Temporal asymmetry of fluctuations in nonequilibrium steady states: Links withcorrelation functions and nonlinear responseCarlo Paneni, Debra J. Searles, and Lamberto Rondoni Citation: The Journal of Chemical Physics 128, 164515 (2008); doi: 10.1063/1.2894471 View online: http://dx.doi.org/10.1063/1.2894471 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/128/16?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Random paths and current fluctuations in nonequilibrium statistical mechanics J. Math. Phys. 55, 075208 (2014); 10.1063/1.4881534 Nonequilibrium multiscale computational model J. Chem. Phys. 126, 124105 (2007); 10.1063/1.2711432 Entropy, thermostats, and chaotic hypothesis Chaos 16, 043114 (2006); 10.1063/1.2372713 Temporal asymmetry of fluctuations in nonequilibrium steady states J. Chem. Phys. 124, 114109 (2006); 10.1063/1.2171964 (Global and local) fluctuations of phase space contraction in deterministic stationary nonequilibrium Chaos 8, 823 (1998); 10.1063/1.166369

Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 130.102.82.118 On: Thu, 01 Sep

2016 03:53:28

Temporal asymmetry of fluctuations in nonequilibrium steady states:Links with correlation functions and nonlinear response

Carlo Paneni,1,a� Debra J. Searles,1,b� and Lamberto Rondoni2,c�

1Nanoscale Science and Technology Centre, School of Biomolecular and Physical Sciences,Griffith University, Brisbane Qld 4111, Australia2Dipartimento di Matematica and INFM, Politenico di Torino, Corso Duca degli Abruzzi 24,10129 Torino, Italy

�Received 28 August 2007; accepted 14 February 2008; published online 25 April 2008�

The presence of temporal asymmetries in fluctuation paths of nonequilibrium systems has recentlybeen confirmed numerically in nonequilibrium molecular dynamics simulations of particulardeterministic systems. Here we show that this is a common feature of homogeneously driven andthermostatted, reversible, deterministic, chaotic, nonequilibrium systems of interacting particles.This is done by expressing fluctuation paths as correlation functions. The theoretical arguments lookrather general, and we expect them to easily extend to other forms of driving and thermostats. Theemergence of asymmetry is also justified using the transient time correlation function expression ofnonlinear response theory. Numerical simulations are used to verify our arguments.© 2008 American Institute of Physics. �DOI: 10.1063/1.2894471�

I. INTRODUCTION

A key issue in the study of nonequilibrium systems is theemergence of thermodynamic irreversibility from reversibleequations of motion. Related to this issue is the question ofthe occurrence of temporally asymmetric behavior in excep-tional fluctuations in nonequilibrium systems. Recently it hasbeen shown that the usual stochastic models for nonequilib-rium thermodynamic systems, such as Langevin dynamics,indicate a break of symmetry in fluctuations of thermody-namic properties in nonequilibrium states,1,2 and those stud-ies have inspired the work presented in this paper. The theo-ries extend the Onsager–Machlup fluctuation theory3 to thenonequilibrium and nonlinear domain. Concisely, they leadto the introduction of a term in the “adjoint” hydrodynamicequations, the equations describing the fluctuations awayfrom the nonequilibrium steady states, that has no counter-part in the usual hydrodynamic equation. That is, for a sys-tem where the equations describing the relaxation toward thenonequilibrium steady states are given by

�t� = D��� , �1�

the adjoint hydrodynamic equations are given by

�t� = D*��� , �2�

where D* is, in general, nonlocal and different from −D. Thetheories apply to stochastic lattice gases that admit a hydro-dynamic description, but are supposed to hold more gener-ally, under some assumptions,1,2 if the mesoscopic evolutionis given by a Markov process. An alternative approach hasbeen to directly monitor the response and relaxation of asystem due to a change in the applied field, and a recent

investigation in that regard was made by Gaspard.4

Temporal asymmetries have also been experimentallyobserved on stochastically perturbed electrical circuits analo-gous to a form of Brownian motion in a force field withweak white noise.5,6 It is therefore clear that stochasticmodels show asymmetric behavior in the fluctuations of theirproperties, and the following question arises: Would asym-metry in fluctuations of dynamical properties occur in theusual molecular-level models of thermodynamic systems,which are reversible and deterministic? In the studies on theperturbed electrical circuits, the symmetry of typical largefluctuations in steady state systems was examined.Previously7,8 we have used a similar approach to verify thatreversible, deterministic systems exhibit temporal asymme-tries for two reversible and deterministic nonequilibriumsimulated systems. We therefore confirmed that the asymme-try is not an artifact of the stochastic representation. This wasalso observed in Refs. 9 and 10.

In our earlier work on homogeneous Couette flow, it wasnecessary to use time dependent periodic boundary condi-tions. This introduces periodic fluctuations to the properties,and their influence needs to be carefully removed.7 Further-more the geometry of this system might be thought to intro-duce artificial asymmetry. For these reasons we decided toconsider a somewhat simpler system: That which involvesthe flow of colored particles due to a color field.

In previous work, we limited ourselves to numerical ob-servation of asymmetry in carefully defined “typical” fluc-tuation paths �FPs� for a number of observables of N-particlesystems, but we gave no theoretical analysis. Here we intro-duce a new definition of FPs that allows us to confirm theubiquity of the results obtained. Using this we justify thepresence of asymmetry by reformulating the FPs as correla-tion functions and also using nonlinear response theory: Wetherefore give a first theoretical explanation of this phenom-

a�Electronic mail: cpaneni@gmail.com.b�Electronic mail: d.bernhardt@griffith.edu.on.c�Electronic mail: lamberto.rondoni@polito.it.

THE JOURNAL OF CHEMICAL PHYSICS 128, 164515 �2008�

0021-9606/2008/128�16�/164515/11/$23.00 © 2008 American Institute of Physics128, 164515-1

Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 130.102.82.118 On: Thu, 01 Sep

2016 03:53:28

enon for deterministic and reversible systems. To substanti-ate our analysis, we use nonequilibrium molecular dynamicssimulations.

As well as considering temporal asymmetries in typicalFPs, as was done previously,7–9 in this work we also look atasymmetry in general time correlation functions, i.e., we ex-amine if Onsager-type relations, �A�t�B�t−s��− �A�t�B�t+s��,become nonzero out of equilibrium for these systems.11–15

We provide some general arguments and present numericalresults.

II. SIMULATED SYSTEM AND DEFINITIONOF FLUCTUATION PATHS

A. System

The purpose of the paper is to understand how asymmet-ric fluctuations can arise in homogeneously driven and ther-mostatted nonequilibrium systems composed of interactingparticles whose positions evolve according to deterministic,reversible dynamics. The theoretical treatment applies to thisclass of nonequilibrium particle systems, and we expect it toextend to more general nonequilibrium systems. For the sakeof illustration, in the numerical work we choose a many-particle system where the particles interact via the Weeks–Chandler–Andersen �WCA� potential,16

VWCA�rij� = �VLJ�rij� + � if rij � rmin

0 if rij � rmin,� �3�

where VLJ�rij� is the two-body Lennard–Jones potential,

VLJ�rij� = 4� �

rij12

− �

rij6 , �4�

� and � are specified parameters, rij is the separation be-tween each pair, rmin=21/6�, and �=−VLJ�rmin�. The WCApotential is therefore the Lennard–Jones potential truncatedat the position of minimum potential energy �rij =21/6�� andthen shifted up so that the potential and its derivative arezero at the cutoff radius.17 In order to efficiently represent amicroscopic part of a macroscopic system without incurringlarge boundary effects, we use periodic boundary conditions.Since our attention is toward phenomena that take place in anonequilibrium steady state, we chose a simple and wellknown system: That in which color diffusion occurs. Thismodels the flow of particles of opposite “color” charge. Thatis, they experience a force due to an external field that de-pends on their color, but the pair interaction potential is in-dependent of their color. The field does work on the systemso, to prevent it from heating and enable a steady state to bereached, a reversible thermostat is applied.18 The system issimilar to a thermostatted system of charged particles towhich a field is applied, except that the particles are notinfluenced by the charges on other particles. The system’sunthermostatted dynamics are driven by the colorHamiltonian,18,19

H = H0 − �i=1

N

ciqxiFc, �5�

where qxi is the x-direction component of the laboratory co-ordinates particle i, H0 is the unperturbed Hamiltonian orinternal energy of the system, ci is the color charge of par-ticle i �we select an equal number of positive and negativecolor charges and ci= �−1�i�, and Fc is the color field, whichis applied in the x-direction in this case. The equations ofmotion that drive the trajectories of the particles are the fol-lowing:

qi =pi

mi, �6�

pi = Fi + ciFcnx − ��pi − P� , �7�

� =� �pi − P� · �Fi + ciFci�

� �pi − P� · pi, �8�

where qi is the laboratory coordinate of particle i, pi its pe-culiar momenta, Fi is the force due to its interaction withother particles, and P= �1 /N��i=1

N pi ensures that the total mo-mentum stays fixed. � is the Gaussian thermostat multiplier,determined using Gauss’s principle of least constraint, whichkeeps the kinetic temperature �i=1

N �pi−P�2 /2mi= �dN−d−1�kBT /2 fixed, where kB is Boltzmann’s constant and d isthe number of Cartesian dimensions. This constraint ensuresthat the trajectories actually followed by our system are thosewhich deviate as little as possible, in a least squares sense,from the unconstrained ones. In the simulations, we choosethe total momentum to be zero initially, and from the equa-tions of motion this means P=0 at all times. The adiabaticheating effect �i.e., the change in internal energy in the ab-sence of the thermostat� or dissipation can be expressed interms of the product of the dissipative flux and the field,

H0ad = − JFcV = JcFcN , �9�

where J is the dissipative flux which is related to the colorcurrent Jc= �1 /N��i=1

N ciqxi=−JV /N. We note that pxi includesa streaming contribution due to the flow of particles of eachcolor. If a realistic thermostatting mechanism was desired,this would be problematic and this contribution should besubtracted. However, it is not possible to know the realstreaming component in this system a priori and, as dis-cussed in Appendix A, there are issues in determining thecorrect nonlinear response expression in small systems whena instantaneous approximation to this contribution is made�as is common practice, see, for example, Chap. 6 of Ref.18�. As the main objective of this paper is to identify asym-metry in a well defined, small, nonequilibrium system, andnot in providing realistic thermostatting mechanism, wechoose the model above. This will not alter our conclusions,and is a reasonable physical model at low fields.

The color diffusion equations of motion above are timereversal invariant under the time reversal mapping iR, whichinverts the signs of the components of the molecularmomenta,

164515-2 Paneni, Searles, and Rondoni J. Chem. Phys. 128, 164515 �2008�

Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 130.102.82.118 On: Thu, 01 Sep

2016 03:53:28

iR�xi,yi,pxi,pyi� = �xi,yi,− pxi,− pyi� . �10�

That is,

iRStiRSt� = � , �11�

where St� is the solution of Eqs. �6�–�8� with initial condi-tion ��M, where M is the phase space of the system.

We use Lennard–Jones reduced units throughout thispaper.17

B. Fluctuation paths

Although our numerical results refer to the specific sys-tem discussed in Sec. II A, the theoretical arguments applymuch more generally. Consider a dissipative system drivenby a field Fe producing a dissipative flux J. Figure 1 displaysa sample of the values of the dissipative flux J over a shortperiod in a typical nonequilibrium system. Several signaturesof irreversibility are clearly observable: The mean value ofthe dissipative flux is lower than zero, the probability ofobserving positive values of J is smaller than that of observ-ing negative values, and the distribution of values of J isskewed toward negative values. Fluctuation relationships andtheorems20,21 have been developed over the past ten yearsthat assist in explanation of these phenomena. Here we focusour attention on another signature of irreversibility: The tem-poral asymmetry in typical FPs. There are many ways ofdefining a FP. However, as discussed previously,7,8 providedthe definition is precise and identifies symmetry at equilib-rium, the most suitable definition will depend on the detailsof the investigation. In this paper we abandon the definitionthat has been considered in the past7–9 and choose one thatseems as intuitive, physically reasonable, and computation-ally efficient, but has the advantage that it can be easily

expressed in a correlation function formalism, as shown inSec. IV.22 In practice, we wish to compare the paths that takethe instantaneous value of the observable away from itsmean value with those that bring it toward this value: Ulti-mately we want to see if the rise of a typical fluctuation ismore or less steep than its decay back to the mean value. Inour construction of the FP, we define a threshold THR �in theexample of Fig. 1 it is set equal to ��−1.5��, where � is themean and � is the standard deviation of the distribution offluxes� and a time interval .23 We consider the stationarypoints �maxima and minima� whose absolute value exceedsthe selected threshold THR. For every such stationary point,we define tstat to be the time at which this stationary point isobserved, and we consider the ordered set of values that theproperty assumes for the time interval before tstat and forthe time interval after tstat �refer again to Fig. 1�. For everystationary point whose absolute value exceeds the selectedthreshold THR, the FP is the set of absolute values that theproperty under interest assumes in the time interval of width2 interval about the stationary point �refer to Fig. 1�. Ofcourse the discrete nature of the simulated dynamics means

that observation of points with X=0 is not possible in prac-tice, so in the simulations we identify stationary points abovethe threshold using the condition that t= tstat whensign�X�St+t��−X�St���=sign�X�St−t��−X�St���, where tis the timestep used in the simulation and sign�X���� is thesign of X���.

More precisely, we define a FP as the series of valuesthat the property assumes in an interval �tstat− ; tstat+ ; �,multiplied by the sign of X�tstat�. So, given an observableX�t� X�St�� with mean �, we select the time width of in-terest, �0, and a positive threshold value THR��, then, for

FIG. 1. �Color online� A sample of the time evolution of the dissipative flux J in a fluid subject to a field Fe. The way in which fluctuation paths are selectedis shown: Troughs are converted to peaks and the average FP ��Y�t��� is constructed as average of such peaks centered in tstat.

164515-3 Fluctuations in nonequilibrium steady states J. Chem. Phys. 128, 164515 �2008�

Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 130.102.82.118 On: Thu, 01 Sep

2016 03:53:28

every tstat �see Fig. 1� such that �X�tstat���THR and X�tstat�=0, we define the observable Yt, for t� �− ,� as Yt

=sign�X�tstat��X�tstat+ t� for t� �− ,�. The time-ordered setof values Yt, with t� �− ,0� defines the path leading towardstationary point or the rise; the time-ordered set of values Yt,with t� �0,�, defines the path leading away from the sta-tionary point or the fall; the combination of the two setsdefines the FPs. For every tstat such that �X�tstat���THR,

X�tstat�=0, and X�tstat��0, we say the observable Yt belongs

to a “peak”, while if X�tstat��0, we say it belongs to“trough”.

In order to identify the typical FP, we consider the aver-age of all the FPs. Evaluating the average FP is computation-ally simpler than evaluating the most probable FP, as hasbeen done in some previous studies.7,8 In the thermodynamiclimit �infinite N�, if a single FP is expected, they will beidentical. Even with a small number of particles �N=8�, themost probable and average FPs have been observed to bevery similar for these systems �see Fig. 2 of Ref. 7�.

The fluctuation relations mentioned above show that, foran odd property, the probability of having values one side ofzero will be greater than the other. Therefore at large fields, ifthe mean is large and positive, it will be by far more likely tofind peaks above the threshold than troughs under the thresh-old �the opposite for mean below zero�. For positive definiteor negative definite properties, only one of the two will bepossible, since the property can only be positive or can onlybe negative. If we set thresholds far from the mean �whichwe must, since we are interested in exceptional fluctuations�,we will be unlikely to find peaks below −THR or troughs

above THR. Ultimately, the average FP so defined will turnout practically to coincide with the average of peak FPs �ifthe mean is above zero� or with the average of trough FPs�if the mean is below zero�. Numerical results confirm thisassertion.

To quantify the asymmetry, we define an asymmetry co-efficient. Given any FP of time width and threshold THR,we define the measure of asymmetry �t for t� �0, +� of theFP as

�t = Y−t − Yt. �12�

In order to identify the asymmetry arising in the FP, theaverage of �t over all the steady state fluctuation paths, ��t�,will be plotted for different values of t� �0, +�. Provided��t� is not zero for every t, the FP is asymmetric. However, ifa null result is obtained we could not definitively rule outasymmetry, as it might be an artifact of the method adopted.We note that in our previous work the time integral of ��t�was used to measure the asymmetry.7,8

III. CORRELATION FUNCTIONS AND NONLINEARRESPONSE THEORY FOR DETERMINATIONOF ASYMMETRY

A. Correlation functions

Using the definition above, we can show how the coef-ficient of asymmetry ��t� can be expressed as a difference ofcross correlation functions. Let X��� be the phase variablewhose asymmetry is being investigated. Let us consider thefollowing functions A��� and B���:

A��� = X��� , �13�

B��;t� = � 0 if �X���� � THR or sign�X��,t�� � sign�X��,− t��sign�X���� if �X���� � THR and sign�X��,t�� = sign�X��,− t�� ,

� �14�

where X�� ,t�=X�St��−X��� and t is the simulationtimestep. An equivalent way of writing this is

B��;t� = sign�X���� ��X���� − THR��sign�X��,t��

+ sign�X��,− t���/2, �15�

where �r� is the Heaviside step function � �r�=1 if r�0and �r�=0 otherwise�. In the limit t→0, B�� ;t�→B��� and is therefore simply equal to +1 for stationarypoints � above THR; it is equal to −1 for stationary pointsbelow −THR; and is it equal to 0 otherwise. We can thereforewrite Yt as

Yt = X�St��B��� , �16�

with t� �− , +�, and this describes a FP if B����0, i.e., if� is a stationary point with �X�����THR.

If we take the �conditional� ensemble average over all �for which B����0, we obtain an average FP,

�Yt�C =�A�St��B����

��B�����= �A�St��B����C = �X�St��B����C,

�17�

where �¯�C denotes a conditional average. Note that��B����� is the proportion of phase points that meet the se-lection criterion. It is necessary to take a conditional averagehere as we wish to obtain a result that is consistent with pastnumerical results for ��t�. In cases where very few peaksmeet the selection criteria, the complete ensemble averagewould be a very small number. Using Eq. �12� and the defi-nition of ��t�, we can write

��t� = �X�S−t��B����C − �X�St��B����C. �18�

The dynamical properties we are interested in are eithersymmetric or antisymmetric under time reversal mapping:X�iR��= pXX���, where pX= �1 is the parity of X���. Weobserve from Eq. �16� that Yt is an even function when A and

164515-4 Paneni, Searles, and Rondoni J. Chem. Phys. 128, 164515 �2008�

Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 130.102.82.118 On: Thu, 01 Sep

2016 03:53:28

B are defined by Eqs. �14� and �13�: That is, pAB=1. Thisresult is obtained by first noting that A�iR��= pXA���. It isstraightforward to see that ��X����−THR� and the delta

function of the time derivative of X, ��X����, do not changewith time reversal symmetry, and that, sign�X���� has thesame parity as X���. Therefore the parity of B will be thesame as that of X���, and pAB= pApB= pXpX=1. We can alsoconsider the behavior of the average FPs on reversal of thesign of Fe. We note that in the definition of Yt, the value ofX�tstat− t� is multiplied by the sign of X�tstat� to ensure thatFPs are always positive at the peak. This could destroy thesmoothness of Yt as a function of Fe at Fe=0 if the fieldchanges the sign of �X� �e.g., if the dependence on Fe islinear�, and therefore in analyzing the behavior of the FPs asa function of Fe we consider the paths defined without thischange in sign �i.e., sign�X�tstat�Yt��. We note that there is noexplicit dependence of A or B on the field, however, thenonequilibrium distribution function will change, and there-fore the ensemble average may change. Therefore, we intro-duce the symbol �A�Fe

to denote the phase space averagecomputed with respect to the steady state distribution ob-tained with field Fe and pA,Fe

for the parity of A with respectto the sign of Fe. We observe that if �A�Fe

= pA,Fe�A�−Fe

then�sign�X�tstat��Yt�Fe

= pA,Fe�sign�X�tstat��Yt�−Fe

.Now that we have the average FP expressed as a corre-

lation function, we can use properties of correlation func-tions to infer some properties of the FP. Table I lists anumber of properties of a correlation function C�z , t�= �A�z�B�z+ t��. These were obtained by considering the sym-metry of the functions on time reversal and change of thesign of the field and assuming that correlations decay in allthe systems considered. At equilibrium and in steady states,the value of C�z , t� is independent of z, but not necessarily oft. Furthermore, at equilibrium the time reversal invariance ofthe distribution function can be used to show that C�z ,0�=0 if pAB=−1 and �d�C�z , t�� /dt�z,t=0=0 if pAB=1.

Using the information in Table I, first we note that theidentification of asymmetry through the examination of�A�z− t�B�z��− �A�z+ t�B�z�� should be carried out with cau-

tion, as in the steady state, selection of some combinations ofA and B will give a value of 0 for this difference. Anotherimportant deduction is that the selection of A and B given inEqs. �13� and �15� will result in ��t� that becomes 0 in thelinear regime if pX,Fe

=1. In contrast, if pX,Fe=−1, the asym-

metry might be linear in Fe. We find that for fixed threshold,our results are consistent with a Fe

2 dependence of ��t� on Fe

for the pressure and with a Fe dependence for the color cur-rent, in accord with this deduction. We also note that in thelimit of long t, our measure of asymmetry will go to zero.

Another important observation can be made when con-sidering steady states. We note that if there is any correlationbetween the functions A and B, then �C�z ,0�− �A��B�� willbe nonzero, but �C�z ,��− �A��B��=0. If correlation betweenA and B exists and the correlation time is finite, in general,

C�z , t��z,t=0�0, although this will not be the case if �A�z,t=0

=0 for all B�z��0 or if A=B, A=B, etc. Similarly,�d2C�z , t� /dt2�z,t=0 , �d3C�z , t� /dt3�z,t=0 , . . . will be nonzero, ingeneral. However, we know that in the limit of large t,C�z , t�= �A��B�=C�z ,−t� because we assumed decay of cor-relations. As demonstrated in Fig. 2, these features and anynonzero value of an odd time derivative of C�z , t� at t=0indicate that asymmetry must exist. This is an importantresult as it means that it is not necessary to monitor the full

FP, it is sufficient to know C�t ,0�, �C�z , t��z,t=0,��d3C�z , t� /dt3��z,t=0, and �A��B�. This is computationallymuch more straightforward, and in most cases the fact that

C�z , t��z,t=0 and ��d3C�z , t� /dt3��z,t=0 are nonzero can be in-ferred from the facts that A and B are correlated and thecorrelation time is finite.

For example, it would be expected that if A is the pres-sure �P� and B is the current �Jc� asymmetry would occur as

�P�z , t�Jc�z���0 when t=0. As mentioned above, if��dnC�z , t� /dtn��z,t=0=0 for all n=1,3 ,5 , . . ., this is insuffi-cient to rule out asymmetry as it might be due to the defini-tion of the FP, but if ��dnC�z , t� /dtn��z,t=0�0, for any n=1,3 ,5 , . . ., asymmetry is present.

For the correlation function developed to produce ��t�

TABLE I. Properties of correlation functions, C�z , t�= �A�z�B�z+ t��, obtained by considering the symmetry of the functions on time reversal and change of thesign of the field, and assuming that correlations decay in all the systems.

Ensemble/Dynamics Condition C�z , t� �C�z , t�−C�z ,−t�� limt→� C�z , t� limt−��C�z , t�−C�z ,−t��a

Equilibrium No special condition �A�z�B�z+ t��eq 0 �A�eq�B�eq 0Equilibrium pAB=−1 0 when t=0 0 0 0Steady state A=B �A�z�A�z+ t�� 0 �A�2 0Steady state A= B or B= A �A�z�A�z+ t�� 0 0 0

Steady state C�z , t� evenfunct. of Fc;linear regime

�A�z�B�z+ t��eq 0 �A�eq�B�eq 0

Steady state No special condition b b �A��B� 0Transient response: z=0;equilibrium at z=0

pA=−1 b b 0 0

Transient response: z=0;equilibrium at z=0

pAB=−1 0 when t=0 b b 0

aThis is expected to be nonzero, in general. For example, in a transient experiment where an equilibrium ensemble average is carried out, but t�0 andz�0.bThese values are expected to be nonzero, in general.

164515-5 Fluctuations in nonequilibrium steady states J. Chem. Phys. 128, 164515 �2008�

Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 130.102.82.118 On: Thu, 01 Sep

2016 03:53:28

�i.e., Eqs. �13� and �14��, it can be seen that �A�z,t=0=0 for allB�z��0 since this is used as a selection criteria in condi-

tional average. Therefore, �C�z , t��z,t=0=0 in this case, andthe argument based on the first derivative in the previousparagraph cannot be used to predict asymmetry in the FPs.However, the third derivative is not subject to such a condi-tion and would be expected to be nonzero at t=0; so ��t�would not vanish and asymmetry in the FPs would exist.This indicates that ��t��0 and asymmetry in the FPs exists.

The above arguments rest on the assumption that corre-lations decay in time, however, the rate of decay is not speci-fied and any decay rate will suffice. In contrast, in the fol-lowing subsection integrals of the correlation functionsappear, and then it will be required that these correlationsdecay sufficiently fast for the integrals to exist.

B. Nonlinear response theory

Since we have expressed the asymmetry coefficient ��t�in terms of a ensemble average of a phase function, we canalso apply nonlinear response theory to its analysis. The ad-vantage of this approach, for the purposes of this work, isthat it allows a nonequilibrium, time dependent phase func-tion to be expressed in terms of an ensemble average overequilibrium initial states.

Using nonlinear response theory, the transient time cor-relation function �TTCF� provides the time dependence of aphase function in a nonequilibrium steady state far fromequilibrium.18 In the long time limit, for a system thatreaches a steady state, it can be considered to be a generali-zation of the Green–Kubo relations24 to states far from equi-librium as it provides an expression for the flux, from whichthe nonlinear transport coefficients can be determined. It al-lows us to express the average value of a function �A�Sz���eq

at time z after equilibrium in terms of its value at equilibrium

�A����eq and an integral of a time correlation function, whichin homogeneously driven and thermostatted systems �seeAppendix A� is given by

�A�Sz���eq = �A����eq − �VFe�0

z

�J���A�Su���eqdu ,

�19�

where J��� is the dissipative flux and A�Su��=A�Su��− �A�eq. The dynamics Su is generated using the nonequilib-rium �thermostatted� equations of motion, but the ensembleaverages use the equilibrium distribution. This can directlybe extended to express the product of two functions A���and B��� separated by a time interval t,25 and Eq. �19� be-comes

�A�Sz+t��B�Sz���eq = �A�St��B����eq

− �VFe�0

z

�J���A�Su+t��B�Su���eqdu .

�20�

Then,25

�A�Sz−t��B�Sz���eq − pAB�A�Sz+t��B�Sz���eq

= pAB�VFe�−z

z

�J���A�Su−t��B�Su���eqdu , �21�

where for A, B given by Eqs. �13� and �14�, pAB=1, asdiscussed above. This is a transient response expression.However, if the system reaches a steady state, it will becomeindependent of z and give the limiting steady state expression�A�S−t��B����Fe

=limz→��A�Sz−t��B�Sz���eq, cf. Eq. �3.57�in Ref. 18 and Appendix B. The above requires the integralsof the correlation functions to exist, hence the correlations todecay faster than 1 / t. This relation, for t� �− , +�, assumesa particular significance if applied to the functions A��� andB��� of Eqs. �13� and �15�, in that the left hand side of Eq.�21� is related to the measure of asymmetry ��t� through Eqs.�17� and �18�. It is this steady state result that we are inter-ested in

��t� = limz→�

�A�Sz−t��B�Sz��� − �A�Sz+t��B�Sz�����B�Sz����

. �22�

We will verify this relationship in numerical simulations inthe next section.

Note that if the function B���=1 for all �, then clearlythe right hand side of Eq. �22� would be zero in the steadystate. If asymmetry is present, it therefore must result fromthe conditional average through our choice of B���. Thisresult might seem surprising on initial consideration, how-ever, the condition that we choose serves to align the peaks:The average of a sawtooth wave will be a constant �andtherefore symmetric� if the phase of the wave is allowed tovary, but it will very asymmetric if the peaks are aligned �orthe phase is fixed�.

Now that we have expressed the asymmetry coefficientin terms of equilibrium ensemble averages and the TTCF, weconsider its properties again. We note from above that the

FIG. 2. �Color online� A schematic illustration of the existence ofasymmetry in systems where correlations decay and ��dnC�z , t� /dtn��z,t=0

�0, for any of n=1,3 ,5,... .

164515-6 Paneni, Searles, and Rondoni J. Chem. Phys. 128, 164515 �2008�

Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 130.102.82.118 On: Thu, 01 Sep

2016 03:53:28

expression �21� with pAB=1 applies to all phase variables A,irrespective of their parity under time reversal. First of all,we observe that Eq. �21� shows that at equilibrium �Fe=0�,the asymmetry coefficient is zero. Although this is not a newobservation, given Eq. �21� it becomes obvious. Similar con-siderations to those in Sec. III A show that the asymmetryvaries as Fe to leading order when pA,Fe

=−1 and as Fe2 when

pA,Fe=1. In addition we trivially observe that if pAB=1, the

left hand side of Eq. �21� is zero, but note that this is alsoevident from consideration of the right hand side of thatequation. In that case, the argument of the time integral is anodd function and the time integral is from −z to z, which willproduce a value of zero for all z. The emergence of asymme-try can then be seen to come from the disturbance of inte-grand when t�0 �i.e., a nonzero value of t results in a shiftin the time at which values of A are determined and those atwhich the values of B are determined�, which will clearlyresult in the argument no longer being an odd function, andtherefore the integral will no longer be zero. This will betrue, in general, but in the special case that z is large, corre-

lations decay, A=B or A=B, etc, the integral can be reformu-lated giving the expected result of zero. We point out thatthis is consistent with the argument of Giberti et al.10 whoargued that lack of interactions and therefore correlationswill lead to temporally symmetric fluctuations.

IV. NUMERICAL RESULTS

Molecular dynamics simulations of color diffusion havebeen carried out in two Cartesian dimensions with a primi-tive cell containing eight particles with kinetic temperaturefixed at 1, a number density of n=0.4, and timestep of 10−3.The equations of motion interaction potential and relation-ship between the color current and dissipative flux are givenin Sec. II A. We consider an equilibrium system as well as arange of nonequilibrium systems �Fc=0.5,1 ,1.5,2 ,3 ,4�, ex-tending from the linear regime to the nonlinear regime andexamine the FPs of Jc. Sets of ten independent simulations ofleast 3�106 timesteps were carried out for each value ofcolor field, in order to determine the mean and the standarddeviations of Jc, so to set the thresholds. These means andstandard deviations of Jc and pressure are reported inTable II.

From the data we can see that the linear regime extendsto approximately Fc=1.5. For large Jc, large fluctuations�several standard deviations from the mean� are biased above

the mean in the nonequilibrium systems; the same does nothappen for the values of Jc at equilibrium, since they aresymmetrically distributed around their mean of zero. Thebias in the mean and deviations from the mean can be un-derstood by considering the microscopic expressions for theproperties considered. On the basis of these data, sets of 45steady state simulations of 3.5�107 timesteps were carriedout for each field. The average the FP of Jc with a thresholdequal to �+2.5� was evaluated for =1 and are shown inFig. 3�a�. Figure 3�b� shows the values of ��t� for the FPs.All values are plotted with error bars being the standard errorof the mean. The average FPs are obtained as averages ofthese runs and so are the ��t�’s. These values are also plottedwith error bars.

TABLE II. Mean � and standard deviation � of color current Jc as averagesfrom ten runs with corresponding standard errors �three times the standarddeviation of the means out of the runs�.

Fc ��Jc� ��Jc�

0 0�0.006 0.341�0.0050.5 0.021�0.007 0.342�0.0041 0.043�0.007 0.343�0.0051.5 0.069�0.005 0.345�0.0032 0.101�0.008 0.35�0.0033 0.196�0.023 0.365�0.0064 0.341�0.019 0.38�0.007

FIG. 3. �Color� �a� Average fluctuation path �FP� of the color current fromthe average of 45 steady state runs with error bars �equal to the standarddeviation of the mean�. �b� ��t� for the average fluctuation path of the colorcurrent from the averages of 45 runs with error bars �equal to the standarderror of the mean�.

164515-7 Fluctuations in nonequilibrium steady states J. Chem. Phys. 128, 164515 �2008�

Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 130.102.82.118 On: Thu, 01 Sep

2016 03:53:28

Importantly, for Fc=0, the value of ��t� is zero to withinnumerical error at all times. This is an independent check ofthe accuracy of our results. Furthermore, at all other Fc,asymmetry is evident: We can see that the ��t� initially in-creases; it reaches a maximum and starts decreasing. Thisbehavior was also observed in our previous work.7,8 Fromthe definition of ��t�, it is obvious that this quantity vanishesat t=0. As t increases, ��t� departs from zero, indicatingasymmetry out of equilibrium and showing that the approachto the peak of the FP is less steep than the departure from it.At a certain time t�0, ��t� starts to decrease due to the decayof the correlations, in the same way as previously describedfor a general odd function. All the previous results confirmthe observations we made for different definitions of FP inour previous papers,7,8 attesting once more the presence ofasymmetry of fluctuations and the independence of this re-sult on the definition of the FP.

The symmetries of various correlation functions werealso considered. Sets of 65 independent simulations of 5�105 timesteps were carried out for �Fc=0.5,1 ,1.5,2 ,3 ,4�in order to compute examples of correlation functions�A�St��B����. The presence of asymmetry in t is expectedfor a generic odd property, such as the correlation betweenthe flux Jc and the pressure P, as discussed above. Figure4�a� displays the cross correlation �Jc�St��P���� as a func-tion of t for our system undergoing color diffusion with Fc

=2. The results are obtained as time averages from steadystate runs and are the average values obtained from 65 runsof 5�105 timesteps with error bars.

We verify that for this function the time dependence ofthe correlation function fulfils the relevant properties ofTable I. At equilibrium, �Jc���P����eq=0, whereas it has afinite positive value when Fc�0. We observe that for allfields, in the limits t→ +� and t→−� the correlation func-tions are equal, and equal to �Jc��P�, as expected for thischaotic dynamic where correlations decay. We also note thatat t=0 the slope is different from zero, and that the value att=0 is different from that at large values of �t. This is con-sistent with the analysis at the end of Sec. III A. If, in thenonequilibrium system, the correlation has a value�Jc���P���� at t=0, with slope different from zero, to thenrelax in both directions to �Jc���P����� �Jc�����P�����0,for intermediate �t�, our cross correlation �Jc�St��P���� mustbe different from �Jc�S−t��P����. For even functions, thepresence of asymmetry can be measured by calculatingXt�A ,B�=C�z , t�−C�z ,−t�. For odd functions, due to theirnonzero time derivative at t=0, we consider the departure ofthe profile from an antisymmetric one by evaluatingXt

odd�A ,B�=C�z , t�+C�z ,−t�−2C�z ,0�. By definition thevalue of Xt

odd is 0 at t=0. We can see from Fig. 4�b� how thenonzero derivative and asymptotic values indicate the pres-ence of a departure from an antisymmetric behavior, which isthe general situation out of equilibrium. There are choices ofparticular functions, such as the correlation of a functionwith itself or with its derivative, for which it can be provenand shown by numerical results, that no asymmetry ispresent. Figure 5�a� shows the color current autocorrelation�Jc�St��Jc���� with Fc=2, and in Fig. 5�b� the correlation

between the color current and its derivative �Jc�St��Jc���� isshown for Fc=2. In the first case the profile is perfectlysymmetric within numerical error, as we can see from Fig.5�c� and in the second case the profile is perfectly antisym-

metric, and this is also shown in Fig. 5�c� where Xtodd�Jc , Jc�

is zero at all t. In contrast, the cross correlation �Jc�St��P����is asymmetric, as confirmed by the numerical results for ��t�in Fig. 5�c�. Note the similarity of the behavior of the FPs inFig. 3 and of the correlation function in Fig. 5. This is anatural consequence of the fact that the system relaxes froma peculiar state to the steady state with timescales that aredictated by the decay of the correlations.

We now provide numerical results confirming the TTCFexpression for the asymmetry �21�, and thus confirmingthe relationship on which we based our main argument in

FIG. 4. �Color online� �a� Cross correlation function of color current Jc andpressure P from the average of 65 steady state runs with error bars �equal tothe standard error of the mean� and with color field Fc=2. �b� Coefficient ofasymmetry Xt

odd, for Fc=2 and the correlation functions shown in �a�, withdeparture from zero indicating asymmetry out of equilibrium.

164515-8 Paneni, Searles, and Rondoni J. Chem. Phys. 128, 164515 �2008�

Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 130.102.82.118 On: Thu, 01 Sep

2016 03:53:28

Sec. III B, i.e., that asymmetry must exist, in general, forcorrelations functions and for the FPs, in particular. Refer-ring to Eq. �21�, we see that if we make z sufficiently largethat a steady state is reached, the right hand side of ourequations gives ��t�, through Eq. �22�. We therefore ran a setof 35 simulations consisting each of 106 transients startingfrom equilibrium values and going forward and backward intime for a period z+ t, with t� �− , +�, =1 and with z=3so as to be far enough from the equilibrium starting point toconsider our system well into the steady state.26 We thenevaluated the right hand side of Eq. �21�. The numerical errorin evaluation of the left hand side of Eq. �21� from the tran-sients is extremely high,27 and therefore these results are notshown. Instead, the left hand side of the equation was evalu-ated from the steady state simulations. Figure 6 shows theright hand side of Eq. �21�, scaled to give ��t� through Eq.�22�, and evaluated from the transient runs is equal to thatevaluated as steady state average, to within numerical errors.The agreement of Figs. 3�b� and 6 within the limits of errorconfirms the link between the presence of asymmetry withTTCF theory for the FPs of a deterministic and reversiblesystem.

Finally, we consider the dependence of ��t� as a functionof the field Fc. We select an arbitrary time t=0.1 close to thepeaks of the plot of ��t� versus time. Figure 7 shows ��t�, att=0.1, of the average fluctuation path of color flux Jc, withTHR=2.5�+�, as a function of the color field Fc applied. Wesee that ��t� grows with the field in the linear regime: Thereare two contributions to the growth of ��t�. First, there is adirect contribution which we would expect to be linear forthe odd property Jc, if the value of the threshold was fixed;then there is a growth due to an increase in the threshold,since � increases linearly with Fc.

FIG. 5. �Color online� �a� Autocorrelation function of color current Jc fromthe average of 65 steady state runs with error bars �equal to the standarderror of the mean� and with color field Fc=2. �b� Correlation function ofcolor current Jc with its derivative from the average of 65 steady state runswith error bars �equal to the standard error of the mean� and with color fieldFc=2. �c� The value of Xt which measures the departure of the cross-correlation function �Jc�St��Jc���� from symmetric behavior, and values ofXt

odd which measures the departures from antisymmetric behavior of

�Jc�St��Jc���� and �Jc�St��P����. All data are computed as average of 65runs with error bars �equal to the standard error of the mean� with color fieldFc=2.

FIG. 6. �Color� The values of ��t� for the average fluctuation path of thecolor current from averages of 35 transient runs with error bars �equal to thestandard error of the mean�, calculated using the right hand side of Eq. �21�with z=3. This can be compared to the results obtained using a steady stateexpression as shown in Fig. 3�b�.

164515-9 Fluctuations in nonequilibrium steady states J. Chem. Phys. 128, 164515 �2008�

Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 130.102.82.118 On: Thu, 01 Sep

2016 03:53:28

V. CONCLUSIONS

The presence of temporal asymmetries has been con-firmed in various nonequilibrium molecular dynamics simu-lations and adopting various methods to detect them. Unlikeearlier studies on particles systems, the model used does nothave complications such as time-varying boundary condi-tions. A new definition of FP is introduced that allows a morerigorous analysis of the emergence of their asymmetry withan application of a field. FPs have been expressed by meansof correlation functions and TTCF theory: Thus for the firsttime we have been able to theoretically justify the asymme-tries that have been observed previously in deterministic, re-versible systems. This has allowed us to show for the firsttime that temporal asymmetries are a common property ofdeterministic and reversible nonequilibrium systems.

ACKNOWLEDGMENTS

The authors wish to thank the Australian ResearchCouncil, The Queensland Parallel Computing Facility, andthe Australian National Facility for support of this work. Wethank Peter J. Daivis, C. Giberti, Stephen R. Williams, andJohn F. Dobson for useful comments and Benjamin V. Cun-ning for discussions.

APPENDIX A: GENERAL TRANSIENT TIMECORRELATION EXPRESSION FOR NONLINEARRESPONSE

In Ref. 18 it has been shown how to obtain the TTCFexpression for some particular cases. Consideration of Eq.�7.29� of that reference provides us with a route to a moregeneral expression,

�B�Sz���eq = �B����eq + �0

z

�B�Su�������eqdu , �A1�

where

����f��� = −�

��· ��f����

= − f����

��· � −

df���dt

+�f���

�t, �A2�

and � is the dissipation function that appears in the Evans–Searles fluctuation relation.21 Although equilibrium ensembleaverages are used in Eq. �A1�, the dynamics represented by Sare the nonequilibrium dynamics. The distribution functionthat is used in the ensemble average in Eq. �A1� and in thedefinition of Eq. �A2� is the distribution function preservedby the field free dynamics, so �f /�t=0.

Interestingly, the isokinetic canonical distribution func-tion with �= �kBT�−1 and using K0= �dN−d−1�kBT /2 is notpreserved when the usual �see Chap. 6 of Ref. 18� colordiffusion equations are employed, and terms of all order in Nare considered. A more problematic, but similar, situation hasbeen observed when �-thermostats have been employed �formore details see Ref. 28�. In the case of color diffusion, theproblem that occurs relates to the subtraction of the stream-ing velocity u. Unlike the approximate streaming velocity,the real streaming velocity is a constant and not dependenton �, so the distribution function could be determined asusual, giving an isokinetic equilibrium distribution function,

feq��� =e−�H0�����K��� − K0�

� e−��H0�����K��� − K0�d�, �A3�

where K0 is the required kinetic energy and the internal en-ergy is given by

H0��� = K��� + ��q� = �i

1

2mi�pi − u�2 + ��q� . �A4�

If u is not known a priori, it can be estimated as 1 /N�i=1N cipi

which fluctuates at equilibrium, even though its average iszero. This leads to a change in the phase space contractionrate that is ON�1� �order 1 in N�, and no longer preserves Eq.�A3�, even when H0 is redefined to account for the alteredstreaming term. An equilibrium distribution can be defined,but it requires that the proportionality constant between K0

and kBT is different. To circumvent this problem in this paper�which does not aim to realistically model a physical prob-lem�, we simply set u=0, obtaining a distribution function�A3� and thermostatting �i�1 /2mi�pi

2=K0= �dN−d−1�kBT /2. A better approach if a physical system was to bestudied would be use an iterative method to determine thestreaming velocity. In this Appendix we show that for thissystem �=−�JFeV, and hence Eq. �19� follows.

In this Appendix we consider general thermostattedequations of motion,

qi =pi

m+ C���Fenx, �A5�

pi = Fi + D���Fenx − ��p − P� . �A6�

The distribution function �A3� is conserved by the equilib-rium equations of motion. That is,

FIG. 7. �Color online� ��t� as a function of the field Fc at arbitrary time t=0.1 close to the peaks of the �t, with THR=2.5�+�.

164515-10 Paneni, Searles, and Rondoni J. Chem. Phys. 128, 164515 �2008�

Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 130.102.82.118 On: Thu, 01 Sep

2016 03:53:28

�

�tfeq��,t� = feq��,t����� − �H0���feq��,t� , �A7�

where the quantity ����=−� /�� · � is the phase space com-pression rate of the system. Using the equations of motion,we can write the phase space compression rate as

���� = �dN − d − 1�� . �A8�

We also find

H0 = − JFeV − 2�K0 = − JFeV − ��dN − d − 1�kBT . �A9�

Clearly Eq. �A7� is zero at equilibrium �Fe=0�.We can now evaluate � using Eq. �A2�, that is,

���� = ���� + �H0��� = − �J���FeV . �A10�

Hence, we obtain the nonlinear response given by Eq. �19�.

APPENDIX B: STEADY STATE AVERAGESFROM NONLINEAR RESPONSE

The equality of Sec. III B,

�A�S−t��B����Fe= lim

z→��A�Sz−t��B�Sz���eq, �B1�

may be obtained from a transformation of coordinates asfollows �cf. Ref. 29�. Let fFe,z be the probability density inphase space at time z, obtained with a field Fe, and assumethat the evolution is such that the probability distributionconverges to a given stationary distribution. Then, the steadystate average obeys

�A�S−t��B����Fe= lim

z→�� A�S−t��B���fFe,z���d� , �B2�

while the probability density obeys fFe,z���= fFe,0�S−z��e−�−z

0�·��Ss��ds, as determined by the Liouville

equation �cf. Eq. �32� in Ref. 29�. The coordinate transfor-mation X=S−z� then yields

�A�S−t��B����Fe= lim

z→�� A�Sz−tX�B�SzX�fFe,0�X�dX

= limz→�

�A�Sz−t��B�Sz���eq, �B3�

since the Jacobian �d� /dX� equals e�0z�·��SsX�ds, which is the

inverse of the factor appearing in the evolved probabilitydensity fFe,z���.

1 L. Bertini, A. De Sole, D. Gabrielli, G. Jona-Lasinio, and C. Landim,Phys. Rev. Lett. 87, 040601 �2001�.

2 L. Bertini, A. De Sole, D. Gabrielli, G. Jona-Lasinio, and C. Landim, J.Stat. Phys. 107, 635 �2002�.

3 L. Onsager and S. Machlup, Phys. Rev. 91, 1505 �1953�; S. Machlup andL. Onsager, ibid. 91, 1512 �1953�.

4 P. Gaspard, Adv. Chem. Phys. 135, 83 �2007�.5 D. G. Luchinsky and P. V. E. McClintock, Nature �London� 389, 463�1997�.

6 M. I. Dykman, P. V. E. McClintock, V. N. Smelyanski, N. D. Stein, andN. G. Stocks, Phys. Rev. Lett. 68, 2718 �1992�.

7 C. Paneni, D. J. Searles, and L. Rondoni, J. Chem. Phys. 124, 114109�2006�.

8 C. Paneni, D. J. Searles, and L. Rondoni, 2006 ICONN Proceedings�IEEE, Piscataway, 2006�.

9 C. Giberti, L. Rondoni, and C. Vernia, Physica A 365, 229 �2006�.10 C. Giberti, L. Rondoni, and C. Vernia, Physica D 228, 64 �2007�.11 H. B. G. Casimir, Rev. Mod. Phys. 17, 343 �1945�.12 M. J. Klein, Phys. Rev. 97, 1446 �1955�.13 H. Ichimura, Chin. J. Phys. �Taipei� 17, 94 �1979�.14 D. Sánchez and M. Büttiker, Phys. Rev. Lett. 93, 106802 �2004�.15 C. A. Marlow, R. P. Taylor, M. Fairbanks, I. Shorubalko, and H. Linke,

Phys. Rev. Lett. 96, 116801 �2006�.16 J. D. Weeks, D. Chandler, and H. C. Andersen, J. Chem. Phys. 54, 5237

�1971�.17 M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids �Clar-

endon, Oxford, 1987�.18 D. J. Evans and G. P. Morriss, Statistical Mechanics of Nonequilibrium

Liquids �ANU, Canberra, 2007�.19 D. J. Evans, Physica A 118, 51 �1983�.20 D. J. Evans and D. J. Searles, Adv. Phys. 51, 1529 �2002�.21 G. M. Wang, E. M. Sevick, E. Mittag, D. J. Searles, and D. J. Evans,

Phys. Rev. Lett. 89, 050601 �2002�.22 This does not exclude that the previously introduced definitions of fluc-

tuation paths may be cast in the present formalism.23 There is no natural criterion to identify THR. However, the choice of THR

and does not affect our theoretical treatment, hence in the numericalwork we chose a value for THR that allows us to sample large fluctuationsand also gather sufficiently good statistics. We typically choose a value of that is equal to several Maxwell times, after which no change in theform of the FP occurs.

24 D. A. McQuarrie, Statistical Mechanics �Harper and Row, New York,1976�.

25 M. G. McPhie, P. J. Daivis, I. K. Snook, J. Ennis, and D. J. Evans,Physica A 299, 412 �2001�.

26 Because of the Lyapunov instability and of the finite precision of numeri-cal simulations, it was not possible, in practice, to run the transientsbackward for more than a time of about 1. Therefore, we exploited thetime reversal properties of the system in order to calculate the valuesassumed by our function for negative times, directly from the valuesobtained in the forward run.

27 The probability of finding a peak exactly at time z of the transient is verylow �at high fields it is much lower than in the corresponding steady statebecause much of the trajectory is in a state close to the initial equilibriumstate where large deviations are even more unlikely to be observed�. Anenormous number of cycles should be run in order to obtain data withreasonable statistics, and while the data we were able to obtain were inagreement with the results, the statistical error was so large that thevalues were not very meaningful.

28 J. N. Bright, D. J. Evans, and D. J. Searles, J. Chem. Phys. 122, 194106�2005�.

29 D. J. Searles, L. Rondoni, and D. J. Evans, J. Stat. Phys. 128, 1337�2007�.

164515-11 Fluctuations in nonequilibrium steady states J. Chem. Phys. 128, 164515 �2008�

Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 130.102.82.118 On: Thu, 01 Sep

2016 03:53:28