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Entropy production fluctuation theorem and the nonequilibrium work relation for free

energy differences

Gavin E. Crooks∗

Department of Chemistry, University of California at Berkeley, Berkeley, CA 94720

(February 1, 2008)

There are only a very few known relations in statistical dynamics that are valid for systems drivenarbitrarily far-from-equilibrium. One of these is the fluctuation theorem, which places conditionson the entropy production probability distribution of nonequilibrium systems. Another recentlydiscovered far-from-equilibrium expression relates nonequilibrium measurements of the work doneon a system to equilibrium free energy differences. In this paper, we derive a generalized version ofthe fluctuation theorem for stochastic, microscopically reversible dynamics. Invoking this generalizedtheorem provides a succinct proof of the nonequilibrium work relation.

I. INTRODUCTION

Consider some finite classical system coupled to a con-stant temperature heat bath, and driven out of equilib-rium by some time-dependent work process. Most rela-tions of nonequilibrium statistical dynamics that are ap-plicable to systems of this sort are valid only in the linear,near-equilibrium regime. One group of exceptions is theentropy production fluctuation theorems [1–12], whichare valid for systems perturbed arbitrarily far away fromequilibrium. Although the type of system, range of appli-cability, and exact interpretation differ, these theoremshave the same general form,

P(+σ)

P(−σ)≃ eτσ. (1)

Here P(+σ) is the probability of observing an entropyproduction rate, σ, measured over a trajectory of time τ .Evans and Searles [2] gave a derivation for driven ther-mostated deterministic systems that are initially in equi-librium, Gallavotti and Cohen [3] rigorously derivedtheir fluctuation theorem for thermostated determinis-tic steady state ensembles, and Kurchan [9], Lebowitzand Spohn [11], and Maes [12] have considered systemswith stochastic dynamics. The exact interrelation be-tween these results is currently under debate [13,14].

In this paper we will derive the following, somewhatgeneralized, version of this theorem for stochastic micro-scopically reversible dynamics:

PF(+ω)

PR(−ω)= e+ω. (2)

Here ω is the entropy production of the driven systemmeasured over some time interval, PF(ω) is the proba-bility distribution of this entropy production, and PR(ω)

∗Electronic address: gavinc@garnet.berkeley.edu

is the probability distribution of the entropy productionwhen the system is driven in a time-reversed manner.This distinction is necessary because we will consider sys-tems driven by a time-dependent process, rather than thesteady perturbation considered elsewhere. The use of anentropy production, rather than an entropy productionrate, will prove convenient.

As a concrete example of a system for which the abovetheorem is valid, consider a classical gas confined in acylinder by a movable piston. The walls of this cylin-der are diathermal so that the gas is in thermal contactwith the surroundings, which therefore act as the con-stant temperature heat bath. The gas is initially in equi-librium with a fixed piston. The piston is then movedinwards at a uniform rate, compressing the gas to somenew, smaller volume. In the corresponding time-reversedprocess, the gas starts in equilibrium at the final volumeof the forward process, and is then expanded back to theoriginal volume at the same rate that it was compressedby the forward process. The microscopic dynamics of thesystem will differ for each repetition of this process, aswill the entropy production, the heat transfer, and thework performed on the system. The probability distri-bution of the entropy production is measured over theensemble of repetitions.

Another expression that is valid in the far-from-equilibrium regime is the recently discovered relationshipbetween the difference in free energies of two equilibriumensembles, ∆F , and the amount of work, W , expended inswitching between ensembles in a finite amount of time[15–19],

〈e−βW 〉 = e−β∆F . (3)

Here β = 1/kBT , kB is the Boltzmann constant, T is thetemperature of the heat bath that is coupled to the sys-tem, and 〈· · ·〉 indicates an average over many repetitionsof the switching process. For the confined gas consideredabove, the free energy depends on the position of thepiston. With Eq. (3), we can calculate the free energychange when the system is compressed to a new volume

1

by making many measurements of the work required toeffect the change, starting each time from an equilibratedsystem, and taking the above average. In the limit of in-stantaneous switching between ensembles, this relationis equivalent to the standard thermodynamic perturba-tion method used to calculate free energy differences withcomputer simulations [20–23].

Equations (2) and (3) are actually closely related. Wefirst note that whenever Eq. (2) is valid, the followinguseful relation holds [8, Eq. (16)]:

〈e−ω〉 =

∫ +∞

−∞

PF(+ω)e−ωdω =

∫ +∞

−∞

PR(−ω)dω

= 1. (4)

We shall show in Sec. III that the generalized fluctuationtheorem, Eq. (2), can be applied to systems that startin equilibrium, and that the entropy production ω forsuch systems is −β∆F + βW . The nonequilibrium workrelation for the free energy change, Eq. (3), can thusbe derived by substituting this definition of the entropyproduction into Eq. (4), and noting that the free energydifference is a state function, and can be moved outsidethe average.

In the following section we will derive the fluctuationtheorem, Eq. (2). Then in Sec. III we will discuss twodistinct groups of driven systems for which the theoremis valid. In the first group, systems start in equilibrium,and are then actively perturbed away from equilibriumfor a finite amount of time. In the second group, systemsare driven into a time symmetric nonequilibrium steadystate. We conclude by discussing several approximationsthat are valid when the entropy production is measuredover long time periods. The fluctuation theorem and itsapproximations are illustrated with data from a simplecomputer model.

II. THE FLUCTUATION THEOREM

As indicated in the Introduction, we consider finite,classical systems coupled to a heat bath of constant tem-perature, T = 1/β. (All entropies are measured in nats[24], so that Boltzmann’s constant is unity.) The stateof the system is specified by x and λ, where x repre-sents all the dynamical, uncontrolled degrees of freedom,and λ is a controlled, time-dependent parameter. Forthe confined gas considered in the Introduction, the heatbath is simply the walls of the cylinder, the state vectorx specifies the positions and momenta of all the parti-cles, and λ specifies the current position of the piston.In computer simulations, the controlled parameter couldbe a microscopic degree of freedom. For example, in afree energy calculation λ could specify the distance be-tween two particles or the chemical identity of an atomor molecule [20–23].

A particular path through phase space will be speci-fied by the pair of functions

(

x(t), λ(t))

. It will prove

convenient to shift the time origin so that the paths un-der consideration extend an equal time on either sideof that origin, t ∈ {−τ, +τ} or t ∈ {−∞, +∞}. Thenthe corresponding time reversed path can be denotedas

(

x(−t), λ(−t))

. The overbar indicates that quanti-ties odd under a time reversal (such as momenta or anexternal magnetic field) have also changed sign.

The dynamics of the system are required to be stochas-tic and Markovian [25]. The fluctuation theorem wasoriginally derived for thermostated, reversible, determin-istic systems [2,3]. However, there are fewer technicaldifficulties if we simply assume stochastic dynamics [9].We will also require that the dynamics satisfy the follow-ing microscopically reversible condition:

P [x(+t)|λ(+t)]

P [x(−t)|λ(−t)]= exp

{

− βQ[x(+t), λ(+t)]}

. (5)

Here P [x(+t)|λ(+t)] is the probability, given λ(t),of following the path x(t) through phase space, andP [x(−t), |λ(−t)] is the probability of the correspondingtime-reversed path. Q is the heat, the amount of energytransferred to the system from the bath. The heat is afunctional of the path, and odd under a time reversal,i.e., Q[x(t), λ(t)] = −Q[x(−t), λ(−t)].

In current usage, the terms “microscopically re-versible” and “detailed balance” are often used inter-changeably [26]. However, the original meaning of micro-scopic reversibility [27,28] is similar to Eq. (5). It relatesthe probability of a particular path to its reverse. Thisis distinct from the principle of detailed balance [26,21],which refers to the probabilities of changing states with-out reference to a particular path. It is the conditionthat P(A −→ B) = P(B −→ A) exp(−β∆E), where ∆Eis the difference in energy between state A and state B,and P (A −→ B) is the probability of moving from stateA to state B during some finite time interval.

The stochastic dynamics that are typically used tomodel reversible physical systems coupled to a heatbath, such as the Langevin equation and MetropolisMonte Carlo, are microscopically reversible in the senseof Eq. (5). Generally if the dynamics of a system obeydetailed balanced locally in time (i.e. each time step isdetailed balanced) then the system is microscopically re-versible even if the system is driven from equilibrium byan external perturbation [19, Eq (9)].

A particular work process is defined by the phase-space distribution at time −τ , ρ(x-τ ), and the value ofthe control parameter as a function of time, λ(t). Eachindividual realization of this process is characterized bythe path that the system follows through phase space,x(t). The entropy production, ω, must be a functionalof this path. Clearly there is a change in entropy dueto the exchange of energy with the bath. If Q is theamount of energy that flows out of the bath and intothe system, then the entropy of the bath must changeby −βQ. There is also a change in entropy associatedwith the change in the microscopic state of the system.

2

From an information theoretic [24] perspective, the en-tropy of a microscopic state of a system, s(x) = − lnρ(x),is the amount of information required to describe thatstate given that the state occurs with probability ρ(x).The entropy of this (possibly nonequilibrium) ensemble

of systems is S = −∑

x ρ(x) ln ρ(x). Thus, for a singlerealization of a process that takes some initial probabil-ity distribution, ρ(x-τ ), and changes it to some differentfinal distribution, ρ(x+τ ), the entropy production [29] is

ω = ln ρ(x-τ ) − ln ρ(x+τ ) − βQ[x(t), λ(t)]. (6)

This is the change in the amount of information requiredto describe the microscopic state of the system plus thechange in entropy of the bath.

Recall that the fluctuation theorem, Eq. (2), comparesthe entropy production probability distribution of a pro-cess with the entropy production distribution of the cor-responding time-reversed process. For example, with theconfined gas we compare the entropy production whenthe gas is compressed to the entropy production whenthe gas is expanded. To allow this comparison of forwardand reverse processes we will require that the entropyproduction is odd under a time reversal, i.e., ωF = −ωR,for the process under consideration. This condition isequivalent to requiring that the final distribution of theforward process, ρ

F(x+τ ), is the same (after a time rever-

sal) as the initial phase-space distribution of the reverseprocess, ρ

R(x+τ ), and vice versa, i.e., ρ

F(x+τ ) = ρ

R(x+τ )

and ρR(x-τ ) = ρ

F(x-τ ). In the next section, we will dis-

cuss two broad types of work process that fulfill this con-dition. Either the system begins and ends in equilibriumor the system begins and ends in the same time symmet-ric nonequilibrium steady state.

This time-reversal symmetry of the entropy productionallows the comparison of the probability of a particularpath, x(t), starting from some specific point in the initialdistribution, with the corresponding time-reversed path,

ρF(x-τ )P [x(+t)|λ(+t)]

ρR(x+τ )P [x(−t)|λ(−t)]

= e+ωF . (7)

This follows from the the conditions that the system ismicroscopically reversible, Eq. (5) and that the entropyproduction is odd under a time reversal.

Now consider the probability, PF(ω), of observing aparticular value of this entropy production. It can bewritten as a δ function averaged over the ensemble offorward paths,

PF(ω)= 〈δ(ω−ωF)〉F

=

∫∫∫ x+τ

x-τρ

F(x-τ )P [x(+t)|λ(+t)]δ(ω−ωF)D[x(t)]dx-τdx+τ .

Here∫∫∫ x+τ

x-τ· · · D[x(t)]dx-τdx+τ indicates a sum or suitable

normalized integral over all paths through phase space,and all initial and final phase space points, over the ap-propriate time interval. We can now use Eq. (7) to con-vert this average over forward paths into an average overreverse paths.

PF(ω)=

∫∫∫ x+τ

x-τρ

R(x+τ )P [x(−t)|λ(−t)] ×

δ(ω−ωF)e+ωF D[x(t)]dx-τdx+τ

= e+ω

∫∫∫ x+τ

x-τρ

R(x+τ )P [x(−t)|λ(−t)] ×

δ(ω+ωR)D[x(t)]dx-τdx+τ

= e+ω 〈δ(ω+ωR)〉R

= e+ω PR(−ω).

The δ function allows the e+ω term to be moved out-side the integral in the second line above. The remainingaverage is over reverse paths as the system is driven inreverse. The final result is the entropy production fluc-tuation theorem, Eq. (2).

The theorem readily generalizes to other ensembles. Asan example, consider an isothermal-isobaric system. Inaddition to the heat bath, the system is coupled to a vol-ume bath, characterized by βp, where p is the pressure.Then the microscopically reversible condition, Eq. (5),becomes

P [x(+t)|λ(+t)]

P [x(−t)|λ(−t)]= exp

{

− βQ[x(t), λ(t)]

−βp∆V [x(t), λ(t)]}

.

Both baths are considered to be large, equilibrium, ther-modynamic systems. Therefore, the change in entropyof the heat bath is −βQ and the change in entropy ofthe volume bath is −βp∆V , where ∆V is the change involume of the system. The entropy production shouldthen be defined as

ω = ln ρ(x-τ ) − ln ρ(x+τ ) − βQ − βp∆V. (8)

The fluctuation theorem, Eq. (2), follows as before. It ispossible to extend the fluctuation theorem to any stan-dard set of baths, so long as the definitions of microscopicreversibility and the entropy production are consistent.In the rest of this paper we shall only explicitly deal withsystems coupled to a single heat bath, but the resultsgeneralize directly.

III. TWO GROUPS OF APPLICABLE SYSTEMS

In this section we will discuss two groups of systemsfor which the entropy fluctuation theorem, Eq. (2), isvalid. These systems must satisfy the condition that theentropy production, Eq. (6), is odd under a time reversal,and therefore that ρ

F(x+τ ) = ρ

R(x+τ ).

First consider a system that is in equilibrium from timet = −∞ to t = −τ . It is then driven from equilibrium bya change in the controlled parameter, λ. The system isactively perturbed up to a time t = +τ , and is then al-lowed to relax, so that it once again reaches equilibriumat t = +∞. For the forward process the system starts inthe equilibrium ensemble specified by λ(−∞), and ends

3

0

5

10

15

1 32x

E(x)

0

0.05

0.1

P(x)

FIG. 1. A very simple Metropolis Monte Carlo simulationis used to illustrate the fluctuation theorem and some of itsapproximations. The master equation for this system can besolved, providing exact numerical results to compare with thetheory. A single particle occupies a finite number of posi-tions in a one-dimensional box with periodic boundaries, andis coupled to a heat bath of temperature T = 5. The energysurface, E(x), is indicated by in the figure. At each discretetime step the particle attempts to move left, right, or stay putwith equal probability. The move is accepted with the stan-dard Metropolis acceptance probability [34]. Every eight timesteps, the energy surface moves right one position. Thus thesystem is driven away from equilibrium, and eventually set-tles into a time symmetric nonequilibrium steady-state. Theequilibrium (◦) and steady state (•) probability distributionsare shown in the figure above. The steady state distributionis shown in the reference frame of the surface.

in the ensemble specified by λ(+∞). In the reverse pro-cess, the initial and final ensembles are exchanged, andthe entropy production is odd under this time reversal.The gas confined in the diathermal cylinder satisfies theseconditions if the piston moves only for a finite amount oftime.

At first it may appear disadvantageous that the en-tropy production has been defined between equilibriumensembles separated by an infinite amount of time. How-ever, for these systems the entropy production has a sim-ple and direct interpretation. The probability distribu-tions of the initial and final ensembles are known fromequilibrium statistical mechanics:

ρeq(x|λ) =e−βE(x,λ)

∑

x

e−βE(x,λ)

= exp{

βF (β, λ) − βE(x, λ)}

. (9)

The sum is over all states of the system, and F (β, λ) =−β−1 ln

∑

x exp{−βE(x, λ)} is the Helmholtz free energyof the system. If these probabilities are substituted intothe definition of the entropy production, Eq. (6), then wefind that

ωF = −β∆F + βW. (10)

Here W is the work and we have used the first law ofthermodynamics, ∆E = W + Q.

0

0.02

0.04

0.06

-8 0 8 16 24βW

∆t = 512

∆t = 256

∆t = 128

∆t = 1024

∆t = 2048

P(βW )

FIG. 2. Work probability distribution (—) for the systemof Fig. 1, starting from equilibrium. The work W was mea-sured over 16, 32, 64, 128, and 256 cycles (∆t = 128, 256, 512,1024, and 2048). For each of these distributions the work fluc-tuation theorem, Eq. (11), is exact. The dashed lines (- - -)are Gaussians fitted to the means of the distributions for ∆t =256 and 1024. (See Sec. IV).

It is therefore possible to express the fluctuation the-orem in terms of the amount of work performed on asystem that starts in equilibrium [30],

PF(+βW )

PR(−βW )= e−∆F e+βW . (11)

The work in this expression is measured over the finitetime that the system is actively perturbed. We have nowestablished the fluctuation theorem for systems that startin equilibrium, and have shown that the entropy produc-tion is simply related to the work and free energy change.Therefore, we have established the nonequilibrium workrelation, Eq. (3), discussed in the Introduction.

The validity of this expression can be illustrated withthe very simple computer model described in Fig. 1. Al-though not of immediate physical relevance, this modelhas the advantage that the entropy production distribu-tions of this driven nonequilibrium system can be cal-culated exactly, apart from numerical roundoff error.The resulting work distributions are shown in Fig. 2.Because the process is time symmetric, ∆F = 0 andP(+βW ) = P(−βW ) expβW . This expression is exactfor each of the distributions shown, even for short timeswhen the distributions display very erratic behavior.

Systems that start and end in equilibrium are not theonly ones that satisfy the fluctuation theorem. Consideragain the classical gas confined in a diathermal cylin-der. If the piston is driven in a time symmetric periodicmanner (for example, the displacement of the piston isa sinusoidal function of time), the system will eventuallysettle into a nonequilibrium steady-state ensemble. Wewill now require that the dynamics are entirely diffusive,so that there are no momenta. Then at any time thatλ(t) is time symmetric, then the entire system is invari-ant to a time-reversal. We start from the appropriatenonequilibrium steady-state, at a time symmetric pointof λ(t), and propagate forward in time a whole number of

4

0

0.02

0.04

0.06

-8 0 8 16 24

∆t = 512

∆t = 256

∆t = 128

∆t = 1024

∆t = 2048

−βQ

P(−βQ)

FIG. 3. Heat probability distribution (—) for the nonequi-librium steady state. The model described in Fig. 1 was re-laxed to the steady state, and the heat Q was then measuredover 16, 32, 64, 128 and 256 cycles (∆t = 128, 256, 512, 1024,and 2048). Note that for long times the system forgets itsinitial state and the heat distribution is almost indistinguish-able from the work distribution of the system that starts inequilibrium.

cycles. The corresponding time reversed process is thenidentical to the forward process, with both starting andending in the same steady state ensemble. The entropyproduction for this system is odd under a time reversaland the fluctuation theorem is valid.

As a second example, consider a fluid under a con-stant shear [1]. The fluid is contained between parallelwalls which move relative to one another, so that λ(t)represents the displacement of the walls. Eventually thesystem settles into a nonequilibrium steady state. A timereversal of this steady-state ensemble will reverse all thevelocities, including the velocity of the walls. The result-ing ensemble is related to the original one by a simple re-flection, and is therefore effectively invariant to the timereversal. Again, the forward process is identical to thereverse process, and the entropy production is odd undera time reversal.

In general, consider a system driven by a time symmet-ric, periodic process. [λ(t) is a periodic, even functionof time.] We require that the resulting nonequilibriumsteady-state ensemble be invariant under time reversal,assuming that we pick a time about which λ(t) is alsosymmetric. This symmetry ensures that the the forwardand reverse process are essentially indistinguishable, andtherefore that the entropy production is odd under a timereversal. It is no longer necessary to explicitly label for-ward and reverse processes. PF(ω) = PR(ω) = P(ω). Forthese time symmetric steady-state ensembles the fluctu-ation theorem is valid for any integer number of cycles,and can be expressed as

P(+ω)

P(−ω)= e+ω. (12)

For a system under a constant perturbation (such as thesheared fluid) this relation is valid for any finite timeinterval.

0.85

1.00

1.15

0 8 16 24 32

r

∆t = 512

∆t = 256

∆t = 128

∆t = 1024

∆t = 2048

βQ

FIG. 4. Deviations from the heat fluctuation theo-rem, Eq. (13), for the distributions of Fig. 3. Ifthe heat fluctuation theorem were exact, then the ratior = βQ/ ln[P(+βQ)/P(−βQ)] would equal 1 for all |βQ|. Forshort times (∆t ≤ 256) the fluctuation theorem is wholly in-accurate. For times significantly longer than the relaxationtime of the system (∆t ≈ 100), the fluctuation theorem isaccurate except for very large values of |βQ|.

IV. LONG TIME APPROXIMATIONS

The steady-state fluctuation theorem, Eq. (12), is for-mally exact for any integer number of cycles, but is of lit-tle practical use because, unlike the equilibrium case, wehave no independent method for calculating the probabil-ity of a state in a nonequilibrium ensemble. The entropyproduction is not an easily measured quantity. However,we can make a useful approximation for these nonequilib-rium systems which is valid when the entropy productionis measured over long time intervals.

From the relation 〈exp(−ω)〉 = 1, Eq. (4), and theinequality 〈exp x〉 ≥ exp〈x〉, which follows from the con-vexity of ex, we can conclude that 〈ω〉 ≥ 0. On av-erage the entropy production is positive. Because thesystem begins and ends in the same probability distri-bution, the average entropy production depends onlyon the average amount of heat transferred to the bath.−〈ω〉 = 〈βQ〉 ≤ 0. On average, over each cycle, energyis transferred through the system and into the heat bath(Clausius inequality). The total heat transferred tends toincrease with each successive cycle. When measurementsare made over many cycles, the entropy production willbe dominated by this heat transfer and ω ≈ −βQ. There-fore, in the long time limit the steady state fluctuationtheorem, Eq. (12), can be approximated as

lim∆t→∞

P(+βQ)

P(−βQ)= exp(βQ). (13)

Because −βQ is the change in entropy of the bath, thisheat fluctuation theorem simply ignores the relativelysmall, and difficult to measure microscopic entropy ofthe system.

Heat distributions for the simple computer model of anonequilibrium steady state are shown in Fig. 3, and the

5

-20

-15

-10

-5

0

-32 -16 0 16 32

log10 P(βW )

βW

FIG. 5. Work distribution (—) and Gaussian approxima-tion (- - -) for ∆t =2048, with the probabilities plotted on alogarithmic scale. The Gaussian is fitted to the mean of thedistribution and has a variance half the mean (see Sec. IV).This Gaussian approximation is very accurate, even to thevery wings of the distribution, for times much longer thanthe relaxation time (∆t ≈ 100) of the system. This samedistribution is shown on a linear scale in Fig. 2.

validity of the above approximation is shown in Fig. 4.As expected, the heat fluctuation theorem, Eq. (13), isaccurate for times much longer than the relaxation timeof the system.

Another approximation to the entropy productionprobability distribution can be made in this long timelimit. For long times the entropy production ω is thesum of many weakly correlated values and its distribu-tion should be approximately Gaussian by the centrallimit theorem. If the driving process is time symmet-ric, then PF(ω) = PR(ω) = P(ω), and the entropy pro-duction distribution is further constrained by the fluc-tuation theorem itself. The only Gaussians that satisfythe fluctuation theorem have a variance twice the mean,2〈ω〉 =

⟨

(ω − 〈ω〉)2⟩

. This is a version of the standardfluctuation-dissipation relation [31]. The mean entropyproduction (dissipation) is related to the fluctuations inthe entropy production. If these distributions are Gaus-sian, then the fluctuation theorem implies the Green-Kubo relations for transport coefficients [1,32,33]. How-ever, we have not used the standard assumption that thesystem is close to equilibrium. Instead, we have assumedthat the system is driven by a time symmetric process,and that the entropy production is measured over a longtime period.

Gaussian approximations are shown in Figs. 2 and 5for the work distribution of the simple computer model.For times much longer than the relaxation time of thesystem these approximations are very accurate, even inthe wings of the distributions. This is presumably dueto the symmetry imposed by the fluctuation theorem.For a nonsymmetric driving process, this symmetry isbroken, and the distributions will not necessarily satisfythe fluctuation-dissipation relation in the long time limit.For example, see Fig. 8 of Ref. [16]. Clearly these dis-tributions will be poorly approximated by the Gaussiandistributions considered here.

V. SUMMARY

The fluctuation theorem, Eq. (2), appears to be verygeneral. In this paper we have derived a version thatis exact for finite time intervals, and which depends onthe following assumptions; the system is finite and clas-sical, and coupled to a set of baths, each characterizedby a constant intensive parameter. The dynamics are re-quired to be stochastic, Markovian, and microscopicallyreversible, Eq. (5), and the entropy production, definedby Eq. (6), must be odd under a time reversal. Thisfinal condition is shown to hold for two broad classesof systems, those that start in equilibrium and those ina time symmetric nonequilibrium steady state. For thelatter systems the fluctuation theorem holds for entropyproductions measured over any integer number of cycles.This generality, and the existence of the nonequilibriumwork relation discussed in the Introduction, suggests thatother nontrivial consequences of the fluctuation theoremare awaiting study.

ACKNOWLEDGMENTS

This work was initiated with support from the NationalScience Foundation, under grant No. CHE-9508336, andcompleted with support from the U.S. Department of En-ergy under contract No. DE-AC03-76SF00098. I am in-debted to C. Jarzynski and C. Dellago for their helpfuldiscussions, and D. Chandler for his advice and guidance.

[1] D. J. Evans, E. G. D. Cohen, and G. P. Morriss, Phys.Rev. Lett. 71, 2401 (1993).

[2] D. Evans and D. Searles, Phys. Rev. E 50, 1645 (1994).[3] G. Gallavotti and E. G. D. Cohen, Phys. Rev. Lett. 74,

2694 (1995).[4] G. Gallavotti and E. G. D. Cohen, J. Stat. Phys. 80, 931

(1995).[5] D. Evans and D. Searles, Phys. Rev. E 53, 5808 (1996).[6] G. Gallavotti, J. Stat. Phys. 84, 899 (1996).[7] E. G. D. Cohen, Physica A 240, 43 (1997).[8] G. Gallavotti, Chaos 8, 384 (1998).[9] J. Kurchan, J. Phys. A 31, 3719 (1998).

[10] D. Ruelle, J. Stat. Phys. 95, 393 (1999), e-print:mp arc/98-770.

[11] J. L. Lebowitz and H. Spohn, J. Stat. Phys. 95, 333(1999), e-print: cond-mat/9811220

[12] C. Maes, J. Stat. Phys. 95, 367 (1999), mp arc/98-754[13] E. Cohen and G. Gallavotti, e-print: cond-mat/9903418.[14] G. Ayton and D. Evans, e-print: cond-mat/9903409.[15] C. Jarzynski, Phys. Rev. Lett. 78, 2690 (1997).[16] C. Jarzynski, Phys. Rev. E 56, 5018 (1997).[17] C. Jarzynski, Acta Phys. Pol. B 29, 1609 (1998).

6

[18] C. Jarzynski, e-print: cond-mat/9802249[19] G. E. Crooks, J. Stat. Phys. 90, 1481 (1998).[20] C. H. Bennett, J. Comput. Phys. 22, 245 (1976).[21] D. Chandler, Introduction to Modern Statistical Mechan-

ics (Oxford University Press, New York, 1987).[22] T. P. Straatsma and J. A. McCammon, Annu. Rev. Phys.

Chem. 43, 407 (1992).[23] D. Frenkel and B. Smit, Understanding Molecular Sim-

ulation: From Algorithms to Applications (AcademicPress, San Diego, 1996).

[24] C. E. Shannon, Bell Syst. Tech. J. 27, 379 (1948).[25] G. R. Grimmett and D. R. Stirzaker, Probability and

Random Processes, 2nd ed. (Clarendon Press, Oxford,1992).

[26] S. R. de Groot and P. Mazur, Non-equilibrium Thermo-

dynamics (North-Holland, Amsterdam, 1962).[27] R. C. Tolman, Phys. Rev. 23, 693 (1924).[28] R. C. Tolman, The Principles of Statistical Mechanics

(Oxford University Press, London, 1938), pp. 163 and165.

[29] This definition of the entropy production is equivalent tothe quantity Σ used by Jarzynski [16].

[30] C. Jarzynski (private communication).[31] H. B. Callen and T. A. Welton, Phys. Rev. 83, 34 (1951).[32] D. Evans and D. Searles, Phys. Rev. E 52, 5839 (1995).[33] D. J. Searles and D. J. Evans, e-print: cond-

mat/9902021.[34] N. Metropolis et al., J. Chem. Phys. 21, 1087 (1953).

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