Time Evolution In Macroscopic Systems.

 III: Selected Applications

 ... W.T. Grandy, Jr.

 ... Department of Physics & Astronomy, University of Wyoming

 ... Laramie, Wyoming 82071

Abstract. The results of two recent articles expanding the Gibbs variational principle to encompass all of statistical mechanics, in which the role of external sources is made explicit, are utilized to further explicate the theory. Representative applications to nonequilibrium thermodynamics and hydrodynamics are presented, describing several fundamental processes, including hydrodynamic fluctuations. A coherent description of macroscopic relaxation dynamics is provided, along with an exemplary demonstration of the approach to equilibrium in a simple fluid.

1. Introduction

In his classic exposition of the Elementary Principles of Statistical Mechanics (1902) Willard Gibbs introduced two seminal ideas into the analysis of many-body systems: the notion of ensembles of identical systems in phase space, and his variational principle minimizing an average index of probability.' With respect to the first, he noted that it was an artifice that may serve to give precision to notions of probability," and was not necessary. It now seems clear that this is indeed the case, and our understanding of probability theory has progressed to the point that one need focus only on the single system actually under study, as logic requires.

Gibbs never revealed the reasoning behind his variational principle, and it took more than fifty years to understand the underlying logic. This advance was initiated by Shannon (1948) and developed explicitly by Jaynes (1957a), who recognized the principle as a Principle of Maximum Entropy (PME) and of fundamental importance to probability theory. That is, the entropy of a probability distribution over an exhaustive set of mutually exclusive propositions {Ai},
 SI ş -k ĺ i PilnPi ,       k > 0,      (1)
is to be maximized over all possible distributions subject to constraints expressed as expectation values
 áAń = ĺ i PiAi

. The crucial notion here is that all probabilities are logically based on some kind of prior information, so that Pi=P(Ai|I), and that information here is just those constraints along with any other background information relevant to the problem at hand. It is this maximum of SI that has been identified in the past with the physical entropy of equilibrium thermodynamics when the constraints involve only constants of the motion.

It may be of some value to stress the uniqueness of the information measure (1) in this respect. In recent years a number of other measures have been proposed, generally on an ad hoc basis, and often to address a very specific type of problem. While these may or may not be of value for the intended purpose, their relation to probable inference is highly dubious. That issue was settled years ago when Shore and Johnson (1980) demonstrated that any function different from (1) is inconsistent with accepted methods of inference unless the two have identical maxima under the same constraints; the argument does not rely on their interpretation as information measures.

There is nothing in the principle as formulated by either Gibbs or Jaynes, other than the constraints, to restrict it to time-independent probabilities; indeed, this had already been noted by Boltzmann with respect to some of his ideas on entropy and expressed by Planck in the form SB=klogW. In turn, the only way probabilities can evolve in time is for I itself to be time dependent, thus freeing the {Ai} from restriction to constants of the motion. In two recent articles (Grandy, 2004 a,b) this notion has been exploited to extend the variational principle to the entire range of statistical physics. 1 It was emphasized there how important it is to take into account explicitly any external sources, for they are the only means by which the constraints can change and provide new information about the system. The results of these steps appear to form a sound basis in probability theory for the derivation of macroscopic equations of motion based on the underlying microscopic dynamics.

The subtitle Gibbs chose for his work was, Developed with Especial Reference to The Rational Foundation of Thermodynamics, the continuation of which motivates the present essay. The intention here is not to develop detailed applications, but only to further explicate their logical foundations. To keep the discussion somewhat self contained we begin with a brief review and summary of the previous results, including some additional details of the equilibrium scenario that provide a framework for the nonequilibrium case. At each stage it is the nature of the information, or the constraints on the PME, that establishes the mathematical structure of the theory, demonstrating how the maximum entropy functional presides over all of statistical mechanics in much the same way as the Lagrangian governs all of mechanics.

The Basic Equilibrium Scenario

A brief description of the elementary structure of the PME was given in I, as well as in a number of other places, but we belabor it a bit here to serve as a general guide for its extension; the quantum-mechanical context in terms of the density matrix r is adopted for the sake of brevity. Information is specified in terms of expectation values of a number of Hermitian operators {fr}, r=1,Ľ,m < n, and the von Neumann information entropy
 SI=-kTr(rlnr)      (2)
is maximized subject to constraints
 Tr(r)=1 ,        áfrń = Tr(rfr) .      (3)
Lagrange multipliers {lr} are introduced for each constraint and the result of the variational calculation is the canonical form
 r = 1 Z e-l·f ,        Z(l1,Ľ,lm)=Tr ćč e-l·f öř ,      (4)
in terms of the convenient scalar-product notation l·f ş l1f1+Ľ+lmfm. The multipliers and the partition function Z are identified by substitution into (3):
 áfrń = - ¶ ¶lr lnZ ,        r=1,Ľ,m ,      (5)
a set of m coupled differential equations. Equation (4) expresses the initial intent of the PME, to construct a prior probability distribution, or initial state from raw data.

The maximum entropy, which is no longer a function of the probabilities but of the input data only, is found by substituting (4) into (2): 2
 S=klnZ +kl·áfń .      (6)
A structure for the statistical theory follows from the analytical properties of S and Z, beginning with the total differential dS=kl·dáfń, as follows from (5). Hence,
 ¶S ¶lr =0 , ¶S ¶áfrń =klr .      (7)
The operators fr can also depend on one or more external' variables a, say, so that
 ¶lnZ ¶a =-l· ¶f ¶a ,     (8)
and because lnZ=lnZ(l1,Ľ,lm,a) we have the reciprocity relation
 ćč ¶S ¶a öř {áfrń} =k ćč ¶lnZ ¶a öř {lr} ,      (9)
indicating which variables are to be held constant under differentiation. When such external variables are present the total differential becomes
 1 k dS
 = ¶lnZ ¶a da+l·dáfń
 = l·dQ ,
(10)
where
 dQr ş dáfrń-ádfrń ,        ádfrń ş ¶fr ¶a da .     (11)
As in I, changes in entropy are always related to a source of some kind, here denoted by dQr.

With (7) the maximum entropy (6) can be rewritten in the form
 1 k S= 1 k ¶S ¶áfń ·áfń+a ćč lnZ a öř .      (12)
If (lnZ/a) is independent of a it can be replaced by lnZ/a, and from (9) we can write
 S=áfń· ¶S ¶áfń +a ¶S ¶a ,      (13)
which is just Euler's theorem exhibiting S as a homogeneous function of degree 1. Thus, under the stated condition the maximum entropy is an extensive function of the input data.

As always, the sharpness of a probability distribution, and therefore a measure of its predictive power, is provided by the variances and covariances of the fundamental variables. One readily verifies that
 áfmfnń-áfmńáfnń
 =- ¶áfmń ¶ln =- ¶áfnń ¶lm
 ş Kmn=Knm ,
(14)
defining the covariance functions whose generalizations appear throughout the theory; they represent the correlation of fluctuations.

In addition to the maximum property of SI with respect to variations in the probability distribution, the maximum entropy itself possesses a variational property of some importance. If we vary the entropy in (6) with respect to all parameters in the problem, including {lr} and {fr}, we obtain an alternative to the derivation of (10):
 dS= ĺ r lrdQr= ĺ r lrTr(frdr) ,      (15)
where dQr is defined in (11). Hence, S is stationary with respect to small changes in the entire problem if the distribution itself is held constant. The difference in the two types of variational result is meaningful, as is readily seen by examining the second variations. For the case of S we compute d2 S from (13) and retain only first-order variations of the variables. If S is to be a maximum with respect to variation of those constraints, then the desired stability or concavity condition is
 d2 S @ dl·dáfń+da·d ćč ¶S ¶a öř < 0 .      (16)
We return to this presently, but it is precisely the condition employed by Gibbs (1875) to establish all his stability conditions in thermodynamics.

So far there has been no mention of physics, but the foregoing expressions pertain to fixed constraints, and therefore are immediately applicable to the case of thermal equilibrium and constants of the motion. In the simplest case only a single operator is considered, f1=H, the system Hamiltonian, and (4) becomes the canonical distribution
 r0= 1 Z0 e-bH ,        Z0(b)=Tre-bH ,      (17)
where b = (kT)-1. When a is taken as the system volume V, (11) identifies the internal energy U=áHń, elements of heat dQ and work dW=ádHń, and the pressure. Because classically the Kelvin temperature is defined as an integrating factor for heat, T must be the absolute temperature and k is Boltzmann's constant. The first term in (11) also expresses the first law of thermodynamics; while this cannot be derived from more primitive dynamic laws, the relation arises here as a result of probable inference. If a second function f2=N, the total number operator, had been included, the grand canonical distribution would have been obtained in place of (17).

In this application to classical thermodynamics Eq.(13) takes on special significance, for it expresses the entropy as an extensive function of extensive variables, if the required condition is fullfilled. In all but the very simplest models direct calculation of lnZ is not practicable, so one must pursue an indirect course. There may be various ways to establish this condition, but with a = V the standard procedure is to demonstrate it in the infinite volume limit, where it is found to hold for many common Hamiltonians. But in some situations and for some systems it may not be possible to verify the condition; then the theory is describing something other than classical thermodynamics. The one remaining step needed to complete the derivation of elementary equilibrium thermodynamics is to show that theoretical expectation values are equal to measured values; the necessary conditions are discussed in the Appendix.

Nonequilibrium States

Although Gibbs was silent on exactly why he chose this variational principle, his intent was quite clear: to define and construct a description of the equilibrium state. That is, the PME provides a criterion for that state. To continue in this vein, then, we should seek to extend the principle to construction of an arbitrary nonequilibrium state. The procedure for doing this, and its rationale, were outlined in II, where we noted that the main task in this respect is to gather information that varies in both space and time and incorporate it into a density matrix describing a nonequilibrium state.

To illustrate the method of information gathering, consider a system with a fixed time-independent Hamiltonian and suppose the data to be given over a space-time region R(x,t) in the form of an expectation value of a Heisenberg operator F(x,t), which could, for example, be a density or a current. We are reminded that the full equation of motion for such operators, if they are also explicitly time varying, is
 i ℏ × F =[F,H]+¶t F ,      (18)
and the superposed dot will always denote a total time derivative. When the input data vary continuously over R their sum becomes an integral and there is a distinct Lagrange multiplier for each space-time point. Maximization of the entropy subject to the constraint provided by that information leads to a density matrix describing this macrostate:
 r = 1 Z exp éë - óő R l(x,t)F(x,t) d3x dt ůű ,      (19a)
where
 Z[l(x,t)]=Trexp éë - óő R l(x,t)F(x,t) d3x dt ůű (19b)
is now the partition functional. The Lagrange multiplier function is identified as the solution of the functional differential equation
 áF(x,t)ń ş Tr[rF(x,t)]=- d dl(x,t) lnZ ,       (x,t) Î R ,      (20)
and is defined only in the region R. Note carefully that the data set denoted by áF(x,t)ń is a numerical quantity that has been equated to an expectation value to incorporate it into a density matrix. Any other operator J(x,t), including J=F, is determined at any other space-time point (x,t) as usual by
 áJ(x,t)ń = Tr éë rJ(x,t) ůű =Tr éë r(t)J(x) ůű .     (21)
That is, the system with fixed H still evolves unitarily from the initial nonequilibrium state (19); although r surely will no longer commute with H, its eigenvalues nevertheless remain unchanged.

Inclusion of a number of operators Fk, each with its own information-gathering region Rk and its own Lagrange multiplier function lk, is straightforward, and if the data are time independent r can describe an inhomogeneous equilibrium system as discussed in connection with (II-10). The question is sometimes raised concerning exactly which functions or operators should be included in the description of a macroscopic state, and the short answer is: include all the relevant information available, for the PME will automatically eliminate that which is redundant or contradictory. A slightly longer answer was provided by Jaynes (1957b) in his second paper introducing information-theoretic ideas into statistical mechanics. He defined a density matrix providing a definite probability assignment for each possible outcome of an experiment as sufficient for that experiment. A density matrix that is sufficient for all conceivable experiments on a system is called complete for that system. Both sufficiency and completeness are defined relative to the initial information, and the existence of complete density matrices presumes that all measurable quantities can be represented by Hermitian operators and that all experimental measurements can be expressed in terms of expectation values. But even if one could in principle employ a complete density matrix, it would be extremely awkward and inconvenient to do so in practice, for that would require a much larger function space than necessary. If the system is nonmagnetic and there are no magnetic fields present, then there is no point to including those coordinates in a description of the processes of immediate interest, but only those that are sufficient in the present context. The great self-correcting feature of the PME is that if subsequent predictions are not confirmed by experiment, then this is an indication that some relevant constraints have been overlooked or, even better, that new physics has been uncovered.

The form (19) illustrates how r naturally incorporates memory effects while placing no restrictions on spatial or temporal scales. But this density matrix is definitely not a function of space and time; it merely provides an initial nonequilibrium distribution corresponding to data áF(x,t)ń Î R. Lack of any other information outside R - in the future, say - may tend to render r less and less reliable, and the quality of predictions may deteriorate. Barring any further knowledge of system behavior this deterioration represents a fading memory, which becomes quite important if the system is actually allowed to relax from this state, for an experimenter carrying out a measurement on an equilibrium sample cannot possibly know the history of everything that has been done to it, so it is generally presumed that it has no memory. Relaxation processes will be discussed in Section 4 below.

With an understanding of how to construct a nonequilibrium state it's now possible to move on to the next stage, which is to steady-state systems in which there may be steady currents, but all variables remain time independent. This is perhaps the most well-understood nonequilibrium scenario, primarily because it shares with equilibrium the property that it is stationary. But in equilibrium the Hamiltonian commutes with the density matrix, [H,r]=0, which implies that r also commutes with the time evolution operator. In the steady state, though, it is almost certain that H will not commute with the operators in the exponential defining r, even if the expectation values defining the state are time independent. While this time independence is a necessary condition, an additional criterion is needed to guarantee stationarity, and that is provided by requiring that [H,r]=0. In II it was shown that this leads to the additional constraint that only the diagonal parts of the specified operators appear in the steady-state density matrix, providing a theoretical definition of the steady state. By diagonal part' of an operator we mean that part that is diagonal in the energy representation, so that it commutes with H. For present purposes the most useful expression of the diagonal part of an operator is
 Fd=F- lim e® 0+ óő 0 -Ą eet ¶t F(x,t) dt ,        e > 0 ,     (22)
where the time dependence of F is unitary: F(t)=eitH/ ℏ F e-itH/ ℏ.

The resulting steady-state density matrix is given by (II-12) and (II-13) and is found by a simple modification of (19) above: remove all time dependence, including that in R, and replace F(x,t) by Fd(x); in addition, a term -bH is included in the exponentials to characterize an earlier equilibrium reference state. Substitution of the resulting rss into (2) provides the maximum entropy of the stationary state:
 1 k Sss = lnZss[b,l(x)] +báHńss + óő l(x)áFd(x)ńss , d3x      (23)
and F is often specified to be a current. This is the time-independent entropy of the steady-state process as it was established and has nothing to do with the entropy being passed from source to sink; entropy generation takes place only at the boundaries and not internally. Some applications of rss were given in II and others will be made below. No mention appears in II, however, of conditions for stability of the steady state, so a brief discussion is in order here.

Schlögl (1971) has studied stability conditions for the steady state in some depth through consideration of the quantum version of the information gain in changing from a steady-state density matrix r˘ to another r,
 L(r,r˘)=Tr éë r ćč lnr-lnr˘ öř ůű ,      (24)
which is effectively the entropy produced in going from r˘ to r. He notes that L is a Liapunov function in that it is positive definite, vanishes only if r = r˘, and has a positive second-order variation. Pfaffelhuber (1977) demonstrates that the symmetrized version, which is more convenient here,
 L*(r,r˘)= 1 2 Tr éë ćč r-r˘ öř ćč lnr-lnr˘ öř ůű ,      (25)
is an equivalent Liapunov function, and that its first-order variation is given by
 -dL*= 1 2 d(Dl DáFdń) .      (26)
The notation is that Dl = l-l˘, for example. If d is taken as a time variation, then Liapunov's theorem immediately provides a stability condition for the steady state,
 - × L * ł 0 .      (27)
Remarkably, (27) closely resembles the Gibbs condition (16) for stability of the equilibrium state, but in terms of
 -D × S ł 0

, as well as the Glansdorff-Prigogine criterion of phenomenological thermodynamics. But the merit of the present approach is that L* does not depend directly on the entropy and therefore encounters no ambiguities in defining a
 D × S

.

Thermal Driving

A variable, and therefore the system itself, is said to be thermally driven if no new variables other than those constrained experimentally are needed to characterize the resulting state, and if the Lagrange multipliers corresponding to variables other than those specified remain constant. 3 As discussed in I, a major difference with purely dynamic driving is that the thermally-driven density matrix is not constrained to evolve by unitary transformation alone. It was argued in I and II that a general theory of nonequilibrium must necessarily account explicitly for external sources, since it is only through them that the macroscopic constraints on the system can change. With that discussion as background let us suppose that a system is in thermal equilibrium with time-independent Hamiltonian in the past, and then at t=0 a source is turned on smoothly and specified to run continuously, as described by its effect on the expectation value áF(t)ń. That is, F(t) is given throughout the changing interval [0,t] and is specified to continue to change in a known way until further notice. We omit spatial dependence explicitly here in the interest of clarity, noting that the following equations are generalized to arguments (x,t) in (II-41)-(II-49). For convenience we consider only a single driven operator; multiple operators, both driven and otherwise, are readily included. Based on the probability model of I, the PME then provides the density matrix for thermal driving:
 rt
 = 1 Zt exp éë -bH- óő t 0 l(t˘)F(t˘) dt˘ ůű ,
 Zt[b,l(t)]
 =Tr exp éë -bH- óő t 0 l(t˘)F(t˘) dt˘ ůű .
(28)
The theoretical maximum entropy is obtained explicitly by substitution of (28) into (2),
 1 k St=lnZt+báH ńt + óő t 0 l(t˘)áF(t˘)ńt dt˘ ;     (29)
it is the continuously re-maximized information entropy. Equation (29) indicates explicitly that áH ńt changes only as a result of changes in, and correlation with F.

The expectation value of another operator at time t is áC ńt=Tr[rt C], and direct differentiation yields
 d dt áC(t)ńt
 =Tr éë C(t)¶trt +rt × C (t) ůű
 =á × C (t)ńt -l(t)KCFt(t,t) ,
(30)
where the superposed dot denotes a total time derivative. We have here introduced the covariance function
 KCFt(t˘,t) ş á F(t˘) C(t)ńt-áF(t˘)ńtáC(t)ńt = - dáC(t)ńt dl(t) ,      (31)
where the overline denotes a generalized Kubo transform with respect to the operator lnrt:
 F(t) ş óő 1 0 e-ulnrt F(t)eulnrt du ,      (32)
which arises from the possible noncommutativity of F(t) with itself at different times. The superscript t in KCFt implies that the density matrix rt is employed everywhere on the right-hand side of the definition, including the Kubo transform; this is necessary to distinguish it from several approximations.

In II we introduced a new notation into (30), which at first appears to be only a convenience:
 sC(t) ş d dt áC(t)ńt-á × C (t)ńt = -l(t)KCFt(t,t) .     (33)
For C=F
 sF(t)
 ş d dt áF(t)ńt-á × F (t)ńt
 =-l(t)KFFt(t,t) ,
(34)
which was seen to have the following interpretation: sF(t) is the rate at which F is driven or transferred by the external source, whereas dáF(t)ńt/dt is the total time rate-of-change of áF(t)ńt in the system at time t, and
 á × F (t)ńt

is the rate of change produced by internal relaxation. Thus, we can turn the scenario around and take the source as given and predict áF(t)ńt, which is the more likely experimental arrangement. This reversal of viewpoint is much like that associated with (4) suggesting that one could as well consider l the independent variable, rather than f, as was discussed in connection with (II-10); in fact, this is usually what is done in practice in applications of (17), where the temperature is specified. If the source strength is given, then the second line of (34) provides a nonlinear transcendental equation determining the Lagrange multiplier function l(t).

An important reason for eventually including spatial dependence is that we can now derive the macroscopic equations of motion. For example, if F(t) is one of the conserved densities e(x,t) in a simple fluid, such as those in (II-66), and J(x,t) is the corresponding current density, then the local microscopic continuity equation
 × e (x,t)+\boldnabla·J(x,t)=0      (35)
is satisfied irrespective of the the state of the system. When this is substituted into (34) we obtain the macroscopic conservation law
 d dt áe(x,t)ńt +\boldnabla·áJ(x,t)ńt = se(x,t) .      (36)
Specification of sources automatically provides the thermokinetic equations of motion, and in II it was shown how all these expressions reduce to those of the steady state when the driving rate is constant.

Everything to this point is nonlinear, but in many applications some sort of approximation becomes necessary, and often sufficient, for extracting the desired physical properties of a particular model. The most common procedure is to linearize the density matrix in terms of the departure from equilibrium, which means that the integrals in (28), for example, are in some sense small. The formalism for this was discussed briefly in II and a systematic exposition can be found elsewhere (Heims and Jaynes, 1962; Grandy, 1988). In linear approximation the expectation value of any operator C(x,t) is given by
 áC(x,t)ń-áC(x)ń0
 =- óő KCF(x,t;x˘,t˘)l(x˘,t˘) d3x˘ dt˘ ,
 (37)
 KCF(x,t;x˘,t˘)
 =á F(x˘,t˘) C(x,t)ń0 -áF(x˘)ń0áC(x)ń0 ,
 (38)
where we have re-inserted the spatial dependence. The integration limits in (37) have been omitted deliberately so that the general form applies to any of the preceding scenarios. The subscripts 0 indicate that all expectation values on the right-hand sides of (37) and (38) are to be taken with the equilibrium distribution (17), including the linear covariance function KCF=K0CF and the Kubo transform (32); KCF is independent of l. It may be useful to note that, rather than linearize about equilibrium, the same procedure can also be used to linearize about the steady state.

The space-time transformation properties of the linear covariance function (38) are of some importance in later applications, so it's a moment well spent to examine these. We generally presume time and space translation invariance in the initial homogeneous equilibrium system, such that the total energy and number operators of (II-67), as well as the total momentum operator P, commute with one another. In this system these translations are generated, respectively, by the unitary operators
 U(t)=e-iHt/ ℏ ,        U(x)=e-ix·P/ ℏ ,      (39)
and F(x,t)=Uf(x)Uf(tF U(t)U(x). Translation invariance, along with (32) and cyclic invariance of the trace, provide two further simplifications: the single-operator expectation values are independent of x and t in an initially homogeneous system, and the arguments of KCF can now be taken as r=x-x˘, t = t-t˘.

Generally, the operators encountered in covariance functions possess definite transformation properties under space inversion (parity) and time reversal. Under the former A(r,t) becomes PAA(-r,t), PA=±1, and under the latter TAA(r,-t), TA=±1. Under inversion the covariance function (38) behaves as follows:
 KFC(-r,-t)
 =KCF(r,t) = PCPFKCF(-r,t)
 =TCTFKCF(r,-t)=PCPFTCTFKCF(-r,-t)
(40)
where the first equality again follows from cyclic invariance. For many operators, including those describing a simple fluid, PT=+1 and the full reciprocity relation holds:
 KCF(r,t)=KFC(r,t) .      (41)
In fact, by changing integration variables in (32) it is easy to show that the nonlinear covariance function (31) also satisfies a reciprocity relation: KtCF(x˘,t˘;x,t)=KtFC(x,t;x˘,t˘).

One further property of linear covariance functions will be found useful. Consider the spatial Fourier transform in which we examine the limit k=|k|® 0,
 lim k® 0 Kab(k,t)= lim k® 0 óő eik·r Kab(r,t) d3r = óő Kab(r,t) d3r .      (42)
That is, taking the limit is equivalent to integrating over the entire volume. But this is also the long-wavelength limit, in which the wavelengths of slowly-decaying modes span the entire volume.

Suppose now that a is a locally-conserved density, such as those describing a simple fluid, whose volume integral A is then a conserved quantity commuting with the Hamiltonian in the equilibrium system. In this event (39) implies that the left-hand side of (42) reduces to KAb(0,0), independent of space, time, and Kubo transform; the covariance function has become a constant correlation function, as in (14), and is just a thermodynamic quantity.

2. Nonequilibrium Thermodynamics

In equilibrium thermodynamics everything starts with the entropy, as in (10), and the same is true here. The instantaneous maximum entropy of thermal driving is exhibited in (29), and with St now a function of time one can compute its total time derivative as
 1 k dSt dt = ćč ¶lnZt ¶a öř × a +b dáHńt dt -l(t) óő t 0 l(t˘)KtFF(t,t˘) dt˘ ,      (43)
the spatial variables again being omitted temporarily. Because H is not explicitly driven its Lagrange multiplier remains the equilibrium parameter b. With a = V, the system volume, the equilibrium expressions (8) and (11) identified the term in lnZ as a work term. In complete analogy, the first term on the right-hand side of (43) is seen to be a power term when the volume is changing; one identifies the time-varying pressure by writing this term as bP(t).

Commonly the volume is held constant and the term containing the Hamiltonian written out explicitly, so that (43) becomes
 1 k dSt dt
 =-bl(t)KtHF(t,0)-l(t) óő t 0 l(t˘)KtFF(t,t˘) dt˘
 =gF(t)sF(t) ,
(44)
where we have employed (34) and defined a new parameter
 gF(t)
 ş b KtHF(t,0) KtFF(t,t) + óő t 0 l(t˘) KtFF(t,t˘) KtFF(t,t) dt˘
 = ćč dSt dáF(t)ńt öř [(thermal) || (driving)] ,
(45)
as discussed in II. The subscript thermal driving' reminds us that this derivative is evaluated somewhat differently than in the equilibrium formalism because the expectation values of H and F are not independent here. When the source strength sF(t) is specified the Lagrange multiplier itself is determined from (34) and gF is interpreted as a transfer potential governing the transfer of F to or from the system. If two systems can exchange quantities Fi under thermal driving, then the conditions for migrational equilibrium at time t are
 gFi(t)1=gFi(t)2 .      (46)

In II it was noted that St refers only to the information encoded in the distribution of (28) and cannot refer to the internal entropy of the system. In equilibrium the maximum of the information entropy is the same as the experimental entropy, but that is not necessarily the case here. For example, if the driving is removed at time t=t1, then St1 in (29) can only provide the entropy of that nonequilibrium state at t=t1; its value will remain the same during subsequent relaxation, owing to unitary time evolution. Although the maximum information (or theoretical) entropy provides a complete description of the system based on all known physical constraints on that system, it cannot describe the ensuing relaxation, for it contains no new information about that process. We return to this in Section 4 below.

Combination of (34) and the second line of (43) strongly suggests the natural expression
 1 k × S t =gF(t) ćč d dt áF(t)ńt -á × F (t)ńt öř .      (47)
in which the first term on the right-hand side represents the total time rate-of-change of entropy
 × S tot

arising from the thermal driving of F(t), whereas the second term is the rate-of-change of internal entropy
 × S int

owing to relaxation. Thus, the total rate of entropy production in the system can be written
 × S tot (t)= × S t + × S int (t) ,      (48)
where the entropy production of transfer owing to the external source,
 × S t

, is given by the first line of (44). This latter quantity is a function only of the driven variable F(t), whereas the internal entropy depends on all variables, driven or not, necessary to describe the nonequilibrium state and is determined by the various relaxation processes taking place in the system. If spatial variation is included the right-hand side of (47) is integrated over the system volume.

It is important to understand very clearly the meaning of Eq.(48), so we re-state more carefully the interpretation of each term. From (44),
 × S t

is the rate of change of the entropy of the macroscopic state of the system due to the source alone; it involves the maximum of the information entropy and is associated entirely with the source. The term
 × S int

is the contribution to the rate at which the entropy of that state is changing due to relaxation mechanisms within the system itself. Thus,
 × S tot

is the total rate of change of the entropy of the macroscopic state. When the source is removed
 × S t =0

, and
 × S tot

is the rate at which the entropy of the macroscopic state changes due to internal relaxation processes. Thus, entropy is always associated with a macroscopic state and its rate of change under various processes; the entropy of the surroundings does not enter into this discussion. Equation (48) does not apply to steady-state processes, in which no entropy is generated internally.

Linear Heating

As a specific example it is useful to make contact with classical thermodynamics and choose the driven variable to be the total-energy function for the system, E(t). 4 In the presence of external forces this quantity is not necessarily the Hamiltonian, but can be defined in the same way as H in the isolated system, (II-67), in terms of the energy density operator:
 E(t) ş óő V h(x,t) d3x      (49)
and the time evolution is no longer unitary. The point is that H does not change in time, only its expectation value.

In the case of pure heating
 × a = × V =0

and (43) and (44) become, respectively,
 1 k × S t =gE(t)sE(t) ,      (50a)

 gE(t)=b KtHE(t,0) KtEE(t,t) + óő t 0 l(t˘) KtEE(t,t˘) KtEE(t,t) dt˘ .      (50b)
The dimension of gE(t) is E-1, so it is reasonable to interpret this transfer parameter as a time-dependent inverse temperature' b(t)=[kT(t)]-1; the temperature must change continuously as heat is added to or removed from the system, though it is difficult to define a measurable quantity like this globally. Hence, in analogy with the equilibrium form S=dQ/T, Eq.(10), the content of (50a) is that
 × S t =kgE × Q ,      (51)
because the rate of external driving is just
 × Q

here. A further analogy, this time with (11), follows from the first line of (34), which extends the First Law to
 × E = × Q + × W ,      (52)
because any work done in this scenario would change only the internal energy. These last two expressions are remarkably like those advocated by Truesdell (1984) in his development of Rational Thermodynamics, and are what one might expect from naïve time differentiation of the corresponding equilibrium expressions. Indeed, such an extrapolation may provide a useful guide to nonequilibrium relations, but in the end only direct derivation from a coherent theory should be trusted.

In (48) the term
 × S int

is positive semi-definite, for it corresponds to the increasing internal entropy of relaxation; this is demonstrated explicitly in Section 4. Combination with (51) then allows one to rewrite (48) as an inequality:
 × S tot =kgE × Q + × S int ł kgE × Q .      (53)
One hesitates to refer to this expression as an extension of the Second Law, for such a designation is fraught with ambiguity; the latter remains a statement about the entropies of two equilibrium states. Instead, it may be more prudent to follow Truesdell in referring to (53) as the Clausius-Planck inequality.

A linear approximation in (50b), as described by (37) and (38), leads to considerable simplification, after which that expression becomes
 b(t) @ b+ óő t 0 l(t˘) KEE(t-t˘) KHH dt˘ ,      (54)
while recalling that b = b(0) is the equilibrium temperature. The static covariance function is now just an equilibrium thermodynamic function proportional to the energy fluctuations (and hence to the heat capacity). In this approximation the expectation value of the driven energy function is
 áE(t)ńt @ áEń0- óő t 0 l(t˘)KEE(t-t˘) dt˘ ,      (55)
so that if, for example, energy is being transferred into the system (sE > 0), then the integral must be negative. We can then write
 b(t) @ b- ęę óő t 0 l(t˘) KEE(t-t˘) KHH dt˘ ęę ,      (56)
and b(t) is decreasing from the equilibrium value. The physical content of (56) therefore is that T(t) @ T(0)+DT(t), as expected. Although it is natural to interpret T(t) as a temperature', we are cautioned that only at t=0 is that interpretation unambiguous.

A complementary calculation is also of interest, in which spatial variation is included and the homogeneous system is driven from equilibrium by a source coupled to the energy density h(x,t). In linear approximation the process is described by
 áh(x,t)ńt-áhń0
 =- óő V d3x˘ óő t 0 dt˘ l(x˘,t˘)Khh(x-x˘,t-t˘) ,
 (57a)
 sh(x,t)
 =- óő V l(x˘,t)Khh(x-x˘,t=0) d3x˘ .
 (57b)
After a well-defined period of driving the source is removed, the system is again isolated, and we expect it to relax to equilibrium (see Section 4); the upper limit of integration in (57a) is now a constant, say t1. Presumably this is a reproducible process.

For convenience we take the system volume to be all space and integrate Eqs.(57) over the entire volume, thereby converting the densities into total Hamiltonians. Owing to spatial uniformity and cyclic invariance the covariance function in (57a) is then independent of the time (the evolution being unitary after removal of the source), and we shall denote the volume integral of sh(x,t) by sh(t). Combination of the two equations for t > t1 yields
 áHń-áHń0= óő t1 0 sh(t˘) dt˘ ,        t > t1 ,     (58a)
or
 áHń = áHń0+DE ,      (58b)
which is independent of time and identical to (55) at t=t1. The total energy of the new equilibrium state is now known, and the density matrix describing that state can be constructed via the PME.

The last few paragraphs provide a formal description of slowly heating a pot of water on a stove, but in reality much more is going on in that pot than simply increasing the temperature. Experience tells us that the number density is also varying, though N/V is constant (if we ignore evaporation), and a proper treatment ought to include both densities. But thermal driving of h(x,t) requires that n(x,t) is explicitly not driven, changing only as a result of changes in h, through correlations. The proper density matrix describing this model 5 is
 rt= 1 Zt exp éë -bH- óő d3x˘ óő t 0 éë
 lh(x˘,t˘)h(x˘,t˘)
 +ln(x˘,t˘)n(x˘,t˘) ůű dt˘ ůű ,
(59)
and the new constraint is expressed by the generalization of (34) to the set of equations
 sh(x,t)
 =- óő lh(x˘,t) Khh(x-x˘;t-t) d3x˘
 - óő ln(x˘,t) Khn(x-x˘;t-t) d3x˘ ,
 0
 =- óő lh(x˘,t) Knh(x-x˘;t-t) d3x˘
 - óő ln(x˘,t) Knn(x-x˘;t-t) d3x˘ ,
(60)
asserting explicitly that sn ş 0.

In this linear approximation ln is determined by lh and we can now carry out the spatial Fourier transformations in (60). The source strength driving the heating is thus
 sh(k,t)=-lh(k,t)Khh(k,0) éë 1- |Knh(k,0)|2 Khh(k,0)Knn(k,0) ůű ,      (61)
where the t=0 values in the covariance functions refer to equal times. For Hermitian operators the covariance functions satisfy a Schwarz inequality, so that the ratio in square brackets in this last expression is always less than or equal to unity; hence the driving strength is generally reduced by the no-driving constraint on n(x,t).

The expression (61) is somewhat awkward as it stands, so it's convenient to introduce a new variable, or operator,
 h˘(k,t) ş h(k,t) - Knh(k,0) Knn(k,0) n(k,t) .     (62)
Some algebra then yields in place of (61)
 sh(k,t)=-lh(k,t)Kh˘h˘(k,0) .      (63)
In the linear case, at least, it is actually h˘ that is the driven variable under the constraint that n is not driven, and the source term has been renormalized.

With (63) the two expectation values of interest are
 áh˘(k,t)ńt-áh˘ń0
 = óő t 0 sh(k,t˘) Kh˘h˘(k,t-t˘) Kh˘h˘(k,0) dt˘ ,
 (64)
 án(k,t)ńt-ánń0
 = óő t 0 sh(k,t˘) Kn h˘(k,t-t˘) Kh˘h˘(k,0) dt˘ .
 (65)
Thus, the number density changes only as a consequence of changes in the energy density. The reader will have no difficulty finding explicit expressions for the new covariance functions in these equations, as well as showing that total particle number is conserved in the process, as expected.

This Section provides explicit procedures for carrying out some calculations in nonequilibrium thermodynamics. Undoubtedly one can develop many more applications of this kind along the lines suggested by Truesdell (1984), and in discussions of so-called extended irreversible thermodynamics (e.g., Jou, et al, 2001).

3. Transport Processes and Hydrodynamics

Linear transport processes in a simple fluid were discussed briefly in II in terms of the locally conserved number density n(x,t), energy density h(x,t), and momentum density mj(x,t), where m is the particle mass and j the current in the fluid that was initially homogeneous; the associated local equations of motion (continuity equations) of the type (35) are given in (II-66). System response to local excesses of these densities was studied in the long-wavelength limit that essentially defines hydrodynamics, and a generic expression for the steady-state expectation values in the perturbed system, in linear approximation, was presented in (II-71). This expression represents the leading term in an expansion in powers of the Lagrange multiplier l, which in this scenario eventually becomes a gradient expansion. In the cases of the densities n and h the procedure led to Fick's law of diffusion and Fourier's law of heat conduction, respectively:
 áj(x)ńss
=-
 óő Ą 0 e-et dt óő v Kjj(x-x˘,t) d3x˘

 óő v Knn(x-x˘) d3x˘
·Ńán(x)ńss
 ş -D(x)·Ńán(x)ńss ,
(66)

 áq(x)ńss
 @ -ŃT· óő v d3x˘ óő Ą 0 e-et Kqq(x-x˘, t) kT2(x˘) dt
 ş -k·ŃT(x) ,
(67)
where j is a particle current, q a heat current, and v is the volume outside of which the covariance function vanishes. The diffusion tensor D and the thermal conductivity tensor k are in many cases considered scalar constants, but both are easily extended to time-dependent coefficients by releasing the steady-state constraint. Correlation function expressions of the type developed here have been found by a number of authors over the years and are often known as Green-Kubo coefficients. Although they are usually obtained by contrived methods, rather than as a straightforward result of probability theory, those results clearly exhibited the right instincts for how the dissipative parameters in the statistical theory should be related to the microscopic physics.

If the spatial correlations in (66) are long range, then v® V and the discussion following (42) implies that Kjj becomes independent of time. In this event the time integral diverges and D does not exist. A similar divergence arises in both (66) and (67) if the time correlations decay sufficiently slowly, possibly indicating the onset of new phenomena such as anomalous diffusion. Of course, one is not obliged to make the long-wavelength approximation, and in these cases it may not be prudent to do so. These observations remain valid even when the processes are not stationary.

The same procedure is readily generalized to more complicated processes such as thermal diffusion, thermoelectricity, and so on. For example, in a system of particles each with electric charge e, and in the presence of a static external electric field derived from a potential per unit charge f(x), the appropriate Lagrange multiplier is no longer the chemical potential m, but the electrochemical potential
 y(x,t)=m(x,t) +ef(x,t) .      (68)
For a steady-state process in the long-wavelength limit and linear approximation one finds for the electric current j and the heat current q the set of coupled equations
 áj(x)ńss
 =- 1 ekT Ńy·Ljj(x)+ 1 kT2 ŃT·Ljq(x) ,
 áq(x)ńss
 =- 1 ekT Ńy·Lqj(x)+ 1 kT2 ŃT·Lqq(x) ,
(69)
where the second-rank tensors can be written generically as
 LAB(x)= lim e® 0+ óő Ą 0 e-et dt óő v á B(x˘) A(x,t)ń0 d3x˘ .      (70)
These are the thermoelectric coefficients: Ljj is proportional to D and Lqq is proportional to k, whereas Ljq is the Seebeck coefficient and Lqj the Peltier coefficient. In an isotropic medium the symmetry properties of the covariance functions imply the Onsager-like reciprocity relation Ljq=Lqj, but this clearly depends strongly on the space-time behavior of the operators involved, as well the specific system under study. An in-depth critique of Onsager reciprocity has been provided by Truesdell(1984).

It remains to examine the momentum density in the simple fluid; although the analysis of transport coefficients is readily applied to other systems, the fluid presents a particularly clear model for this exposition. Within the steady state context, the linear approximation, and the long-wavelength limit, (II-71) for a perturbation in mj leads to the following expression for the expectation of another operator C(x):
 áDC(x)ńss
 = -m óő V li(x˘)KCji(x-x˘) d3x˘
 + ćč Ńkli öř óő v d3x˘ óő Ą 0 e-et KCTik(x-x˘,t) dt ,
(71)
where here and subsequently we adopt the notation DC(x)=C(x)-áCń0, and the
 lim e® 0+

is understood. In addition, sums over repeated indices are implied.

First consider C as the total momentum operator P. Then, in the given scenario, we know that
 á j(x˘) P(x)ń0=(n0/b)dij

, where n0 and b are equilibrium values and KPTij(x-x˘,t)=0 from the symmetry properties (40). Hence (71) reduces to
 óő V ámji(x)ńss d3x=- n0 m b óő V li(x) d3x .      (72)
A convenient notation emerges by defining a fluid velocity vi(x) by writing ámji(x)ńss=mn0vi(x), so that the Lagrange multiplier is identified as l=-bv.

Now take C in (71) to be a component of the energy-momentum tensor. By the usual symmetry arguments,
 áTki(x)ńss = áTkiń0+Ńmln óő Ą 0 e-et dt óő v Kki,mn(x-x˘,t) d3x˘ ,      (73)
where the covariance function is that of Tmn and Tki, and the equilibrium expectation value áTkiń0=P0dki is the hydrostatic pressure. With the identification of l, (73) is the general equation defining a fluid; in fact, we can now show that it's a Newtonian fluid.

There are numerous properties of Tmn and the covariance functions involving it that are needed at this point; they will simply be stated here and can be verified elsewhere (e.g., Puff and Gillis, 1968). As expected, Tmn is symmetric and the space integral in (71) can be carried out immediately; the space-integrated covariance function has only a limited number of components in the isotropic equilibrium medium, in that nonzero values can only have indices equal in pairs, such as áT12T12ń0 and áT22T33ń0; in addition, the space-integrated tensor itself has the property
 T11=T22=T33=1/3 T ,      (74)
where T is the tensor trace. If we adopt the notation vi,k ş vi/xk and convention that sums are performed over repeated indices, then (73) can be rewritten as
 áTki(x)ńss = P0dki-h ćč vk,i+vi,k-2/3vl,ldki öř -zvl,ldki ,      (75)
where we have identified the shear viscosity
 h ş b V óő Ą 0 e-et Kmn,mn(t) dt ,        m ą n ,      (76a)
and the bulk viscosity
 z ş b 9V óő Ą 0 e-et KTT(t) dt .      (76b)
In (76a) any values of m ą n can be used, but are generally dictated by the initial current; the indices are not summed. Note that both h and z are independent of space-time coordinates.

Equations of Motion

Local conservation laws of the form (35) for a simple fluid were displayed in (II-66), and their conversion into macroscopic equations of motion for expectation values of densities and currents takes the form (36) under thermal driving. For simplicity we shall only consider the momentum density to be driven here, although most generally all three densities could be driven. With mj(x,t) as the driven variable, the exact macroscopic equations of motion are
 d dt án(x,t)ńt+Ńiáji(x,t)ń
 =0 ,
 (77a)
 m d dt áji(x,t)ńt+ŃkáTki(x,t)ń
 =sji ,
 (77b)
 d dt áh(x,t)ńt+Ńiáqi(x,t)ń
 =0 .
 (77c)
The rate sj that the source drives the momentum density can equally be written as a force density nF. As for other driving possibilities, an example would be to drive the number density in an ionized gas, rather than the momentum, and study electron transport. The source in this case would be se=nemeE, where E is a static field and me is the dc electron mobility.

The nonlinear equations of motion (77) are independent of the long-wavelength approximation and are valid arbitrarily far from equilibrium. They represent five equations for the five densities, but it is necessary to employ specific models for áTki(x,t)ń and áqi(x,t)ń, and these are most often taken as the linear approximations to the heat and momentum currents, as in (67) and (75), respectively. For example, under thermal driving the linear approximation (73) is replaced by
 áTki(x,t)ń = áTkiń0+ óő v d3x˘ óő t 0 ŃmlnKki,mn(x-x˘,t-t˘) dt˘ .      (78)
The long-wavelength limit is still appropriate and the pressure remains hydrostatic. While the Lagrange multiplier is now given by (34), it is basically still a velocity, so that one can proceed in the same manner as above and find for the dynamic viscosities
 h(t)= b V óő t 0 Kmn,mn(t-t˘) dt˘       m ą n ,      (79a)

 z(t)= b 9V óő t 0 KTT(t-t˘) dt˘ .      (79b)

In linear approximation the solutions to the equations of motion (77) are just the linear predictions of the type (37). With the notation Dn(x,t)=n(x,t)-ánń0, etc., for the deviations, these are
 áDn(x,t)ń
 =- óő v óő t 0 l(x˘,t˘)·Knj(x-x˘,t-t˘) d3x˘dt˘ ,
 (80a)
 áDh(x,t)ń
 =- óő v óő t 0 l(x˘,t˘)·Khj(x-x˘,t-t˘) d3x˘dt˘ ,
 (80b)
 áDTmn(x,t)ń
 =- óő v óő t 0 l(x˘,t˘)·KTmnj(x-x˘,t-t˘) d3x˘dt˘ ,
 (80c)
 áDj(x,t)ń
 =- óő v óő t 0 l(x˘,t˘)·Kjj(x-x˘,t-t˘) d3x˘dt˘ ,
 (80d)
 áDq(x,t)ń
 =- óő v óő t 0 l(x˘,t˘)·Kqj(x-x˘,t-t˘) d3x˘dt˘ .
 (80e)
In these equations l is obtained from (34),
 l(x,t)=- sj(x,t) Kjj(x-x˘) ,      (81)
where the denominator is an equal-time covariance. To verify that these are solutions, first take the time derivative of (80a),
 - d dt án(x,t)ń
 = óő v l(x˘,t)·Knj(x-x˘,0) d3x˘
 + óő v óő t 0 l(x˘,t˘)·K[(n)\dot]j(x-x˘,t-t˘) d3x˘dt˘ .
(82)
Employ the microscopic conservation law
 × n +Ń·j=0

to write K[(n)\dot]j=-Ń·Kjj and note that the first term on the right-hand side of (82) vanishes by the symmetry properties (40); the result is just what one finds from taking the divergence of (80d), thereby verifying (77a). A similar calculation verifies (77c), but verification of (77b) contains a different twist. In the time derivative of (80d) the term analogous to the first term on the right-hand side of (82) is, from (II-44),
 óő v l(x˘,t)·Kjj(x-x˘,0) d3x˘ = sj(x,t) .      (83)
Thus, at least in linear approximation, the statistical predictions (80) are completely consistent with, and provide the first-order solutions to the deterministic equations (77).

Fluctuations

The statistical fluctuations are determined, as always, by the correlation of deviations, or covariance functions. In general these are the nonlinear covariances (31), or possibly those defined by the steady state distribution. In any case, they are usually quite difficult to evaluate other then in very approximate models. For the moment, then, attention will be focused on the linear covariance functions, not only because they are readily evaluated in the long-wavelength approximation, but that and the linear approximation are well controlled. Thus, in linear hydrodynamics a first approach to fluctuations is a study of Knn, Kjj, Khh, Kqq, and Kki,mn, the last referring to the energy-momentum tensors. One should note that these only describe statistical fluctuations; whether they can be equated with possible physical fluctuations is another matter and is discussed in the Appendix.

It is well known (e.g, Grandy, 1988) that the Fourier-transformed quantity KAB(k,w) is the dissipative part of the physical response of the system, whereas the ordinary (i.e., no Kubo transform) correlation function CAB ş áDADBń0, in terms of the deviations defined above, describes the actual fluctuations. The two are related by a fluctuation-dissipation theorem,
 KAB(k,w)= 1-e-b ℏw b ℏw CAB(k,w) ®   b ℏw << 1 CAB(k,w) .      (84)

The covariances of the densities n, h, and j are obtained in a straightforward manner in the long-wavelength limit and are simple (equilibrium) thermodynamic functions (e.g., Puff and Gillis, 1967; Grandy, 1987). They are:
 lim k® 0 Knn(k,t)
 = n02 kT b ,
 (85a)
 lim k® 0 Khh(k,t)
 = kT2CV V + kT b éë h0+P0- aT kT ůű 2 ,
 (85b)
 lim k® 0 Kjijk(k,t)
 = n0 b dik ,
 (85c)
where kT is the isothermal compressibility, a the coefficient of thermal expansion, CV the constant-volume heat capacity, and subscripts 0 refer to equilibrium quantities. Expressions on the right-hand sides of (85) are most readily obtained from familiar thermodynamic derivatives, as suggested in (14), but correlations for the dissipative currents require a little more work.

Consider first
 Kql qm(x˘,t˘;x,t)= qm ql(r,t) 0

, which will be abbreviated Klm temporarily. The space-time Fourier transform is
 Klm(k,w)= óő d3r e-ik·r óő Ą -Ą dt eiwt Klm(r,t) .      (86)
Introduce a bit of dispersion into the integral, to insure convergence, by replacing w with w±ie, e > 0. The properties (40) imply that this covariance function is invariant under time reversal, so that in the limits k® 0, w® 0 (86) becomes 6
 lim [(k® 0) || (w® 0)] Klm(k,w)=2 lim e® 0 óő d3r óő Ą 0 dt e-|e|t Klm(r,t) .      (87)
But this last expression is just the thermal conductivity k in (67), if we note that the factor T(x˘) ~ T(x) can be extracted from the integral in that expression, as is usual. Symmetry again implies that only the diagonal components contribute here, so the inverse transformation of (87) yields in the long-wavelength limit
 Kql qm(r,t) @ 2kkT2d(r)d(t)dlm .      (88)

Similarly, fluctuations in the energy-momentum tensor are described by the covariance function
 Kkl,mn(r,t) = Tmn Tkl(r,t) 0 -P02dkldmn

. With the definitions (76) the above procedure leads to
 Kkl,mn(r,t) @ 2kT éë h ćč dkmdln+dkndlm öř + ćč z-2/3h öř dkldmn ůű d(r)d(t) .      (89)

The expressions (88) and (89) for fluctuations in the dissipative currents are precisely those found by Landau and Lifshitz (1957) in their study of hydrodynamic fluctuations. Here, however, there is no need to introduce fictitious random' forces, additional dissipative stresses, or extraneous averaging processes, for these are just the straightforward results expected from probability theory. Hydrodynamic fluctuations apparently have been observed in convection waves in an isotropic binary mixture of H2O and ethanol (Quentin and Rehberg, 1995).

When a system is farther from equilibrium, but in a steady state, the deviations from that state can be studied in much the same way. In accordance with the stationary constraint, when the current is driven at a constant rate the density matrix rss depends only on the diagonal part of the current operator, jd(x). By definition this operator commutes with the Hamiltonian, as does the total-momentum operator, which again leads to some simplifications in the long-wavelength limit. But the fact remains that expectation values and covariance functions are very difficult to evaluate with a density matrix rss, let alone in an arbitrary nonequilibrium state, which has led to alternative means for estimating fluctuations about these states.

In fluid mechanics it is customary to reformulate the equations of motion (77) in terms of the fluid velocity v by introducing a notation áj(x,t)ń = án(x,t)ńv(x,t); this is motivated in part by Galilean invariance. Transformation of the densities and currents can be performed by a unitary operator U(G)=exp[iv·G/ ℏ], where G is the generator of Galilean transformations and v is a drift velocity. The results for the expectation values are, in addition to that for j,
 áTij(x,t)ń
 =áTij(x,t)ń0+mnvivj ,
 áh(x,t)ń
 =áh(x,t)ń0+1/2mnv2 ,
 áqi(x,t)ń
 = ćč áh(x,t)ń0+1/2mnv2 öř vi+vk áTkj(x,t)ń0  ,
(90)
where here the subscript 0 denotes values in the rest frame. This, of course, is merely a re-description of the system in the laboratory frame and contains no dynamics; dissipation cannot be introduced into the system by a unitary transformation! But one now sees the view from the laboratory frame.

With the additional definition of the mass density r = mán(x,t)ń, not to be confused with the density matrix, (77) can be rewritten as
 ¶tr+Ń·(rv)
 =0 ,
 r ćč ¶tvj+v·Ńvj öř +¶iTij
 =nFj ,
 ¶t(rh)+¶i ćč rhvi+qi öř
 =-Tij¶ivj ,
(91)
where every quantity with the exception of Fj is actually an expectation value in the state described by rt. For large macroscopic systems we expect these values to represent sharp predictions but still possess small deviations from these values. Such a deviation can excite the mass density r to a temporary new value r˘=r+Dr, for example.

By direct substitution (91) can be converted into a set of equations for the deviations of all the variables, and by retaining only terms linear in those deviations they become linear equations that have a good chance of being solved in specific scenarios. The statistical fluctuations for the dissipative currents have the same forms as those of (88) and (89), but T, h,and z are now space and time dependent 7 (e.g., Morozov, 1984); in addition, the stress-tensor model of (75) now contains deviations in the fluid velocity as well. These equations will not be developed here since they are given in detail elsewhere (e.g., Fox, 1984); an application to the Rayleigh-Bénard problem has been given by Schmitz and Cohen (1985).

Ultrasonic Propagation

Two special cases of thermal driving arise when the rate is either zero or constant, describing equilibrium or a steady-state process, respectively. Another occurs when sF can usefully be replace by a time-dependent boundary condition. For example, a common experimental arrangement in the study of acoustical phenomena is to drive a quartz plate piezoelectrically so that it generates sound waves along a plane. Thus it is quite realistic to characterize the external source by specifying the particle current on the boundary plane at z=0. The system excited by the sound wave is then described by the density matrix
 r = 1 Z exp ěí î -bH+ óő dx˘ óő dy˘ óő dt˘ l(x˘,y˘,t˘)·J(x˘,y˘,0;t˘) üý ţ ,         (92)
and Z is the trace of the exponential.

To keep the model simple the volume is taken to be the entire half-space z > 0, and we presume the z-component of current to be specified over the entire xy-plane for all time. Although there are no currents in the equilibrium system, current components at any time in the perturbed system in the right half-space are given by áJa(x,y,z;t)ń = \Tr[r Ja(x,y,z;t)]. Restriction to small-amplitude disturbances, corresponding to small departures from equilibrium, implies the linear approximation to be adequate:
 áJa(x,y,z;t)ń = óő Ą -Ą dx˘ óő Ą -Ą dy˘ óő Ą -Ą dt˘ l(x˘,y˘,t˘)KJaJz(x˘,y˘,0,t˘;x,y,z,t) ,     (93)
and consistency requires this expression to reproduce the boundary condition at z=0:
 áJz(x,y,0;t)ń = óő Ą -Ą dx˘ óő Ą -Ą dy˘ óő Ą -Ą dt˘ l(x˘,y˘,t˘)KJzJz(x-x˘,y-y˘,0;t-t˘) .     (94)
We are thus considering low intensities but arbitrary frequencies.

Linearity suggests it is sufficient to consider the disturbance at the boundary to be a monochromatic plane wave. Thus,
 áJz(x,y,0;t)ń = Je-iwt ,      (95)
where J is a constant amplitude. Substitution of this boundary value into (94) allows one to solve the integral equation for l(x˘,y˘,t˘) immediately by Fourier transformation, and the Lagrange-multiplier function is determined directly by means of the driving term, as expected. We find that
 l(x,y,t)=lw e-iwt ,     (96)
with
 lw-1 ş J-1 óő Ą -Ą dx óő Ą -Ą dy KJzJz(x,y,0;w) ,     (97)
so that l is independent of spatial variables. Given the form of the covariance function in (94), the current in the right half-space will also be independent of x and y:
 áJa(x,y,z;t)ń = lw e-iwt óő Ą -Ą dx˘ óő Ą -Ą dy˘ óő Ą -Ą dt˘ eiwt˘ KJaJz(x˘,y˘,z;t˘) .      (98)

Define a function
 KJaJz(z,w) ş KJaJz(0,0,z;w) = óő Ą -Ą dkz 2p eikzz KJaJz(kz,w) ,      (99)
where KJaJz(kz,w) ş KJaJz(0,0,kz;w). Then (93) for the current in the perturbed system can be rewritten as
 Ja(z,t) ş áJa(0,0,z;t)ń = Ja(z)e-iwt ,      (100)
and
 Ja(z) ş lwKJaJz(z,w) .      (101)
In the same notation, lw-1=J-1 KJzJz(0,w) . Thus, the amplitude of the sound wave relative to that of the initial disturbance is
 Ja(z) J = KJaJz(z,w) KJzJz(0,w) .      (102)

So, application of a monochromatic plane wave at the boundary results in a disturbance that propagates through the system harmonically, but with an apparent attenuation along the positive z-axis given by Ja(z). Analysis of the spatial decay depends on the detailed structure of the current-current covariance function, and only on that; this remains true if we synthesize a general wave form.

As an example, suppose that
 KJzJz(kz,w)=2pg(w) d(kz-k0) .     (103a)
From(102) the z-component of current is then
 Jz(z)=J eik0z ,      (103b)
and the initial plane wave propagates with no attenuation.

More interesting is a covariance function of Lorentzian form, such as
 KJzJz(kz,w)= af(w) a2+(kz-k0)2 .      (104a)
A similar calculation yields
 Jz(z)=J eik0z e-a|z|  ,      (104b)
which exhibits the classical exponential attenuation. Although the Lorentzian form provides at least a sufficient condition for exponential decay of the sound wave, there clearly is no obvious requirement for the attenuation to be exponential in general.

Note that the number density itself could have been predicted in the above discussion, merely by replacing Ja with n. In a similar manner we find that
 n(z,t) ş án(0,0,z;t)ń-ánń0 = lw KnJz(z,w) e-iwt .      (105)
But the covariance function KnJz is directly proportional to only the density-density covariance function Knn, and therefore
 KnJz(z,w)= w 2p óő Ą -Ą dkz kz eikzz Knn(kz,w) .      (106)
The variation in density, n(z,t), is directly related to the correlated propagation of density fluctuations, and it is precisely this correlation of fluctuations that makes intelligible speech possible.

The preceding model has been adapted by Snow (1967) to an extensive study of free-particle systems. Although there is essentially no propagation in the classical domain, the quantum systems do exhibit interesting behavior, such as second sound.

4. Relaxation and The Approach To Equilibrium

In the thermally driven system the instantaneous nonequilibrium state is described by the density matrix (28), with associated entropy (29). If the driving source is removed at time t=t1 the macroscopic nonequilibrium state rt1 at that time has maximum information entropy
 1 k St1=lnZt1+báH ńt1 + óő t1 0 l(t˘)áF(t˘)ńt1 dt˘ ,      (107)
which is fixed in time. From (48) we note that the total rate of entropy production is now
 × S int (t)

and we expect the system to relax to equilibrium; that is, we want to identify and study the relaxation entropy Sint(t). In the discussion following (46) it seemed that not much could be said about this quantity in general because there was no information available at that point to describe the relaxation. But now it appears that there are indeed two cogent pieces of information that change the situation greatly. The first new piece is that the source has been removed, so that the system is now isolated from further external influences; the second is the set of equations of motion (77), in which the source term is zero. Although St1 cannot evolve to the canonical entropy of equilibrium, we can now construct an Sint(t) that does, because the relaxation is deterministic.

The absence of external driving assures us that the microscopic time evolution now takes place through unitary transformation, and that the density matrix develops by means of the equation of motion i ℏtr = [H,r]. Unquestionably, if the system returns to equilibrium at some time t in the future, the density matrix r(t) evolved in this way will correctly predict equilibrium expectation values. As discussed at length in I, however, there is virtually no possibility of carrying out such a calculation with the complete system Hamiltonian H; and even if that could be done it is not possible for r(t) to evolve into the canonical equilibrium distribution, because the eigenvalues of rt1 remain unchanged under unitary transformation. In addition, the information entropy is also invariant under this transformation, simply because there is no new macroscopic information being supplied to it. But this last observation is the key point: it is not the microscopic behavior that is relevant, for we have no access to that in any event; it is the macroscopic behavior of expectation values that should be the proper focus.

Some very general comments regarding relaxation were made in II, primarily in connection with (II-88), but now it is possible to be much more specific in terms of a definite model. In the following the simple fluid is chosen as a relaxation model because it is both familiar and elementary in structure. Nevertheless, the procedure should apply to any system with well-defined equations of motion analogous to (77).

At t1 total system quantities such as E, N, V, etc., are fixed and define the eventual equilibrium state; indeed, if we knew their values - and surely they could be predicted quite accurately with rt1 - that state could be constructed immediately by means of the PME. But varying densities and currents continue to exist in the still-inhomogeneous system for t ł t1, and will have to smooth out or vanish on the way to equilibrium. As an example, the number density satisfies (77a) and, for t > t1, there are no external forces so that the relaxing particle current is found from Fick's law: áj(x,t)ń = -D Ńán(x,t)ń, in which it is presumed that D is independent of space-time coordinates. In both these expressions one can always replace ánń by dn(x,t) ş án(x,t)ń-n0, where n0 is a final value; it is the density of the eventual equilibrium state and is given in principle by the above values N/V. Combination of the two yields the well-known diffusion equation
 ¶tdn(x,t)=D Ń2dn(x,t) ,      (108)
which is to be solved subject to knowing the initial value dn(x,t1); this value can be taken as that predicted by rt1.

The next step is to construct the relaxation density matrix rr at t=t1+e in terms of the densities án(x,t)ń and áh(x,t)ń in the simple fluid. Both ánń and áhń are now solutions of ddeterministicequations 8 and do not depend on any previous values except the last instant - only initial values are required. The counterpart of this in probability theory is a Markov process, which is exactly what rr will describe. The relaxation density matrix is then
 rr(t)= 1 Zr(t) exp éë - óő V b(x,t)h(x,t) d3x + óő V l(x,t)n(x,t) d3x ůű ,     (109)
and the Lagrange multipliers are now formally determined from
 áh(x,t)ń = - d db(x,t) lnZr(b,m) ,       án(x,t)ń = - d dl(x,t) lnZr(b,m) .      (110)
The positive sign in front of the second integral in (109) is taken because in the Boltzmann region the chemical potential is negative. Also, the momentum density could have been included here as well, but these two operators are sufficient for the present discussion.

The form of (109) requires further comment to avoid confusion. Contrary to the belief expressed in some works, it is not a sensible procedure to construct a density matrix from data taken at a single point in time and expect the result to adequately describe nonequilibrium processes. Not only is all history of what has been done to the system in the past disregarded, but no information is provided to r about if, why, or how the input data are actually varying in time; the resulting probability distribution cannot possibly have anything to say about time-varying quantities. But this is not what has been done in (109); rr can be constructed accurately at any instant for t > t1 because the input information is available at any instant as a solution to (108) and its counterpart for dh. As a consequence one can consider rr(t) to evolve deterministically as well.

When the source is removed the total system maximum entropy for t > t1 is the relaxation entropy
 1 k Sint(t)
 = lnZr(t)+ óő V b(x,t)áh(x,t)ńr d3x
 - óő V l(x,t)án(x,t)ńr d3x .
(111)
This is now the physical internal entropy of the system, for this is one of the few instances when the entropy of the system can be disentangled from St.

Formal solutions to (108) and the corresponding heat equation are well known; we write the generic solution as u(x,t) and express the initial value as u(x,0)=f(x). For a large volume, which we may as well take as all space, a short calculation via Fourier analysis yields two equivalent forms for the solution:
 u(x,t)
 = 1 (4pDt)3/2 óő f(x˘)e-(x-x˘)2/4Dt d3x˘
 = óő d3k (2p)3 eik·x f(k)e-Dk2t ,
(112)
both of which vanish in the limit t®Ą. The first form has the merit of demonstrating that it also vanishes as |x|®Ą, indicating that the densities smooth out at the distant boundaries at any time; in addition, it readily reproduces the initial condition as t® 0. The second form, however, is a bit more transparent and, depending on the spectral properties of f(k), reveals that the dominant wavelengths l will determine the relaxation time t ~ l2/D. Of course, t will be slightly different for dn and dh, but they should be of the same order of magnitude.

These results demonstrate that in the system after the source is removed at t=t1 the macroscopic densities evolve to constants:
 án(x,t)ńr
 ®   t®Ą n0=N/V ,
 áh(x,t)ńr
 ®   t®Ą h0=E/V .
(113)
In turn, this implies that in (111) those constants can be extracted from the integrals to yield the additional constants b ş ňb(x,Ąd3x, bm ş ňl(x,Ąd3x. 9 Because rr can be reconstructed at any time, it must be that h(x,t)® H/V, n(x,t)® N/V, as well, where H and N are the Hamiltonian and number operator, respectively. Our conclusion is that
 rr(t) ®   t®Ą req= 1 ZG e-b(H-mN) ,      (114)
and the relaxation entropy increases monotonically to the equilibrium value:
 1 k Sint(t) ®   t®Ą 1 k Seq=lnZG+báHń0 -bmáNń0 ,      (115)
where b and the chemical potential m are the equilibrium parameters of the grand canonical distribution. To see that this is actually an increase, integrate both quantities in (113) over all space, which provides upper bounds N and E for each integral, respectively. But for any two integrable functions f1, f2, with |f1| Ł C, an upper bound, it is a theorem that
 ęę óő b a f1(x)f2(x) dx ęę Ł C óő b a |f2(x)| dx .      (116)
Hence, Sint(t) in (111) can never be larger than Seq in (115), and we have demonstrated the monotonically increasing approach to equilibrium. Note that this represents a complete equilibrium in that the system retains no memory of anything done to it in the past; this memory loss obviously comes about because the evolution has been deterministic. We have only a theoretical prediction, of course, and only experiment can confirm that it actually is an equilibrium state. While (115) is in agreement with the Second Law, it is certainly not a statement of that law, for we started from a nonequilibrium state.

The crucial point in this demonstration must be emphasized. What makes the procedure possible is the ability to construct a density matrix at any time using the information available at that time. It is the context based on probability theory and the PME that allows introduction of rt1 and rr above, thereby providing a cohesive description of relaxation.

5. Some Final Comments

A major aim of this exposition has been to demonstrate the broad applicability of Gibbs' variational principle in governing all of statistical mechanics, once the notion of thermal driving by external sources is brought into the picture. But what may tend to get lost here is the fundamental role of probability in illuminating the way to a coherent theory, as was discused at length in I and II. When the entropy concept is realized to begin with a rule for constructing prior probability distributions its physical manifestation becomes much clearer, in that it is now seen, not as a property of the physical system, but of the macroscopic state of that system, or of a process that is occurring in it. The singular nature of the equilibrium state has obscured this feature heretofore because these different views coalesce there. This leads us to comment on a point often missed in phenomenological theories of nonequilibrium thermodynamics, where there is a tendency to consider entropy as just another field variable and posit the existence of an entropy density' as if it were a conserved dynamical variable. While macroscopic dynamical variables such as energy, particle number, angular momentum, etc., indeed emerge as average values of their microscopic counterparts, entropy is only a macroscopic quantity that connects the two domains through probability theory. Were it otherwise, the implication would be that when external sources are removed the total entropy is fixed as that of the nonequilibrium state at that time; as we have seen, that cannot be so. This doesn't mean that entropy is nonphysical when it stands still long enough to be measured in an equilibrium state, or when characterizing the rate of relaxation to that state, but it does mean that it is not a dynamical variable; it is, however, a functional of dynamical variables and changes in time only because they do. Boltzmann, Gibbs, and Planck understood this point long ago, but somehow early in the 20th century it seems to have gotten lost, and continues to be misconstrued.

With a straightforward demonstration of the approach to equilibrium in hand, it remains to address the question of irreversibility. In the model above the relaxation is driven by density gradients in the system, whose decays are described by the diffusion and heat equations; similar equations govern the relaxation in other models. Clearly these equations are not time-reversal invariant, and in equilibrium there are no gradients or external forces to drive the system in reverse - the state is stable under small fluctuations. From a macroscopic point of view, then, one can see why the relaxation process is irreversible, but this does nothing to explain why the microscopic equations of motion, which are invariant under time reversal, cannot conspire to bring the system back to the original nonequilibrium state it was in at t=t1. After all, Poincaré's recurrence theorem is certainly true and a many-body system left to itself will return to its initial microscopic state at some time in the future; that time turns out to be something like 101023 years, but the point remains.

It has long been understood, though apparently not widely appreciated, that the necessary microscopic initial conditions for the reversed microscopic motions have probability close to zero of being realized in an equilibrium system with N >> 1 degrees of freedom, which does fit in nicely with that huge recurrence time. The tool for proving this was provided long ago by Boltzmann in what may have been the first connection between entropy and information. In the form given it by Planck, the maximum entropy is written as SB=klnW, where W is the measure of a volume in phase space or of a manifold in Hilbert space; it measures the size of the set of N-particle microstates compatible with the macroscopic constraints on the system. As common sense would dictate, the greater the number of possibilities, the less certain we are of which microstate the system might occupy; conversely, more constraints narrow the choices and reduce that uncertainty. Subject to those constraints, a system will occupy the macrostate that can be realized in the greatest number of ways, the state of maximum entropy; barring external intervention, microscopic dynamics will keep it there. In addition, Boltzmann noted that there was nothing in this relation restricting it to equilibrium states.

Boltzmann's beautiful qualitative insight is readily quantified. Consider a system in a macrostate A1 with entropy S1=klnW1, where W1 is the size of the set of states C1 compatible with the constraints defining A1. Now expose it to a positive-source process that carries it into a new macrostate A2 with entropy S2=klnW2 and set of compatible states C2; by this is meant a source that adds thermal energy or particle number, or perhaps increases the volume. Although unnecessary, to keep the argument simple we consider these to be initial and final equilibrium states. If this is a reproducible experiment - in the sense that A1 can be reproduced, but certainly not any particular microstate in C1 - then it is surely necessary that W2 ł W1, which is already an elementary statement of the Second Law. But just how much larger is the set C2? From Boltzmann's expression the ratio of phase volumes can be written
 W1 W2 =exp ćč - S2-S1 k öř .      (117)
If the difference in entropies is merely a nanocalorie at room temperature this number is
 ~ exp ćč -1012 öř

, and the number of microstates compatible with A1 is vanishingly small compared with those compatible with A2. Therefore, to have any reasonable probability of being realized the initial microstates required to regain the macrostate A1 would have to be contained in the high-probability manifold C2, whose microstates are all compatible with the constraints defining the macrostate A2, and not A1; these requisite initial microstates must therefore lie in the complement of C2 and have very low probability. Moreover, this manifold of requisite initial microstates must have dimension about the same as that of C1, but from (117) this dimension is immeasurably smaller than that of C2, so that it is even less probable that these initial states could be realized. As understood so clearly by Gibbs and Boltzmann, the time-reversed evolution is not impossible, it's just extremely improbable. At this point it is difficult to see what more there is to say about the origin of thermodynamic irreversibility in macroscopic systems.

Appendix

The derivation of thermodynamics from probability theory cannot be considered complete without establishing the relationship of expectation values to physically measurable values of the associated macroscopic variables. To address this point, consider any time-dependent classical variable f(t), where it suffices to suppress any other independent variables in this discussion, and consider just the equilibrium system. Given any equilibrium probability distribution for f(t), the best prediction we can make for the variable, in the sense of minimum expected square of the error, is the expectation value
 áf(t)ń = áfń ,      (A.1)
independent of time. The reliability of this prediction is determined, as always, by the expected square of the deviation of f from the value (A.1), or the variance
 [Df(t)]2
 ş á(f(t)-áfń)2ń
 =áf2ń-áfń2 ,
(A.2)
again independent of time. Only if |Df/áfń| << 1 is the distribution making a sharp prediction, which is to expected for N >> 1 degrees of freedom.

Now (A.1) just reflects the value of f predicted by the probability distribution, and is not necessarily the same as the value actually measured for the single physical system being studied. Similarly, (A.2) is only a measure of how well the distribution is predicting that expectation value; it represents the statistical fluctuations, and may or may not correspond to possible fluctuations of the physical quantity. Certainly knowledge that the value of f is known only to ±1% does not imply that the physical value actually fluctuates by ±1%. This is a point stressed repeatedly by E.T. Jaynes in several different contexts, and we follow his arguments here (e.g., Jaynes, 1979). Nevertheless, the reality of physical fluctuations is not in doubt, as evidenced by the phenomena of Brownian motion, ccriticalopalescence, and spontaneous voltage fluctuations in resistors at constant temperature, so that we might expect a relationship between the two types.

To uncover possible connections of this kind we note that the value measured in the laboratory is not an expectation value, but a time average:
 f ş 1 T óő T 0 f(t) dt ,      (A.3)
where the averaging time T will be left unspecified for the moment. The best prediction we can make for this measured value, then, is
 á f ń = 1 T óő T 0 f(t) dt = 1 T óő T 0 áfń dt ,      (A.4)
or in equilibrium,
 á f ń = áfń .      (A.5)
This is a rather general rule of probability theory: an expectation value áfń is not equivalent to a time average
 f

, but it is equal to the expectation value of that average.

Thus it seems that the predictions of statistical mechanics are clearly related to measured physical values, if the prediction (A.5) is reliable. This again is determined by the variance,
 (D f )2= 1 T2 óő T 0 dt óő T 0 dt˘[áf(t)f(t˘)ń-áf(t)ńáf(t˘)ń] .      (A.6)
In equilibrium with time-independent Hamiltonian the expectation values depend only on t˘-t. With some judicious changes of variable the last expression can then be reduced to a single integral:
 (D f )2 = 1 T2 óő T 0 (T-t) Kff(t) dt ,     (A.7)
in terms of the covariance function
 Kff(t) ş áf(0)f(t)ń-áfń2 .      (A.8)
If the integrals in (A.7) converge as T®Ą, then the correlations will die off as 1/ÖT and hence
 D f ® 0

in the limit. When this is true we can assert with some confidence that the expected measurable value equals the expectation value; otherwise, there is no sharp relation between expectation values and time averages. Note, however, that there is no guarantee that the measured value will actually be the same as that in (A.5); only experiment can verify that.

In the same vein, we can ask how the measurable mean-square fluctuations are related to the statistical fluctuations. The time-average of the deviation from the measured mean is
 (df)2
 ş 1 T óő T 0 [f(t)- f ]2 dt
 = f2 - f 2 ,
(A.9)
and the expectation value is found to be
 á(df)2ń = (Df)2 -(D f )2 .      (A.10)
Hence, measurable fluctuations can be the same as the statistical fluctuations only if the distribution is such, and the averaging time so long, that
 |D f /Df| << 1

. Invariably this is presumed to be the case.

Reliability of the prediction (A.10) is determined, as usual, by the variance á(df)4ń-á(df)2ń2, which reduces to a 4-fold time average of a 4-point correlation function. Clearly, verification of any statement equating physical fluctuations with statistical fluctuations will involve some decidedly nontrivial calculations.

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Footnotes:

1These references will be denoted as I and II, respectively, in what follows, the corresponding equations then referred to as (I-n) and (II-n).

2In this subsection we denote the maximum-entropy function by S, without subscripts.

3This characterization of thermal driving was first introduced by Mitchell (1967).

4The energy density is usually what is driven; this will be considered presently when spatial variation is included.

5This model was presented years ago by Mitchell (1967) as an early example of mode-mode coupling.

6The limit w® 0 eliminates the 3-point-correlation term contributions that are regular in w.

7These quantities can be obtained by employing expressions such as (37) and (38), rather that adapting the steady-state scenario.

8This is not entirely accurate; while (108) is mathematically deterministic, it has a strong inferential component in Fick's law and its variables are (sharp) expectation values. A better choice might be `quasi-deterministic'.

9More directly, the Lagrange multiplier functions are evolving in parallel with the extensive variables and at the same rate.

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