{\partial}\over{\partial \alpha^i}}
\left[ r^ie^S\right]=0$ and $d^i=L^{ij}_s{{\partial S}\over{\partial
\alpha^j}},$ then $Q^{ij}=L^{ij}_s$ implies that $P_o=e^S$ is stationary.
Furthermore, when also $\partial L^{ij}_s/\partial \alpha^j=0,$ then $P_0=e^S$
stationary and non-degenerate implies conversely that $Q^{ij}=L^{ij}_s.$
\end{Prop}
Here ``non-degeneracy'' of $P_0$ just signifies that
${{\partial^2 P_0}\over{\partial \alpha^i\partial\alpha^j}}$ is a
nonsingular matrix for almost all $\balpha$ in the state space.
We leave the elementary proof of this proposition to the reader. We note
only that its statement specifies---under the expressed conditions---the {\em
unique} choice of the noise strength to obtain the stationary measure
$P_0=e^S.$
In his original work \cite{17}, Graham obtained the equivalent
decomposition as that above in the setting of the most general
Fokker-Planck equation, without the ``divergence-free'' condition
Eq.(\ref{158}). His strategy was to redefine the notion of
``derivative'' in order to make the condition always true! In fact,
it is possible to take the Fokker-Planck diffusion $Q^{ij}(\balpha)$
as a contravariant ``metric'' tensor defining a Riemannian geometric
structure in the state space of the $\balpha$'s. In that case, one
can take the associated definition of ``connection'' and ``covariant
derivative'' so that the metric tensor itself has vanishing covariant derivative:
\be Q^{ij}_{;k}=0, \lb{175} \ee
expressing that ``lengths'' are invariant under the ``parallel transport''
in state space. (We follow the standard device of denoting
the covariant derivative $D/D\alpha^k$ by $(\cdot)_{;k}.$)
This stronger condition implies at once the zero covariant-divergence
result $Q^{ij}_{;j}=0.$ In that case, it is not hard to put the
Fokker-Planck equation into a completely covariant version
\be \partial_t P=-\left(J^iP-Q^{ij}P_{;j}\right)_{;i}, \lb{176} \ee
identical in structure to Eq.({\ref{161}) above. Note here that $P$ and
$J^i$ {\em are not the same as those above}, but correspond also to
covariant versions.
Having arrived at the equation (\ref{176}), Graham then derived the
same decomposition as ours above, with exactly the same argument,
simply replacing ordinary derivatives with covariant derivatives
everywhere. Note that Graham's decomposition is the full nonlinear
generalization of that introduced by Tomita and Tomita \cite{29}
for linear Langevin dynamics. While Graham did not explicitly state that
the identity between $Q^{ij}$ and $L^{ij}_s$ was a type of FDR nor even mention
the relation to ``Onsager form,'' he did make a special emphasis of the fact
that the {\em FDR of second type} extended to this general Fokker-Planck
context without, in particular, any assumptions on time-reversibility.
This required a special coupling of the external force, constructed
according to the principle that the stationary measure in the presence
of forces $F_i$ should change by the exponential modification $\propto
\exp[F_i\alpha^i]$ (see also \cite{32}, Section 5.5.1). The correct way
to achieve this is by introducing the force into the Onsager form of $J^i,$ as
\be J^i=-L^{ij}(X_j-F_j)+(L^{ij}_a)_{;j}. \lb{176a} \ee
%*Averages taken outside derivatives
If one defines as usual the response function $G^{ij}(t)=
\langle {{\delta\alpha^i(t)}\over{\delta F_j(0)}}\rangle$ to this imposed
force, then Graham showed that
\be \langle\dot{\alpha}^i(t)\alpha^j(0)\rangle=
-\left[G^{ij}(t)-G^{ji}(-t)\right], \lb{177} \ee
just as the usual FDR of second type. (In fact, we can obtain the same
result also in our restricted context.) Observe that it is necessary in
general to know the stationary meaure $P_0$ in order to make the
appropriate coupling of $F_i,$ since the decomposition of $J^i$ into
$r^i+d^i$ used $S.$ As Graham observed, the only role of
time-reversibility ---when it is present---is that it allows this
decomposition to be made {\em a priori} on the basis that $r^i$ is
odd and $d^i$ even under time-reversal. These statements are equivalent
to the well-known ``potential conditions'' of Graham and Haken for detailed
balance \cite{38}, and they imply that the Onsager matrix $L^{ij}(\balpha)$
obeys the Onsager reciprocal relations $\en_i\en_jL^{ij}(\en\cdot\balpha)
=L^{ji}(\balpha).$ \footnote{Without the reversibility assumption, rather formal
``generalized'' OR and potential conditions can be formulated also in
the nonlinear Fokker-Planck context. E.g. see Section VI.B of \cite{39}.}
\noindent {\em (4.2) Fluctuating Equations for the Driven Diffusion Model}
We now turn to the problem of constructing the nonlinear fluctuation theory
corresponding to the drift-diffusion equation Eq.(\ref{58}). As discussed
above, we make no attempt here to derive it microscopically but simply
assume it to exist and then use the previous results to restrict its form.
To be more precise, we make the following:
\begin{Hyp}
There exists a Langevin description of the nonlinear fluctuations of the form
\be \partial_\tau n= -\partial_m[\hat{\j}_m(n)-\en D_{ml}(n)
\partial_l n+\en^{d/2} j'_m], \lb{178} \ee
with the stationary measure
\be P_o^\en[n]\propto \exp\left\{\en^{-d}S[n]\right\}. \lb{179} \ee
\end{Hyp}
Let us say a few words to justify some particular aspects of the
hypothesis. First, it is essential that the equation preserve the feature
of local particle conservation, since that is a fundamental property of
the microscopic dynamics. Therefore, the Langevin force ought to be added
as an additional ``noisy current'' into the driven-diffusion equation
(which must be recovered in the $\en\rightarrow 0$ limit.) The basic
assumption is that the correct stationary measure of the Langevin
dynamics is that given in Eq.(\ref{179}). It seems reasonable from the
way in which $s(n)$ was defined in terms of the fluctuation probabilities
for density $n$ in the stationary states of the microscopic dynamics.
Since $S[n]=\int_\Lambda d^d\br\,\,s(n(\br)),$ the main assumption involved
here is a ``locality property'' of static fluctuations in the extended system,
which ought at least to be valid in the ``fast-decay'' case. From a
physical point of view, the hydrodynamical density which appears in these
equations is supposed to represent ``coarse-grained values'' obtained by
averaging over ``mesoscopic'' cells of size intermediate between the scale of
the particles and the system size. \footnote{More formally, the cells are of
linear dimension $\ell_0\sim\en^\gamma$ with some $0<\gamma<1$ in the limit
$\en\rightarrow 0,$ with the macroscopic domain $\Lambda$ fixed.}
For such variables there will be simultaneously nonlinear effects of
the dynamics and strong stochastic influence of the molecular noise.
Because of the coarse-graining procedure these hydrodynamic fields should
contain no modes smaller than the cell-size, and it is simplest to represent
them mathematically by Fourier series truncated at a wavenumber $k_0$ the
inverse of the cell-diameter $\ell_0,$ as we do here, or alternatively by
discretization on a lattice with grid-unit $\ell_0.$ Since the theory of stochastic
PDE's is difficult, it is desirable even from a purely mathematical point of
view to define the nonlinear Langevin dynamics instead as a stochastic ODE
for a finite set of variables (Fourier modes or lattice field-variables).
For a further discussion of the microscopic derivation of such equations by
``coarse-graining'' see Zubarev and Morozov, Section 2.1 of \cite{6}. We
shall also comment further at the end of this section upon the physical
uses (and limitations) of our nonlinear fluctuation equations for the DDS.
The basic problem is to guess the form of the noisy currents in Eq.(\ref{178}).
We shall first present here the result, and afterward give its
justification. The Langevin currents are assumed in the particular form
\be j'_m(\br,\tau)=\sqrt{\en}g_m^\sigma(n(\br,\tau))\eta_\sigma(\br,\tau),
\lb{180} \ee
$\sigma=1,...,d,$ where $\eta$ is a spacetime white-noise with covariance
\be \langle \eta_\sigma(\br,\tau)\eta_\rho(\br',\tau')\rangle
=2\delta_{\sigma\rho}\delta^d(\br-\br')\delta(\tau-\tau'). \lb{181} \ee
(Observe that $\delta(\br-\br')$ here denotes a ``coarse-grained'' delta
functional, or $\Delta(\br-\br')$ in the notation of Zubarev and Morozov
\cite{6}.) This choice of the noisy currents is not entirely necessary but does
have some convenient features. For example, the discussion of Morozov in
\cite{37} implies that for this form the Ito and Stratonovich interpretations coincide. This allows
us to use ordinary rules of calculus. With this choice of noise, our proposed Langevin equation
then leads to the Fokker-Planck equation
\begin{eqnarray}
{{\partial}\over{\partial\tau}}P(n,\tau)
& = & -\int d^d\br\,\,{{\delta}\over{\delta n(\br)}} J(\br;n)P(n,\tau) \cr
\,& & \,\,\,\,\,\,\,\,\,+\en^{d+1}\int d^d\br\int d^d\br'\,\,
{{\delta^2}\over{\delta n(\br)\delta n(\br')}}
Q(\br,\br';n)P(n,\tau), \lb{182}
\end{eqnarray}
where
\be J(\br;n)=-\partial_m[\hat{\j}_m(n(\br))-\en D_{ml}(n(\br))\partial_l n(\br)], \lb{183} \ee
and
\be Q(\br,\br';n)= -\partial_m[Q_{ml}(n(\br))\partial_l\delta(\br-\br')], \lb{184} \ee
with
\be Q_{ml}(n)=g_m^\sigma(n)g_l^\sigma(n). \lb{185} \ee
Any other choice of the Langevin equation which leads to the same
Fokker-Planck equation is stochastically equivalent and would
suffice just as well. For example, other square roots of the operator Q
could be employed in the noise term, such as the positive one
(e.g. see Section 6 of \cite{24}), but it is more convenient to choose the local form above.
The essential part of our choice is that the matrix $\bQ$ must be taken as
\be Q_{lm}(n)=L_{lm}^s(n), \lb{186} \ee
with $\bL^s$ the Onsager matrix calculated in Section 2. This is given, for example,
as $\chi(n)$ times $\bD^s$ written as the Green-Kubo formula in Eq.(144).
Therefore, the $\{g_m^\sigma\}$ are introduced here as an orthogonal set
of eigenvectors of $\bQ$ (normalized to have length given by the
square root of the corresponding eigenvalues.)
The motivation for this prescription is that the systematic part of the
equation, the driven diffusion dynamics, may be written in the form
\be J(\br;n)= r(\br;n)+d(\br;n), \lb{187} \ee
with
\be r(\br;n)=-\bc(n)\cdot\grad_\br n
+\en\int d^d\br'\,\,L_a(\br,\br';n)
{{\delta S(n)}\over{\delta n(\br')}}, \lb{188} \ee
and
\be d(\br;n)= \en\int d^d\br'\,\,L_s(\br,\br';n)
{{\delta S(n)}\over{\delta n(\br')}}, \lb{189} \ee
with
\be L(\br,\br';n)= -\partial_m[L_{ml}(n(\br))\partial_l\delta(\br-\br')].
\lb{190} \ee
(Actually, as already observed, the $\bL_a$ term in (\ref{188}) vanishes for the
one-component system, but we shall not make use of this fact.) We shall
show that the $r$ term leaves invariant the proposed stationary measure:
\be \int d^d\br\,\,{{\delta}\over{\delta n(\br)}}\left(r(\br;n)e^{\en^{-d}S(n)}\right)=0.
\lb{190a} \ee
Furthermore, we shall show that the fundamental property of the noise
covariance and the Onsager matrix (kernel) are satisfied:
\be \int d^d\br'\,\,{{\delta L(\br,\br';n)}\over{\delta n(\br')}}
= \int d^d\br'\,\,{{\delta Q(\br,\br';n)}\over{\delta n(\br')}}=0.
\lb{191} \ee
>From these facts we can infer by our previous proposition that the chosen
form of the noise will lead uniquely to our desired stationary measure.
It will turn out, incidentally, that the factor $\en^{-d}$ in
$\exp\left(\en^{-d}S\right)$ is fixed only by the requirement that
\be d(\br;n)= \en^{d+1}\int d^d\br'\,\,Q(\br,\br';n)
\cdot\en^{-d}{{\delta S(n)}\over{\delta n(\br')}},
\lb{191a} \ee
while all the other terms satisfy invariance with an arbitrary factor.
The fundamental property of the Onsager matrix, Eq.(\ref{191}), was already
demonstrated in the work of Zubarev and Morozov, Appendix E of \cite{6}.
In fact, it is a general property of kernels of the form in
Eqs.(\ref{184}),(\ref{190}). For completeness, we will repeat the proof
here. The argument uses the fact that the UV regularization procedure
is space-reflection symmetric. By explicit calculation,
\be {{\delta L(\br,\br';n)}\over{\delta n(\br'')}}=
-\partial_m[L_{ml}'(n(\br))\delta(\br-\br'')\partial_l\delta(\br-\br')].
\lb{192} \ee
Setting $\br''=\br'$ in this expression gives then
\be {{\delta L(\br,\br';n)}\over{\delta n(\br')}}=
-\partial_m[L_{ml}'(n(\br))\delta(\br-\br')]\cdot\partial_l\delta(\bo).
\lb{192a} \ee
However,
\begin{eqnarray}
\grad\delta^d(\bo) & =& {{1}\over{|\Lambda|}}\sum_{kFrom these results we see finally that the most probable history subject to a
given set of imposed constraints will be determined by a variational
principle of {\em least excess dissipation,} which generalizes the famous
``Onsager principle of least dissipation'' to fluctuations in
homogeneous nonequilibrium steady-states. One should be wary
in interpreting such variational principles. The meaning of ``constraint''
here is ``passive'' rather than ``active'' \cite{45}. That is, the principle
does not apply to experimental situations where the constraints are
enforced by some external means, but rather to the---exceedingly
rare!---subensemble of fluctuation histories appearing spontaneously
in the steady-state which satisfy the constraints. The most plausible
application of the principle is instead to characterize the (absolute)
most probable state without any constraints whatsoever, especially when there are
multiple solutions of the deterministic hydrodynamic equations and internal
noise is the physical selection mechanism.
\section{Conclusions}
\noindent {\em (5.1) Outstanding Problems for DDS}
Without a doubt the most outstanding issue for general DDS left unresolved in this work is the validity of the
drift-diffusion equation (\ref{58}). The main concern is consistency with the long-range correlations
discussed in Section 3.3. We have seen that these correlations are likely to affect the thermodynamics
of homogeneous steady-states of the DDS, making it shape-dependent. Therefore, several of the properties
implicitly assumed in the derivation of hydrodynamics appear to fail. The doubt centers mostly about
the {\em nonlinear} drift-diffusion equation. As discussed in Appendix VI, the correlation-function derivation of the
{\em linearized} hydrodynamics appears to generalize to the ``slow decay'' case, and yields consistent results. Furthermore,
as discussed in \cite{20} (also Section 3.3) the linear drift-diffusion equation with a white-noise force actually
{\em predicts} the expected power-law decay if the ``naive'' Price FDR is abandoned.
Even if the deterministic equation is valid, there is still a question whether the
stochastic version Eq.(\ref{201}) proposed in Section 4.2 gives the correct description
of fluctuations in general. This nonlinear Langevin equation always leads to the naive FDR upon linearization.
The general fluctuation dissipation theorem which we proved in Section 4.1 established that this nonlinear Langevin equation
is {\em unique} subject to the conditions of Hypothesis 2 in Section 4.2. If the long-range correlations occur, then one of
these conditions must be false in general. We strongly suspect that it is the assumed form of the stationary distribution,
Eq.(\ref{179}), which is wrong. In fact, we have seen in Section 3.3 that there is considerable delicacy in the
description of the steady-state fluctuations even for the homogeneous state. The function $s(n),$ if
it exists, may depend upon the way in which volumes go to infinity. This indeterminacy should be reflected
in the stationary distribution of the nonlinear Langevin equation, although we do not have a definite proposal
alternative to Eq.(\ref{179}).
Perhaps it is fair to emphasize here the successes of our analysis: we have derived the
drift-diffusion equation (\ref{58}) by a fully microscopic procedure, yielding in principle
explicit expressions for all the quantities involved (entropy function, drift velocities,
and Onsager coefficients). Furthermore, we have established for this equation an $H$-theorem
governing the approach of solutions to homogeneous steady-states. A nonlinear Langevin theory
of fluctuations was proposed, in which a multiplicative white-noise force is added to the Eq.(\ref{58}),
unique subject to the condition of having the desired stationary distribution Eq.(\ref{179}).
Although we lack a rigorous proof, we are quite confident that these results are correct as stated
in the ``fast decay'' case. Examples of the latter type are known---such as the ASEP model of Appendix III
---so that these results have some nontrivial domain of validity.
\noindent {\em (5.2) Remarks on the Nonequilibrium Distribution Function Method}
Our analysis of the DLG models was meant in part to be a case study of the nonequilibrium distribution
method of \cite{2,3,4}. One of the main questions we set out to answer in this work was: {\em Where does
the ``nonequilibrium distribution function'' method apply?} Our primary focus was on the class of systems not
in local thermodynamic equilibrium, but we believe that the discussion has clarified the conditions for its
applicability also in the standard situations. The key requirement is the condition of ``fast decay''
of correlations in space-time, and when that holds the method seems to be well-founded even if
local states are not reversible. Therefore, there seems to be a basis to apply the method to a
wide range of externally-driven nonequilibrium systems, discussed further in subsection 5.3 below.
The primary caveat concerns the long-range correlations, which seem to be ubiquitous in such non-equilibrium
systems. The slow correlation decay makes the derivation by the nonequilibrium distribution suspect for two
reasons: (i) the integrals over space-time of correlations no longer obviously converge, and (ii) there
is no longer necessarily a unique relationship between density and chemical potential. It is not clear
that these difficulties represent a {\em failure} of the method, since, as discussed above, it is possible
that the drift-diffusion equation (\ref{58}) itself is no longer valid. This is one of two alternative
scenarios. The other is that the nonlinear equation is valid but requires a more refined derivation. A possibility
worth exploring is the ``level-2'' approach of Zubarev-Morozov \cite{6}, since that method does not assume
a chemical potential as a function of density but rather a {\em chemical potential functional} $G[f]$
required to produce a given {\em statistical distribution} $f[\rho]$ in the space of density fields $\rho(\br).$
The functional relationship is elementary, $G[f]= -\log f, $ so that no ambiguities from correlation
effects appear at that level.
\noindent {\em (5.3) Other Applications}
Other possible applications of the methods discussed in this paper are reaction-diffusion systems (RDS) with local
sources of reagents, plasmas in external electric and magnetic fields, and granular flow under the action of
gravity, etc. Any system with hydrodynamic behavior can in principle be considered. Here we will just say a
few words about the former two applications. The last subject is discussed in \cite{45c}.
A hydrodynamic description of chemical systems is possible if the reaction rates are sufficiently slow,
leading to reaction-diffusion equations. For example, see \cite{45d} in the context of stochastic lattice
gas models. The homogeneous case was already treated by Zubarev with the distribution function method in
Section 23.5 of his book \cite{2}. It should be possible to treat also the inhomogeneous case with diffusion
by his methods. A crucial difference with the DDS discussed in the present work is the non-conservative
character of RDS. The reaction-diffusion equation is not of the form of a pure continuity equation, but contains
also sources and sinks. Therefore, a number of aspects of our previous discussion would need to be changed.
It should be noted that rigorous results are available for hydrodynamic large-deviations in the
lattice-gas models of RDS \cite{45e,45f}. These can provide checks of formal theory.
Another interesting area of application is plasma systems, where the local equilibrium description also
generally breaks down. Magnetohydrodynamics (MHD) is the generally proposed hydrodynamic description.
A comprehensive discussion in the context of transport equations is contained in \cite{45g}.
Here we will just make a few remarks pertinent to the issue of Onsager reciprocity in plasmas, which is
reviewed by Krommes and Hu in \cite{39}. It should be clear that our perspective largely agrees with
theirs, since both works agree that OR is intrinsically connected with time-reversibility. It may be
true for other reasons that a transport matrix is symmetric, such as self-adjointness of a collision
operator \cite{45h}, but this is not OR. In fact, as we have emphasized the {\em dissipative} Onsager
matrix is always the symmetric part, but this is {\em not} a consequence of OR and has nothing to do with
time-reversibility. We depart slightly from \cite{39} in regarding GOR as a rather formal generalization
of the original OR, and not nearly as useful, since it relates a physical situation to an unphysical
one, the time-reversed state. The latter cannot be prepared in the laboratory since all reservoir
dynamics would also need to be reversed. It is only theoretically possible to consider such reversal
in the nonequilibrium steady-state for some artifical dissipative dynamics of the Nos\'{e}-Hoover type or for randomly
modelled heat baths with the stochastic notion of time-reversal discussed in this work.
\section{Appendices}
\noindent {\bf Appendix I: Are Stationary Measures of the DLG Non-Gibbsian?}
The reversible ($E=0$) lattice gas is constructed in such a way that the canonical Gibbs measures
$P_{\Lambda,N}(\eta)=Z^{-1}_{\Lambda,N}e^{-\beta H_\Lambda(\eta)}\delta_{N_\Lambda,N}$ are stationary
for each $N=0,1,...,|\Lambda|,$ where the short-range Hamiltonian $H_\Lambda$ is prescribed in the detailed
balance condition for the exchange rates. Under finite drive, $E\neq 0,$ the canonical Gibbs measures will
in general no longer be stationary. This leads to the question whether there exists some other
Hamiltonian, $\widetilde{H}_\Lambda,$ such that the stationary measures are of the form $\widetilde{Z}^{-1}_{\Lambda,N}
e^{-\beta \widetilde{H}_\Lambda(\eta)}\delta_{N_\Lambda,N}.$ In finite volume, the answer is clearly positive,
since the theory of finite-state Markov jump processes guarantees that for fixed $N$ the stationary measure
does not vanish on any configuration. Thus the question only becomes meaningful at infinite volume. Put
differently, one has to understand how $\widetilde{H}_\Lambda$ depends on the volume $\Lambda,$ for a given
$E$ and $n.$
Let $P$ be a stationary, translation-invariant state of the DLG in infinite volume. We consider the
distribution of the configuration $\eta_B$ in the bounded domain $B$ conditioned on the
outside configuration $\eta_{B^c}.$ The formal requirement for $P$ to be a {\em Gibbs measure}
is that there exist a set of interaction potentials $\{J_A:A\subset\bZ^d\}$ on finite subsets of the
lattice, which is translation-invariant, $J_A+\bx=J_A,$ and summable, $\sum_{A\ni\bo}|A||J_A|<\infty,$
such that the conditional expectations are of the form
\be P(\eta_B|\eta_{B^c})=Z^{-1}_B(\eta_{B^c})
\exp\left[-\sum_{A\cap B\neq\emptyset}\,J_A\eta^A\right]. \lb{I1} \ee
Here $\eta^A=\prod_{\bx\in A}\eta_\bx.$ Essentially, (\ref{I1}) states that conditional expectations
do not vanish and depend continuously on $\eta_{B^c}.$ See \cite{19} for an extensive account of
Gibbs measures and their properties. We only mention here that under suitable conditions on the potential---
amounting to conditions of high temperature and low density---the Gibbs measure will be unique and exhibit exponential
clustering. That is, for any local functions $f,g$ on $\Omega,$ the truncated correlation will obey
\be |\la f\cdot\tau_\bx g\ra-\la f\ra \la g\ra|\leq C_{f,g}e^{-\gamma|\bx|}. \lb{I2} \ee
The constant $C_{f,g}$ depends upon the functions $f,g$ but the decay constant $\gamma$ does not.
Since there is evidence, discussed in Section 3.3, that this clustering is not obeyed by the
stationary meaures of the DLG, it is suggested that the invariant measures may not be Gibbs.
However, even if the long-range decay could be established rigorously, this would not imply the measures
are non-Gibbs. In other words, it would not be ruled out that the invariant measures are Gibbs states
with long-range interactions (as happens for dipole systems) or are critical, where such power-law decays
are commonly observed. Nevertheless, there are situations in physics where non-Gibbsian measures, violating Eq.(\ref{I1}),
do appear. A notable example is the ``effective distributions'' produced by some real-space RG operations, treated very
exhaustively in \cite{45a}. Section 4.5 of that work also discusses nonequilibrium steady-states.
However, for the DLG we simply do not yet know the answer. This is related to the fact that we have no convergent
cluster expansion for the stationary measures (from which presumably Eq.(\ref{I1}) would follow.) There is only
one system for which non-trivial steady-states have been constructed. It is a mixture of ``first class'' and
``second class'' particles in 1D. First and second class particles both jump only to the right, respecting
exclusion at each site. However, a second class particle must always give way to a first class one. In this
case, the stationary measures are non-Gibbs \cite{45b}. For $B=\{\bo\}$ it is checked by explicit
computation that the continuous dependence Eq.(\ref{I1}) fails.
\noindent {\bf Appendix II: Chemical Potential and Stationary Measures}
We argued in Section 2.1 that for the DLG stationary measures, if not more
generally, a variation in the density amounts to a variation in the chemical
potential {\em only}. Here we state a theorem which supports this claim under
a reasonable assumption. Let us assume that we have found a measure $P$ on $\Omega=\{0,1\}^{\bZ^d}$
which is time-invariant for the DLG ($P\in{\cal I}$) and invariant under spatial shifts ($P\in{\cal S}$). This
means that
\be \int dP\,Lf=0 \lb{II1} \ee
and
\be \int dP\,\tau_{\bx}f=\int dP\, f,\,\,\,\,\,\,\,\,\,\,\,\bx\in\bZ^d, \lb{II2} \ee
for all local functions $f.$ In addition, let us suppose there exists a potential $\{J_A: A\subset \bZ^d\}$
which obeys the summability condition
\be \sum_{A\ni \bo}\,|A|\cdot |J_A|<\infty, \lb{II3} \ee
such that for each finite $B\subset \bZ^d,$
\be P(\eta_B|\eta_{B^c})=Z_B(\eta_{B^c})^{-1}
\exp\left[-\sum_{A\cap B\neq \emptyset}J_A\eta^A\right]. \lb{II4} \ee
Comparing with Eq.(\ref{I1}), we see that this amounts to $P\in {\cal G}(J).$ A related notion is that of
{\em canonical Gibbs measures} for the potential $J,$ denoted ${\cal G}_c(J),$ with conditional expectations
\be P(\eta_B|n_B,\eta_{B^c})=Z_B(n_B,\eta_{B^c})^{-1}
\exp\left[-\sum_{A\cap B\neq \emptyset}J_A\eta^A\right]. \lb{II5} \ee
The difference here is that the conditioning is also with respect to $n_B(\eta),$ the number of
particles in $B.$ The standard reference on this subject is \cite{51}. The result we need for
the DLG is the following
\begin{Th}
Suppose for some summable potential $J,$ that ${\cal I}\cap{\cal S}\cap{\cal G}(J)\neq\emptyset.$ Then,
${\cal I}\cap{\cal S}\subseteq{\cal G}_c(J).$
\end{Th}
This theorem, due to A. Asselah, extends a previous result of K\"{u}nsch \cite{23} for
non-reversible spin-flip processes to DLG. Our main interest here is in its consequence
\begin{Cor}
Under the assumption of the previous theorem, if $Q\in{\cal I}$ is also ergodic to space-translations,
then there exists a chemical potential $\lambda=J_{\{\bo\}}$ such that $Q\in{\cal G}(J_{\{\bo\}},\{J_A:|A|\geq 2\}).$
\end{Cor}
This follows from the previous result based upon two well-known facts: for a translation-invariant potential $J,$
(i) the boundary set ${\rm ex}{\cal G}_c(J)$ of extremal elements of the Choquet simplex ${\cal G}_c(J)$ is characterized
as the subset of elements ergodic under the space-shifts \cite{19}, and (ii) ${\rm ex}{\cal G}_c(J)=\bigcup_{\lambda}\,
{\rm ex}{\cal G}(\lambda,\{J\})$ (Theorems 5.14-5.15, \cite{51}.) Hence, we obtain our desired characterization of
the invariant measures $Q$ as Gibbs for the potential $\{J_A:|A|\geq 2\}$ and for some chemical potential $\lambda.$
%*I've expanded the discussion of Amineh's proof. Is it now adequate?
We here only sketch the proof of the Theorem, which is technical because one wants to get away with
minimal assumptions on the potential $J.$ The main idea is to consider the time-derivative
of the {\em relative entropy} per unit volume, $h(Q|P).$ Since we are in the translation-invariant set up,
this quantity is well-defined. Moreover, because $Q$ is assumed stationary,
\be \left. {{d}\over{dt}}h(Q e^{Lt}|P)\right|_{t=0}=0. \lb{II6} \ee
Let us define for any pair of sites $\bx,\by$ the measure $Q\Delta_{\bx\by}$ by
\be \int d(Q\Delta_{\bx\by})\,f=\int dQ\,(\Delta_{\bx\by}f), \lb{II7} \ee
for all local functions $f.$ Writing out the time derivative in Eq.(\ref{II6}) explicitly, one deduces
after a careful estimate of boundary terms the following result: For each of the bounded sets of lattice sites
$B_n=[-2^n,2^n]^d$ and each nearest-neighbor pair $\langle \bx,\by\rangle\subset\tilde{B}_n=[-2^n+n,2^n-n]^d,$
\be Q_{B_n}\left(\eta^{\bx\by}_{B_n}\right)
=\int\,{{d(P\Delta^{\bx\by})}\over{dP(\eta)}}Q(\eta_{B_n},d\eta_{B_n^c}). \lb{II6a} \ee
Therefore, $\int\,\Delta^{\bx\by}f\,dQ=\int\,{{d(P\Delta^{\bx\by})}\over{dP}}f\,dQ$ for every local function $f$
and every nearest-neighbor pair $\langle \bx,\by\rangle.$ The conclusion is that
\be {{d(Q\Delta_{\bx\by})}\over{dQ(\eta)}}={{d(P\Delta_{\bx\by})}\over{d P(\eta)}} \lb{II6b} \ee
for all $\eta\in\Omega.$ On the other hand, if $B$ is any bounded set of lattice sites and
${\cal B}(n_B,\eta_{B^c})$ is the $\sigma$-field generated by the random variables $n_B,\{\eta_\bx|\bx\in B^c\},$
then a Borel function $f$ is ${\cal B}(n_B,\eta_{B^c})$-measurable if and only if $\Delta_{\bx,\by}f=f$
for all nearest-neighbor pairs $\langle \bx,\by\rangle$ with $\bx,\by\in B.$ In that case it is easy to see that
\be {{d(Q(\cdot|n_B,\eta_{B^c})\Delta_{\bx\by})}\over{dQ(\eta_B|n_B,\eta_{B^c})}}
={{d(Q\Delta_{\bx\by})}\over{d Q(\eta)}}. \lb{II6c} \ee
Because any two configurations $\eta_B,\eta_B'$ in $B$ with $n_B,\eta_{B^c}$ fixed can be obtained by a sequence
of nearest-neighbor exchanges, $\eta_B'=\Delta^{\bx_M\by_M}\cdots\Delta^{\bx_1\by_1}\eta_B,$ it follows from
the combination of Eqs.(\ref{II6c}) and (\ref{II6b}) that
\be Q(\eta_B|n_B,\eta_{B^c})=P(\eta_B|n_B,\eta_{B^c}) \lb{II6d} \ee
for all bounded sets $B.$ Therefore, $Q\in {\cal G}_c(J).$
There is a slightly different way to state the Theorem, which may lend plausibility to its hypothesis.
According to Eq.(\ref{63}), the rates of the time-reversed process with respect
to $P\in{\cal I}$ are given by
\be c^r(\bx,\by,\eta)=c(\bx,\by,\eta^{\bx\by}){{d(P\Delta_{\bx\by})}\over{dP(\eta)}}, \lb{II8} \ee
which is defined for all $\eta,$ if $P$ is Gibbs. However, it is then true by the Theorem (cf. Eq.(\ref{II6b})) that
the righthand side is the {\em same }for all $Q\in {\cal I}.$ Therefore, the time-reversed process does not depend
upon the stationary measure which is used to define it when one of the measures (and thus each of them) is Gibbs.
It would be of interest---in particular in view of our discussion in Appendix I---to avoid the assumption
of $P$ being Gibbs altogether. However, if also non-translation invariant states are permitted, then there
must be some condition which excludes the ``blocked states'' $P_b,$ defined by the conditional expectations
\be P_b(\eta_B|\eta_{B^c})=Z^{-1}_B(\eta_{B^c})
\exp\left[-\sum_{A\cap B\neq\emptyset}\,\overline{J}_A\eta^A
+\sum_{\bx\in B}q\beta(\bE\cdot\bx)\eta_\bx\right], \lb{II10} \ee
where $\overline{J}$ is the short-range potential defining the Hamiltonian in Eq.(\ref{3}). The measures
$P_b$ are stationary and reversible.
\noindent {\bf Appendix III: The Asymmetric Simple Exclusion Process}
The asymmetric exclusion process (ASEP) is defined by the rates
\begin{equation}
c(\bx,\by,\eta) = p(\by-\bx)\eta_\bx(1-\eta_\by) + p(\bx-\by)(1 - \eta_\bx)\eta_\by, \lb{III1}
\end{equation}
where $p(\bx) \geq 0$ with compact support, $p(\bo) = 0$, and $\sum_\bx p(\bx) =1$
as a normalization. Thus after an exponentially distributed waiting time a
particle at site $\bx$ jumps with probability $p(\by-\bx)$ to site $\by$ provided
this site is empty. If
\begin{equation}
p(\bx) = p(-\bx) e^{\beta q \bE\cdot \bx}, \lb{III2}
\end{equation}
then Eq.(3) is satisfied with $H=0$. For nearest neighbor jumps only, i.e.
$p(\bx) =0$ whenever $|\bx| >1$, Eq.(\ref{III2}) can always be fullfilled. In general,
we do not impose such a restriction. The very welcome simplification of
ASEP originates in the assumption that the exchange rates depend
{\it only} on the occupation variables at sites $\bx$ and $\by$. For the ASEP
a number of rigorous results are available which will be briefly reviewed here.
The Bernoulli measures, i.e. the $\eta_\bx$'s are independent and
$\la\eta_\bx\ra_n = n$, are stationary for the rates (A.1). In fact, these are
the only stationary and translation invariant measures in infinite volume
\cite{46}. If Eq.(\ref{III2}) holds, then also the inhomogeneous blocked states
\begin{equation}
Z^{-1}\exp[\beta q \sum_\bx (\bE \cdot \bx)\eta_\bx] \lb{III3}
\end{equation}
and their translates are stationary. They play no role in hydrodynamics.
The two point function $S(\bx) = \la\eta_\bx\eta_\bo\ra_n - n^2 = \delta_{\bx\bo} n(1-n)$
and consequently the structure function $\hat{S}(\bk) = n(1-n),$ which is independent
of $\bk$. Of course, large deviations Eq.(\ref{11}) are satisfied with the
entropy function $s(n) = -[n\log n + (1-n)\log(1-n) + \log 2]$
relative to density $n^* = 1/2$. The average current
\begin{equation}
\hat{\jj}(n) = n(1 - n) \sum_\bx \bx p(\bx). \lb{III4}
\end{equation}
The rates of the time reversed process are given by
\begin{equation}
c^r(\bx,\by,\eta) = p(\bx-\by)\eta_\bx(1-\eta_\by) + p(\by-\bx)(1-\eta_\bx)\eta_\by, \lb{III5}
\end{equation}
which reflects the simplicity of the invariant measure. In particular, if Eq.(\ref{III1})
holds, the reversed rates also satisfy Eq.(\ref{III1}) with $\bE$ replaced by $-\bE$.
The exclusion process is reversible iff $p(\bx) = p(-\bx)$ in which case it
is called symmetric. This corresponds to the equilibrium situation.
The average current vanishes, $\sum_\bx \bx p(\bx) = \bo$. Conductivity, Onsager
matrix, and diffusion matrix are given by
\begin{equation}
(q\beta)^{-1} \sigma_{ml} = n(1-n) \frac{1}{2} \sum_\bx x_mx_lp(\bx)
= L_{ml} = n(1-n)D_{ml}. \lb{III6}
\end{equation}
The latter is independent of $n$ and the hydrodynamic equation becomes
linear. Note that we could impose $\hat{\jj}(n) = n(1-n) \sum_\bx \bx p(\bx) = \bo$
and still $p(\bx) \neq p(-\bx)$. This is an example of a zero current, but
not time-reversal invariant lattice gas.
For the ASEP the Euler equations read
\begin{equation}
\partial_\tau n(\br,\tau) + \partial_m\hat{\j}_m(n(\br,\tau)) = 0 \lb{III7}
\end{equation}
which are established in full generality by Rezakhanlou \cite{47}, with prior
results detailed in \cite{14a}, Notes to Section 4.2. He proves that on the time
scale $\sim\epsilon^{-1}$ and the spatial scale $\sim\epsilon^{-1}$ in microscopic
units a typical density profile follows the unique entropic solution to Eq.(\ref{III7}).
In particular, his results cover times beyond the first shock and entropy may no longer be conserved.
Next we turn to the Onsager matrix. The static part reads
\begin{equation}
L^{stat}_{ml} = n(1-n)\frac{1}{2} \sum_\bx x_mx_lp(\bx) \lb{III8}
\end{equation}
which is clearly symmetric. Note that the second jump moments are {\it not}
truncated. For the dynamic part we use
\begin{eqnarray}
\bj(\bx) = \frac{1}{2} \sum_\by \by[p(\by)\eta_\bx(1-\eta_{\bx+\by}) -
p(-\by)(1-\eta_\bx)\eta_{\bx+\by}],\nonumber\\
\bj^r(\bx) = \frac{1}{2} \sum_\by \by[p(-\by)\eta_\bx(1-\eta_{\bx+\by}) -
p(\by)(1-\eta_\bx)\eta_{\bx+\by}],\nonumber\\
\bj^s(\bx) = \frac{1}{2}[\bj(\bx) + \bj^r(\bx)] = \frac{1}{2} \sum_\by \by(p(\by) +
p(-\by))(\eta_\bx - \eta_{\bx+\by}),\nonumber\\
\bj^a(\bx) = \frac{1}{2}[\bj(\bx) - \bj^r(\bx)] = \frac{1}{2} \sum_\by \by
((p(\by) - p(-\by))(\eta_\bx - \eta_{\bx+\by})^2, \lb{III9}
\end{eqnarray}
and we note that
\begin{equation}
\sum_\bx \bj^s(\bx) = 0. \lb{III10}
\end{equation}
Therefore
\begin{eqnarray}
L^{dyn}_{ml} = -\int_0^{\infty}dt\left[
\sum_\bx\la j^r_m(\bx)e^{tL}j_l(\bo)\ra^{\top}_n -
\chi(n)\hat{\j}_m'(n)\hat{\j}_l'(n) \right] \nonumber\\
= \int_0^{\infty} dt \left[ \sum_\bx \la j^a_m(\bx)e^{tL}j^a(\bo)\ra^{\top}_n
- \chi(n)\hat{\j}_m'(n)\hat{\j}_l'(n) \right]. \lb{III11}
\end{eqnarray}
Since the RHS of Eq.(\ref{III11}) is an autocorrelation, $\bL^{dyn} \geq \bo$ as a matrix. Together
with Eq.(\ref{III8}) this implies $\bD(\bE) \geq \bD(E=0)$. Thus the driving tends to
increase diffusion. In fact, it could increase it so much that $L=\infty$.
According to the standard physical picture \cite{48,49}
this will be the case whenever $\hat{\jj}(n) \neq \bo$ and $d=1,2$. If $\hat{\jj}
(n)=\bo$, then the integral Eq.(\ref{III11}) should exist in any dimension.
In his thesis Lin Xu \cite{50} considers the non-reversible ASEP with zero
current $\hat{\jj}(n) = \bo$ in one spatial dimension. Although not carried
out explicitly, the results are asserted for any dimension. He proves
that on the (microscopic) time scale $\epsilon^{-2}$, i.e. on the
diffusive time scale, the density is approximated with probability one by
the solution of the nonlinear diffusion equation
\begin{equation}
\partial_{\tau}n(\br,\tau) = \partial_mD_{ml}(n(\br,\tau))\partial_ln(\br,\tau) \lb{III12}
\end{equation}
with $D_{ml} = (n(1-n))^{-1}(\bL^{stat} + \bL^{dyn})_{ml}$ from Eqs.(\ref{III8}), (\ref{III11}).
Lin Xu also provides a lower bound on $\bD$ which in our notation reads $\bL^{dyn} > \bo$.
Note that while the ASEP is nonreversible the Gaussian fluctuations in the steady-state are
given by a reversible Ornstein-Uhlenbeck process. Thus, on the large scale reversibility is regained.
Esposito, Marra, and Yau (EMY) \cite{13} study the ASEP with $\hat{\jj}(n) \neq \bo$
in three and more dimensions. One immediate difficulty stems from the
difference in Euler and Navier-Stokes time scale, equivalently, from
their different scaling under spatial dilations. EMY decided to adopt a
procedure familiar from incompressible Navier-Stokes equations for fluids.
They fix a reference density $n^*$ and take a small density deviation of
order $\epsilon$, i.e. the chemical potential in the local equilibrium
state Eq.(\ref{24}) is of order $\epsilon$. Expanding Eq.(\ref{III4}) there is a
constant drift velocity $\bc(n^*) = (1 - 2n^*)\sum_\bx\bx p(\bx)$. Thus one is led
to consider density deviations of order $\epsilon$ in a moving frame of
reference. EMY prove that the average density, on the Navier-Stokes time
scale,
\begin{equation}
\lim_{\epsilon \rightarrow 0} \epsilon^{-1}[\bar{n}_{\epsilon}(\br -
\epsilon^{-1}c(n^*)\tau, \epsilon^{-1}\tau) -n^*] = n(\br,\tau) \lb{III13}
\end{equation}
has a limit as $\epsilon \rightarrow 0$. The limit density satifies
\begin{equation}
\partial_\tau n(\br,\tau) = \partial_m[-(\sum_{\bx}x_m p(\bx))n(\br,\tau)^2 + D_{ml}(n^*)
\partial_l n(\br,\tau)] \lb{III14}
\end{equation}
with $D_{ml} = (n(1-n))^{-1}(\bL^{stat} + \bL^{dyn})_{ml}$ from Eqs.(\ref{III8}), (\ref{III11}).
Again it is established that $\bL^{dyn}>\bo$. EMY also prove that, on the scales
considered, typical density profiles follow the solution to Eq.(\ref{III14}).
%* The following paragraph discusses the new work of LOY
Finally, Landim, Olla and Yau (LOY) \cite{76} have studied more recently in the same model as EMY the problem
equivalent to ours in the text. Namely, they prove that the average density $\bar{n}_\en(\br,\tau)$
on the Euler time-scale $\sim \en^{-1}$ is given correctly to order $\en$ by the solution $n_\en(\br,\tau)$ of the
drift-diffusion equation
\be \partial_\tau n(\br,\tau) + \partial_m\hat{\j}_m(n(\br,\tau))=
\en \partial_mD_{ml}(n(\br,\tau))\partial_ln(\br,\tau) \lb{III15} \ee
with again $D_{ml} = (n(1-n))^{-1}(\bL^{stat} + \bL^{dyn})_{ml}$ from Eqs.(\ref{III8}), (\ref{III11}).
This is proved in the precise sense that
\be \int_\Lambda \,d^d\br \phi(\br)\left[\bar{n}_\en(\br,\tau)-n_\en(\br,\tau)\right]=o(\en), \lb{III16} \ee
for any nice test function $\phi$ (i.e. $\phi\in H^1$) and for each fixed $\tau < T_0,$ the time of appearance of the
first shock in the solution of Eq.(\ref{III7}) for the same initial data. This is a result of the same type,
just slightly weaker, as that conjectured in Eq.(\ref{59}) in the text. LOY prove no results for typical density
profiles in this setting.
%* I've kept this Appendix, but substantially rewritten it based upon Herbert's remarks
\noindent {\bf Appendix IV: Steady-State Stability and the FDR of Kraichnan}
In a very interesting work \cite{36}, which is unfortunately little referenced outside the turbulence literature,
R. H. Kraichnan in 1959 derived a microscopic version of the FDR of 2nd type for a general classical dynamics
\be dx_i(t)/dt= V_i(x(t)) \lb{V1} \ee
with a conserved quantity, or``energy'' $H(x),$ and a Liouville theorem for conservation of phase volume.
These guarantee that the ``Gibbs measures'' $\propto e^{-\beta H}$ are stationary for the dynamics. Under these same two
conditions, Kraichnan established that the mean response function in the stationary ``Gibbs state''
\be G_{ij}(t,s)=\langle{{\delta x_i(t)}\over{\delta f_j(s)}}\rangle \lb{V2} \ee
associated to a change in the external force $f_j$ coupled as:
\be {{dx_i}\over{dt}}(t)= V_i(x(t))+f_i(t), \lb{V3} \ee
is determined by the FDR
\be G_{ij}(t,s)= \beta\langle x_i(t)\cdot {{\partial H}\over{\partial x_j}}(x(s))\rangle. \lb{V4} \ee
His two page derivation is interesting because it shows that the FDR arises as the consequence of ``stability'' of the
Gibbs state to coupling of two independent, identical systems in a general way preserving conservation of total energy
$H(x)+H(y)$ and a Liouville theorem for the coupled dynamics. This is very reminiscent of the standard argument
for positive equilibrium response, or susceptibility, which we gave in Eqs.(\ref{23}),(\ref{23a}). In fact,
if the distribution on ``histories'' is regarded as a formal Gibbs distribution in one higher dimension, then
the arguments are quite analogous. Kraichnan's method yields also the corresponding quantum KMS conditions
and 2nd type FDR's in the context of stochastic Langevin equations.
Actually, it is possible to derive Kraichnan's FDR in an even simpler way as a direct corollary of the
{\em classical KMS condition}
\be \langle\{f,g\}\rangle= \beta\langle g\{f,H\}\rangle, \lb{V5} \ee
in which $\{f,g\}$ is the canonical Poisson bracket; see \cite{90,91}. This condition is easily proved at finite-volume
by a simple integration by parts, and also at infinite-volume for a suitable class of functions $f,g.$
If one takes $f=p_j$ one gets
\be \langle {{\partial g}\over{\partial x_j}}\rangle=\beta\langle g{{\partial H}\over{\partial x_j}}\rangle, \lb{V6} \ee
which is also true (by the same arguments) even if the dynamical vector field $V$ is not Hamiltonian. It is easy
to see that Eq.(\ref{V6}) with $g=x_i(t)$ gives Kraichnan's FDR. In fact,
\be {{\delta x_i(t)}\over{\delta f_j(0)}}={{\partial x_i(t)}\over{\partial x_j}}, \lb{V7} \ee
because making the small perturbation to the force $f_j(s)\rightarrow f_j(s)+\en\delta(s)$ is the same
as making the small perturbation $x_j\rightarrow x_j+\en$ to the initial data. It is rather natural to see
that Kraichnan's FDR is a consequence of the KMS condition, because the latter is a more general expression
of ``stability'' of the state. It is known, for example, that the classical KMS condition Eq.(\ref{V5})
arises as a consequence of stability of the Gibbs measure to small perturbations of the Hamiltonian dynamics \cite{92}.
In the case of a linear Langevin dynamics such as our Eq.(\ref{96}), it is easy to check that Kraichnan's method
gives a result
\be \langle{{\delta \alpha_i(t)}\over{\delta \eta_j(0)}}\rangle=
\la \alpha_i(t)X_j(0)\ra, \lb{V8} \ee
equivalent to Eq.(\ref{114}). However, note that while Kraichnan's argument applies also to the
nonlinear Langevin equations, like our Eq.(\ref{155}), it does not yield a relation obviously equivalent
to Graham's 2nd type FDR in that context. At least it does not in Kraichnan's original form of the argument.
In fact, the two FDR's of Kraichnan and Graham for this case deal with the response functions to differently imposed
forces.
\noindent {\bf Appendix V: Microscopic Current Correlations in the DLG Models}
For $t>s$ we calculate that
\begin{eqnarray}
\sE_n(J_m(\bx,t)J_l(\by,s)) & = & \lim_{\delta\rightarrow 0}{{1}\over{\delta}}
\int_0^\delta du\,\,\sE_n(J_m(\bx,t-s)J_l(\by,u)) \cr
\,& = & \lim_{\delta\rightarrow 0}\int P_n(d\eta)\,\,
\sE^\eta\left[\sE^\eta\left(J_m(\bx,t-s)\big|{\cal F}_\delta\right)
\cdot{{1}\over{\delta}}
\int_0^\delta du\,\, J_l(\by,u)\right] \cr
\,& = & \lim_{\delta\rightarrow 0}\int P_n(d\eta)\,\,
\sE^\eta\left[\sE^{\eta^{\by,\by+\ve_l}}\left(J_m(\bx,t-s-\delta)\right)
\cdot{{1}\over{\delta}}
\int_0^\delta du\,\, J_l(\by,u)\right] \cr
\,& = & \lim_{\delta\rightarrow 0}\int P_n(d\eta)\,\,
\left(\Delta_{\by,\by+\ve_l}e^{(t-s-\delta)L}j_m(\bx)\right)(\eta)
\cdot{{1}\over{\delta}}\int_0^\delta du\,\, \left(e^{uL}j_l(\by)\right)(\eta) \cr
\,& = & \int P_n(d\eta)\,\,
\left(\Delta_{\by,\by+\ve_l}e^{(t-s)L}j_m(\bx)\right)(\eta)
\cdot j_l(\by,\eta). \lb{VI1}
\end{eqnarray}
We can say a few words to explain and justify each line. The first uses just stationarity. The second
line introduces the conditional expectation $\sE^\eta(\cdot)$ given that the configuration at time zero is $\eta$
and the sigma algebra ${\cal F}_t=\sigma\{N(\cdot,s):st$ that
\be \sE_n(J_m(\bx,t)J_l(\by,s))= \int P_n(d\eta)\,\, j_m(\bx,\eta)\cdot
\left(\Delta_{\bx,\bx+\ve_m}e^{(s-t)L}j_l(\by)\right)(\eta). \lb{VI3} \ee
Another contribution appears if $s=t$ which is calculated as
\begin{eqnarray}
\lim_{\delta\rightarrow 0}{{1}\over{\delta}}
\int_0^\delta du\int_0^\delta du'\,\,\sE_n(J_m(\bx,u)J_l(\by,u'))
\,& = & \delta_{\bx,\by}\delta_{mn}\lim_{\delta\rightarrow 0}{{1}\over{\delta}} \sE_n\left[{\cal N}_{\bx,\bx+\ve_m}([0,\delta])\right] \cr
\,& = & \delta_{\bx,\by}\delta_{mn}\langle c(\bx,\bx+\ve_m)\rangle_n. \lb{VI4}
\end{eqnarray}
We introduced the random variable ${\cal N}_{\bx,\bx+\ve_m}([s,t])$ which counts the absolute number
of exchanges between sites $\bx$ and $\bx+\ve_m$ in the time-interval $[s,t].$ Altogether we see that
\begin{eqnarray}
\sE_n(J_m(\bx,t)J_l(\by,s)) & = & \langle c(\bo,\ve_m)\rangle_n\delta_{ml}\delta_{\bx\by}\delta(t-s) \cr
\, & & \,\,\,\,\,\,\,\,\,\,+\theta(t-s)\langle\Delta_{\by,\by+\ve_l}e^{L(t-s)}j_m(\bx)
\cdot j_l(\by)\rangle_n \cr
\, & & \,\,\,\,\,\,\,\,\,\,+\theta(s-t)\langle\Delta_{\bx,\bx+\ve_m}e^{L(s-t)}j_l(\by)
\cdot j_m(\bx)\rangle_n. \lb{VI5}
\end{eqnarray}
On the other hand, using the identity Eq.(\ref{64}), with $f=e^{tL}j_m(\bo)\cdot (\eta_\bx-\eta_{\bx+\ve_l}),$
we can rewrite this as
\begin{eqnarray}
\sE_n(J_m(\bx,t)J_l(\by,s)) & = & \langle c(\bo,\ve_m)\rangle_n\delta_{ml}\delta_{\bx\by}\delta(t-s) \cr
\, & & \,\,\,\,\,\,\,\,\,\,-\theta(t-s)\langle e^{L(t-s)}j_m(\bx)\cdot j_l^r(\by)\rangle_n \cr
\, & & \,\,\,\,\,\,\,\,\,\,-\theta(s-t)\langle e^{L(s-t)}j_l(\by)\cdot j_m^r(\bx)\rangle_n. \lb{VI6}
\end{eqnarray}
This is exactly the result claimed in Eq.(\ref{137}) of Section 3.3.
\noindent {\bf Appendix VI: Kadanoff-Martin Approach to Hydrodynamics in DLG}
We shall repeat the argument in Appendix 3 of the KLS paper, but making only the weaker regularity assumption that
\be \chi(n,\hat{\bk})\equiv \lim_{k\rightarrow 0}S_n(\bk) \lb{VII1} \ee
exists. By integration we obtain distinct relations $n(\lambda,\hat{\bk})$ between density and chemical
potential, depending upon the wavevector direction of the density modulation in space. Since it is still
supposed to be true that
\be {{S(\bk,t)}\over{S(\bk)}}\approx \exp\left[ i(\bc\cdot\bk)t-(\bk\cdot\bD\cdot\bk)t\right] \lb{VII2} \ee
at long times and low wavenumbers, we may still define
\be \hat{\bk}\cdot\bc(\hat{\bk})\equiv \lim_{k\rightarrow 0}\lim_{t\rightarrow \infty}
{{1}\over{t}}{{\partial}\over{\partial k}}
\left[ {{S(\bk,t)}\over{S(\bk)}}\right], \lb{VII3} \ee
and
\be \hat{\bk}\cdot\bD\cdot\hat{\bk}\equiv\lim_{k\rightarrow 0}\lim_{t\rightarrow \infty}
{{-1}\over{2t}}\left\{{{\partial^2}\over{\partial k^2}}
\left[{{S(\bk,t)}\over{S(\bk)}}\right]-(\hat{\bk}\cdot\bc(\hat{\bk}))^2t^2\right\}.
\lb{VII4} \ee
Let us examine each of these in turn.
For the first, we calculate that
\be \hat{\bk}\cdot\bc(\hat{\bk})=\lim_{k\rightarrow 0}\lim_{t\rightarrow \infty}{{1}\over{t}}\cdot
{{1}\over{S(\bk)}}\sum_\bx e^{i\bk\cdot\bx}(\hat{\bk}\cdot\bx)\sE_n(N(\bx,t)N(\bo,0))^\top,
\lb{VII5} \ee
if we assume that limits such as $\lim_{k\rightarrow 0}\sum_\bx e^{i\bk\cdot\bx}x_m\sE_n(N(\bx,t)N(\bo,0))^\top$
exist and are finite. In that case, all the contributions from the derivative ${{\partial}\over{\partial k}}$
of $S^{-1}(\bk)$ vanish. Then, using the conservation law, we obtain
\begin{eqnarray}
\hat{\bk}\cdot\bc(\hat{\bk}) & = & {{\hat{\bk}}\over{\chi(\hat{\bk})}}\cdot\left[
\lim_{k\rightarrow 0}\lim_{t\rightarrow \infty}
{{1}\over{t}}\int_0^tds\,\,\sum_\bx e^{i\bk\cdot\bx}\sE_n(\bJ(\bx,s)N(\bo,0))^\top \right]
\cr
\,& = & {{\hat{\bk}}\over{\chi(\hat{\bk})}}\cdot
\left[\lim_{k\rightarrow 0}
\sum_\bx e^{i\bk\cdot\bx}\langle\bj(\bx)\eta_\bo\rangle_n^\top \right]. \lb{VII6}
\end{eqnarray}
In fact, we shall show that $\bc$ is actually $\hat{\bk}$-{\em independent} under the assumption that
{\em there is a unique stationary state} $P_n$ {\em for each density} $n.$ The result follows from the chain of equalities:
\begin{eqnarray}
\bc(n,\hat{\bk}) & = & {{1}\over{\chi(n,\hat{\bk})}}\lim_{k\rightarrow 0}
{{\partial}\over{\partial\lambda}}\left[
{{1}\over{Z(\lambda)}}\langle\bj(\bo)\exp\left(\lambda\sum_\bx e^{i\bk\cdot\bx}\eta_\bx\right)
\rangle_n \right]_{\lambda=0} \cr
\,& = & {{1}\over{\chi(n,\hat{\bk})}}{{\partial}\over{\partial\lambda}}
\hat{\jj}(n(\lambda,\hat{\bk}))\big|_{\lambda=\lambda(n,\hat{\bk})} \cr
\,& = & {{\partial\hat{\jj}}\over{\partial n}}(n). \lb{VII7}
\end{eqnarray}
Therefore, the same formula holds for $\bc$ as before.
Actually, the previous argument applies generally if we define ``$\hat{\bk}$-dependent Zwanzig-Mori spaces,'' via
the inner-product
\be \langle A|B\rangle_{n,\hat{\bk}}=\lim_{k\rightarrow 0}\sum_\bx e^{i\bk\cdot\bx}
\sE_n(A(\bx)B(\bo))^\top_n, \lb{VII8} \ee
and establishes that
\be \langle A|N\rangle_{n,\hat{\bk}}=\chi(n,\hat{\bk}){{\partial\hat{A}}\over{\partial n}}(n). \lb{VII9} \ee
Therefore, all of the $\hat{\bk}$-dependence is through the susceptibility in this expression. Furthermore,
if we assume an ergodic property of the dynamics, then it is generally true as well that
\be \lim_{t\rightarrow \infty}\langle A(t)|B\rangle_{n,\hat{\bk}}=\chi(n,\hat{\bk})
{{\partial\hat{A}}\over{\partial n}}(n){{\partial\hat{B}}\over{\partial n}}(n). \lb{VII10} \ee
The proof is just that
\begin{eqnarray}
\lim_{t\rightarrow \infty}\langle A(t)|B\rangle_{n,\hat{\bk}}
\,& = & \lim_{k\rightarrow 0}\lim_{t\rightarrow \infty}
\sum_\bx e^{i\bk\cdot\bx} \sE_n(A(\bo,t)B(\bx,0))^\top_n \cr
\,& = & \lim_{k\rightarrow 0}\lim_{t\rightarrow \infty}
{{\partial}\over{\partial\theta}}\left[
{{1}\over{Z(\theta)}}\sE_n\left(A(t)\exp\left(\theta\sum_\bx e^{i\bk\cdot\bx}B(\bx)\right)\right)
\right]_{\theta=0} \cr
\,& = & \left.{{\partial}\over{\partial\theta}}\hat{A}(n(\theta,\hat{\bk}))\right|_{\theta=0} \cr
\,& = & \left. {{\partial\hat{A}}\over{\partial n}}(n)\cdot
{{\partial n}\over{\partial \theta}}(\theta,\hat{\bk})\right|_{\theta=0} \cr
\,& = & {{\partial\hat{A}}\over{\partial n}}(n)\cdot
\lim_{k\rightarrow 0}\sum_\bx e^{i\bk\cdot\bx}\sE_n(B(\bx)N(\bo))^\top_n \cr
\,& = & \chi(n,\hat{\bk}){{\partial\hat{A}}\over{\partial n}}(n)
{{\partial\hat{B}}\over{\partial n}}(n). \lb{VII11}
\end{eqnarray}
(The ergodic assumption was used in the third line.)
>From these considerations it seems reasonable to make the following {\em hypothesis}: namely, that
$\langle A|B\rangle_{n,\hat{\bk}}$ {\em is independent of} $\hat{\bk}$ {\em whenever} $A,B\in\{N\}^\perp.$
The motivation of this hypothesis is that only the conserved variable (here, the particle number) is
expected to show long-range correlations and the orthogonal variables (the ``fast'' subspace) should
exhibit rapid decay.This hypothesis can be restated as saying, for arbitrary variables $A,B,$ that
$\langle A|B\rangle_{n,\hat{\bk}}-\chi(n,\hat{\bk}){{\partial\hat{A}}\over{\partial n}}(n)
{{\partial\hat{B}}\over{\partial n}}(n)$ will be $\hat{\bk}$-independent. In particular, under this
hypothesis the covariance of the ``fast component'' of the microscopic electric currrent
\be R_{ml}\equiv{{1}\over{2}}\int_{-\infty}^{+\infty}dt\left[\langle J_m(t)|J_l\rangle_{n,\hat{\bk}}
-\chi(n,\hat{\bk}){{\partial\hat{\j}_m}\over{\partial n}}(n)
{{\partial\hat{\j}_l}\over{\partial n}}(n)\right], \lb{VII12} \ee
is {\em constant} (i.e. $\hat{\bk}$-independent) for $k\rightarrow 0.$
We can now calculate the diffusion tensor defined in Eq.(\ref{VII4}) above, as
\begin{eqnarray}
\hat{\bk}\cdot\bD\cdot\hat{\bk} & = &
\lim_{k\rightarrow 0}\lim_{t\rightarrow +\infty}{{1}\over{2t}}
\left[{{1}\over{\chi(n,\hat{\bk})}}\sum_\bx e^{i\bk\cdot\bx}(\hat{\bk}\cdot\bx)^2
\sE_n(N(\bx,t)N(\bo,0))^\top-(\hat{\bk}\cdot \bc(n))^2\right] \cr
\,& = & \lim_{k\rightarrow 0}\lim_{t\rightarrow +\infty}\hat{k}_m\hat{k}_l{{1}\over{2t}}
\int_0^t ds\int_0^t ds' \times\left[{{1}\over{\chi(n,\hat{\bk})}}\sum_\bx e^{i\bk\cdot\bx}
\sE_n(J_m(\bx,s)J_l(\bo,s'))^\top-c_m(n)c_l(n)\right] \cr
\,& = & \hat{k}_m\hat{k}_l\cdot \lim_{k\rightarrow 0}{{1}\over{2}}
\int_{-\infty}^{+\infty}dt \left[{{1}\over{\chi(n,\hat{\bk})}}\sum_\bx e^{i\bk\cdot\bx}
\sE_n(J_m(\bx,t)J_l(\bo,0))^\top-c_m(n)c_l(n)\right], \lb{VII13}
\end{eqnarray}
or,
\be \hat{\bk}\cdot\bD(n)\cdot\hat{\bk}={{\hat{\bk}\cdot\bR(n)\cdot\hat{\bk}}\over{\chi(n,\hat{\bk})}}. \lb{VII14} \ee
Observe that Eq.(\ref{VII13}) is precisely a microscopic version of the FDR Eq.(\ref{145}) obtained at the macroscopic
Langevin level in Section 3.3. It is a weaker version of the Price noise-diffusion relation consistent with
the long-range density correlations in the steady-state.
\noindent {\bf Acknowledgements.} We thank A. Asselah, R. Esposito, M. K.-H. Kiessling,
Y. Oono, E. Speer, H.-T. Yau, and R. K. P. Zia for conversations on the subject of this work.
The research was supported in part by AFOSR Grant AF-0115, 4--28296, by NSF--DMR--134244--20946,
and by a Max Planck Research Award.
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\end{document}
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