An integral fluctuation theorem for systems with unidirectional transitions
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] S e p An integral fluctuation theorem for systems with unidirectional transitions
Saar Rahav and Upendra Harbola Schulich Faculty of Chemistry, Israel Institute of Technology, Haifa 32000, Israel. and Department of inorganic and physical chemistry,Indian Institute of Science, Bangalore, 560012, India.
The fluctuations of a Markovian jump process with one or more unidirectional transitions,where R ij > R ji = 0, are studied. We find that such systems satisfy an integralfluctuation theorem. The fluctuating quantity satisfying the theorem is a sum of the entropyproduced in the bidirectional transitions and a dynamical contribution which depends on theresidence times in the states connected by the unidirectional transitions. The convergenceof the integral fluctuation theorem is studied numerically, and found to show the samequalitative features as in systems exhibiting microreversibility. I. INTRODUCTION
The last two decades have seen substantial advancement in our understanding of the thermo-dynamics of small out-of-equilibrium systems. Much of the progress was related to the study offluctuations in such systems. In particular, it was found that many out-of-equilibrium processessatisfy fluctuation theorems [1–8]. These celebrated relations compare the probabilities to observea realization of a process and its time-reversed symmetry related counterpart. The ratio of theseprobabilities is expressed in terms of thermodynamic quantities such as entropy production or heat.A closely related set of results is termed work relations [9, 10]. The latter focus on the fluctuationsin the work done on the system when it is driven away from equilibrium.Fluctuation theorems are valid for systems which are driven arbitrarily far from their thermalequilibrium. The discovery of fluctuation theorem has opened up new research directions andenhanced our qualitative and quantitative understanding of small systems in contact with thermalenvironments. Some of the progress made is summarized in several review articles [11–23].One of the important concepts underlying fluctuation theorems is the ability to meaningfullyand consistently assign thermodynamic interpretation to a single realization of a stochastic out-of-equilibrium process. This approach is sometimes referred to as stochastic thermodynamics.Sekimoto has demonstrated that heat and work can be defined for a single realization of a process sothat the first law is satisfied [24]. Seifert has introduced the concept of a fluctuating system entropyand demonstrated that it allows to obtain an exact fluctuation theorem [8]. Fluctuation theoremscan be viewed as replacing the inequality of the second law of thermodynamics by an equalityfor the exponential average of a realization dependent fluctuating quantity [21]. The inequality isrestored for the ensemble average of this quantity with the help of the Jensen inequality. Moreinformation about stochastic thermodynamics can be found in Seifert’s comprehensive review [23].Derivations of fluctuation theorems commonly use a direct comparison of the probabilities of arealization and that of its time reversed counterpart. For systems driven by a given force protocol,one similarly compares probabilities of realization of a forward process to that of its time reversed(backward) process driven by a time reversed force protocol. The same considerations can beapplied for systems in absence of time reversal symmetry, for instance due to the presence of amagnetic field, see Ref. [25] for a recent review. All of those approaches employ microreversibility,namely the fact that time inversion (combined with an inversion of momenta, driving protocol, andpossibly magnetic field) maps between allowed realizations of a forward and a backward process.In jump processes microreversibility means that if the transition from state i to state j has a finiterate, R ji >
0, then so does the reversed transition, R ij >
0, where we assumed absence of magneticfields.There are instances however where the simplest description of a natural process is one wheremicroreversibility is violated. Consider an atom in an excited state which decays via spontaneousemission of a photon which escapes from the system. In many situations it is useful to derive areduced description for the atom in which the field serves as an external reservoir. For spontaneousemission the empty field modes can be interpreted as a zero temperature reservoir. The reversedprocess, namely an excitation of the atom, requires presence of photons, but when there are no suchphotons, this reverse process will not occur. As a result the reduced description has R emission > R absorption = 0. When the field modes are in a thermal state with finite temperaturethe presence of stimulated processes restore microreversiblity. Unidirectional transitions can beincorporated into models of heat engines and machines which also include reversible transitions,such as the model of a photosynthetic reaction center studied by Dorfman et. al. [26] (See Fig. 2there). Irreversible jump processes are also used to model biological enzymes which break down thesubstrate they move on, such as cellulase, see e.g. [27]. In the following we use this as motivationto study jump processes which violate microreversibility. We focus on the fluctuations in suchsystems and ask whether they satisfy a fluctuation theorem.Fluctuations of systems with unidirectional transitions have generated limited interest so far.Ohkubo derived a fluctuation theorem which holds also for irreversible systems [28]. It is basedon a posterior transition rates obtained with the help of Bayes’ theorem. A fluctuation theoremfor a system of soft spheres with dissipative collisions was derived by Chong et. al. [29]. Itapplies to systems with continuous dynamics and moreover requires thermal initial conditions.Other approaches used to investigate fluctuation theorems in systems with an irreversible transitionreplace the vanishing rate by an effective finite rate using some coarse-graining of the dynamics.Ben-Avraham, et. al. have suggested to measure the state of the system in fixed time intervals[30]. This coarse graining in time allows for an effective backward rate which is actually obtainedfrom the combined contributions of allowed transitions which take the system to the other side ofthe irreversible transition. Zeraati et. al. chose to view the vanishing transition rate as being thelimit of a very small but finite rate, which is small enough to be unlikely to be observed in a finitetime experiment [31]. This effective rate was then estimated using Bayes theorem which in turnis then used to obtain a lower bound for the entropy production that depends on the observationtime. Both of those approaches exhibit logarithmically diverging quantities which are ill definedin the limit where the coarse-graining is removed, namely for vanishing time intervals betweenmeasurements or infinite observation time.Here we employ an approach which does not suffer from such difficulties, and show that anintegral fluctuation theorem holds for systems with unidirectional jumps. This fluctuation theoremis based on a different treatment of reversible and irreversible transitions. It holds for a fluctuatingquantity which is a sum of two contributions. The first is the usual fluctuating entropy productiondue to reversible transitions. The second is an unusual dynamical term which depends on thefluctuating residence times in the states connected by the irreversible (unidirectional) transitions.This prescription avoids the difficulties in defining an entropy production for the irreversible termsthat led to diverging expressions in the coarse-grained approaches.The structure of the paper is as follows. In Sec. II we consider a simple example of a jump pro-cess with a single irreversible transition and derive the integral fluctuation theorem. The derivationis simple and can be easily applied to systems with more states, irreversible transitions, or time de-pendent transition rates. Such generalizations are straightforward, and are stated without detailedproof in Sec. III. In Sec. IV we discuss the number of realizations needed for convergence of theexponential average appearing in the integral fluctuation theorem. We point out that estimatesbased on the identification of typical and dominant realizations which were developed for systemswith reversible rates are also applicable here. We summarize our results in Sec. V. R FIG. 1: A graph representation of the jump process with one unidirectional transition studied here.
II. A SIMPLE MODEL
We introduce the integral fluctuation theorem with the help of a simple example of a Markovianjump process. The use of an example allows us to present the derivation without using unneces-sarily complicated notations. The choice of the example is based on two requirements. We wantthe system to include one irreversible transition. In addition, we want at least one closed cycleof reversible transitions, to allow for a steady state flux even in the absence of the irreversibletransition. These considerations lead us to study a system with four states and one irreversibletransition, which is the minimal model that has no more that one transition between states andsatisfies the requirements. Generalizations to more general jump processes are possible and willbe described In Sec. III. The possible transitions between the four states are characterized by thetransition rates, with R ij ≥ j → i (for i = j ). When the transi-tion between i and j is reversible the combination ln R ij R ji is interpreted as the entropy change in thereservoir during the transition [5]. This identification is motivated by the fact that for thermallyactivated rates this term commonly has the form E i − E j T , with E i the energy of state i and T thetemperature of the reservoir. This is an ill-defined quantity for unidirectional rates, where R ij > R ji = 0.The simple model investigated in this section can be conveniently represented using a graph,which is depicted in Fig. 1. The black solid lines represent reversible (bidirectional) transitions.In contrast, the transition from state 2 to state 4 is irreversible, with rates R > R = 0.The system’s probability distribution evolves according to a master equation, d p dt = Rp , with R ii ≡ − P j = i R ji = − r i . Here P is the vector containing population of the four states and R isthe transition rate matrix. If left alone the system relaxes to a steady state, p ss . A history, orequivalently a realization, of the system is a list detailing the state of the system at any given timeincluding also the specific transitions it made during the entire realization. We denote a givenrealization by γ . For example γ = n t −→ t −→ · · · t n −→ o (1)corresponds to a realization where the system was initially at state 2, and stayed there until time t . (This can be denoted equivalently by γ ( t ) = 2 for 0 ≤ t < t .) At time t the system makes atransition to state 3, etc. Eventually the system makes a transition from state 4 to state 1 at time t n . The system then stays in that state until the end of the observation at time t f . Since this is aMarkovian jump process the probability density of such a history is P ( γ ) = p i (2) e − r t R e − r ( t − t ) · · · R e − r ( t f − t n ) , (2)where p i denotes the initial probability distribution. We will denote by p f the final probabilitydistribution, namely the solution of the master equation at time t f , given the initial condition p i .Let us denote by γ the time reversed realization of γ . For γ of Eq. (1) this is clearly γ = (cid:26) t f − t n −−−−→ · · · t f − t −−−→ t f − t −−−→ (cid:27) . (3)We note that time reversal is a one-to-one mapping between realizations. However, many of thetime reversed realizations are not allowed under the dynamics of the irreversible jump processdepicted in Fig. 1 because they would make the forbidden 4 → γ as obtained from an auxiliarydynamics in which one reverses the direction of the irreversible transition. For the simple exampleconsidered in this section this auxiliary dynamics has R = R and R = 0 and otherwise R ij = R ij (for i = j ). We intentionally avoid the more common terminology of ”forward” and”backward” processes, which are best left for cases in which the backward process has a meaningfulphysical interpretation.Importantly, the time reversal mapping between γ in the physical dynamics and γ in the aux-iliary dynamics is one-to-one. One can assign a probability density P ( γ ) for realizations of theauxiliary dynamics. For the realization in Eq. (3) one finds P ( γ ) = p i (1) e − r ( t f − t n ) R · · · e − r ( t − t ) R e − r t . (4)We note that r i = r i for the states linked by the unidirectional transition. It is crucial to point outthat with the interpretation of irreversibility as coming from spontaneous emission the auxiliarydynamics is not physical. It involves transitions taking the system from a low energy state to anexcited state without energy input from the environment.The auxiliary dynamics allows us to obtain a formal integral fluctuation theorem by definingΣ( γ ) ≡ ln P ( γ ) P ( γ ) , (5)and noting that (cid:10) e − Σ (cid:11) = X γ e − Σ( γ ) P ( γ ) = X γ P ( γ ) = 1 . (6)Equation (6) is a mathematical identity. Fluctuation theorems of this type are of interest whenthey can be given an appealing physical interpretation. We show that it is possible to expressΣ( γ ) using only properties of the physical dynamics, suppressing any explicit dependence on theauxiliary dynamics.To do so we note that Eq. (6) is valid for any choice of the initial condition for the auxiliarydynamics. We therefore choose p i = p f , namely the initial distribution of the auxiliary dynamicsis the same as the final distribution of the physical dynamics. A short calculation then findsΣ( γ ) = ∆ S rev ( γ ) + R ( τ ( γ ) − τ ( γ )) . (7)Here ∆ S rev ( γ ) = X i, rev ln R γ i +1 γ i R γ i γ i +1 + ln p i ( γ (0)) p f ( γ ( t f )) , (8)where the first term on the right hand side is the sum of contributions to the medium entropyproduction from all the reversible transitions during the realization γ ( t ), while the second is thechange of the fluctuating system entropy [8]. Note that the quantity Σ( γ ) fluctuates from onerealization to another. τ i ( γ ) in Eq. (7) denotes the (fluctuating) time that the system spends instate i during the realization. It can be written as τ i ( γ ) = R dtχ i ( γ ( t )) where χ i = 1 for γ ( t ) = i and 0 otherwise.The jump process depicted in Fig. 1 therefore satisfies the integral fluctuation theorem D e − ∆ S rev + R ( τ − τ ) E = 1 , (9)which is accompanied by a second law like inequality h ∆ S rev + R ( τ − τ ) i ≥ . (10)The quantity Σ satisfying the integral fluctuation theorem (9) has a simple physical interpretationwhich is an interesting mixture of dynamical and thermodynamic quantities. The thermodynamicpart, ∆ S rev , includes the change in medium entropy in the finite temperature reservoirs. Thethermodynamic interpretation of this entropy production as resulting from e.g. energy exchangedwith finite temperature reservoirs is well understood, in contrast to the absence of a similar inter-pretation for irreversible transitions. The contribution of the irreversible transitions is dynamicaland depends on residence times, namely the time that the system spends in the states connectedby the transition. The structure of this dynamical term, a product of the transition rate timesresidence times, is similar to the so called traffic which contributes to the linear response of jumpprocesses [33, 34]. Both the dynamical term in Eq. (9) and the traffic as considered in Refs.[33, 34] are given by time integrals of escape rates, but only the irreversible transition contributesto the dynamical part of Σ and furthermore the sign of the contribution depends on whether theirreversible transition points into, or out of, the state.It is worthwhile to examine the inequality (10) more closely at steady state, where all the termsappearing in the inequality are linearly proportional to time, h ∆ S rev i = q rev t f and h τ i i = p ss ( i ) t f .The inequality (10) can be rewritten as q rev + R ( p ss (4) − p ss (2)) ≥ . (11)Here q rev is the reversible entropy production rate at steady state. Each reversible transition i → j contributes ln( R ji /R ij ) to the entropy production. At steady state the mean rate of the i → j transition is R ji p ss ( i ). Therefore q rev = X ( i,j ) , rev [ R ji p ss ( i ) − R ij p ss ( j )] ln( R ji /R ij ) (12)where the sum is over all unordered pairs of states which are connected by a reversible transition.(For simplicity we assume here that at most one transition connects any give pair of states.) Finally,with the help of the conservation laws for steady state fluxes this inequality can be recast as J ss ln R p ss (4) R p ss (1) + J ss ln R p ss (1) R p ss (2) + J ss ln R p ss (4) R p ss (3) + J ss ln R p ss (3) R p ss (2)+ R p ss (2) (cid:20) ln p ss (2) p ss (4) + p ss (4) p ss (2) − (cid:21) ≥ . (13)The terms of the form J ssij ln R ij p ss ( j ) R ji p ss ( i ) = ( R ij p ss ( j ) − R ji p ss ( i )) ln R ij p ss ( j ) R ji p ss ( i ) ≥ x + x − ≥ x . R p ss (2) is clearly the steady state flux of irreversibletransitions. However, it is not clear whether ln p ss (2) p ss (4) + p ss (4) p ss (2) − h τ ih τ i + h τ ih τ i − , which dependson the ratio of likelihood to find the system at both sides of the unidirectional transition, can bemeaningfully interpreted as some generalized affinity. III. POSSIBLE GENERALIZATIONS
In this section we briefly describe various generalizations of the integral fluctuation theorem,Eq. (9). The derivations are straightforward and most of the details are omitted.We first note that there is some mathematical freedom in the choice of possible auxiliary dy-namics. One can modify the magnitude of various transition rates in the auxiliary dynamics, andEq. (6) will still hold, as long as the correct transitions are prohibited. However, this freedomto play with the magnitude of rates results in a Σ whose ”entropic” and ”dynamical” parts havedubious physical interpretation. For instance, using an auxiliary dynamics with a modified valueof R would result in contributions of ln R R to the ”entropy production”. But such a physicalinterpretation is unjustified since the rate R has nothing to do with the dynamics of the physicalsystem. Moreover, with this choice of auxiliary dynamics Σ depends on the rate R , and the re-sulting integral fluctuation theorem no longer depends only on properties of the physical dynamics.The auxiliary dynamics used in Sec. II was chosen to prevent the appearance of such difficulties,and keep the physical interpretation of Σ transparent. In that sense the demand for consistentphysical interpretation suggests that the auxiliary dynamics should be the one that was used inSec. II.The derivation of Eq. (8) presented in Sec. II never made use of the fact that there are onlyfour states in the system. It applies to a jump process with any finite number of states as long asthe initial and final probability distributions have finite values for all states. Even when some ofthe probabilities p f , p i vanish, it is possible that the approach developed by Murashita et. al. [32]may be of use, but this is beyond the scope of the current paper.Another possible generalization is to a system with several unidirectional transitions. In thiscase the auxiliary dynamics is one where all the irreversible rates have been reversed. Their contri-bution to the fluctuation theorem enters through the escape rates r i and r i . These escape rates aresums over all the rates of transitions leaving a state, and the different irreversible transitions musttherefore contribute additively to the escape rates. The result is that several irreversible rates con-tribute additively to Σ, and the dynamic contribution has the form P α R α + α − [ τ α + ( γ ) − τ α − ( γ )],where α runs over different irreversible transitions, which connect state α − to state α + , and havethe rate R α + α − .The last generalization we consider is to systems with time dependent rates. For such sys-tems the conditional probability factors expressing the probability to stay in a state have the formexp h − R t i +1 t i dtr α ( t ) i , in contrast to factors of exp [ − r α ( t i +1 − t i )] appearing in autonomous sys-tems. The resulting contribution to Σ has dynamical terms of the form R dtR α + α − ( t ) χ α − ( γ ( t ))replacing the terms R α + α − τ α − .Based on these considerations the integral fluctuation theorem (6) holds for time dependentjump processes with several irreversible transitions, and, as long as the probability distribution isnon-vanishing, Σ takes the formΣ( γ ) = ∆ S rev ( γ ) + X α Z dtR α + α − ( t ) [ χ α + ( γ ( t )) − χ α − ( γ ( t ))] . (14) IV. CONVERGENCE OF THE EXPONENTIAL AVERAGE
Exponential averages, such as the one in Eqs. (6) and (9), often exhibit poor convergence. Theunderlying reason is the difference between typical and dominant realizations. Typical realizationsare the ones which are likely during the process of interest, and correspond to Σ values in the vicinityof the maximum of P (Σ). In contrast, the dominant realizations are those for which e − Σ P (Σ) ismaximal. Jarzynski has discussed the convergence of exponential averages of this type using a gasin an expanding piston as an example [35]. He used the detailed version of the fluctuation theoremto argue that the dominant realizations are actually the (time-reversed) typical realizations of thecorresponding reversed process. In addition he has derived a simple estimate for the number ofrealizations needed for convergence of the exponential average. The purpose of this section is todemonstrate that these considerations also apply to systems with unidirectional transitions, andalso to numerically verify the validity of Eq. (9).To do so we simulate the jump process of Sec. II using the Gillespie algorithm. This algorithmefficiently generates stochastic trajectories with the correct distribution by determining the timeof the next transition, making use of the fact that the waiting times between jumps are distributedexponentially [36]. The transition rates were taken to be R = 3, R = 0 . R = 4, R = 1, R = 0 . R = 2 . R = 1, R = 0 .
78 and R = 2 .
3. The jump process is assumed to be atsteady state. For the parameters above we find p ss (1) ≃ . p ss (2) ≃ . p ss (3) ≃ . p ss (4) ≃ . -20 -10 0 10 20 Σ Σ )e - Σ P( Σ )P(- Σ) -40 -30 -20 -10 0 10 20 30 40 Σ Σ )e −Σ P( Σ )P(- Σ ) FIG. 2: (Color online) The distribution of Σ in the jump process depicted in Fig. 1 (black solid line). Thedashed red line depicts e − Σ P (Σ) calculated from P (Σ). The dashed-dotted green line correspond to thedistribution P ( − Σ) obtained from simulations of the auxiliary dynamics. The results in the left panel arefor t f = 5, whereas the right panel is calculated for t f = 20. The numerically computed probability distribution of Σ is depicted in the left panel of Fig. 2for t f = 5. This distribution was generated from 10 different realizations of the process. Thispanel also depicts e − Σ P (Σ), which was calculated from P (Σ). In addition it shows the distribution P ( − Σ), which was calculated by numerical simulation of the auxiliary dynamics with the suitableinitial condition ( p ss ). The latter two curves are expected to be identical. They are indeed veryclose to each other, and the differences between them are possibly due to a combination of imperfectsampling of the tail of P (Σ) and of errors introduced by binning the sparsely sampled region in thetail of P (Σ). The spike at Σ = 0 is due to a discrete contribution to the probability density fromthe trajectories which start at states 1 or 3 and never make a jump during the whole process. Thereare also discrete contributions at R t f and − R t f from trajectories spending the whole time atstates 4 and 2 respectfully. The weights of these contributions is smaller compared to the one atΣ = 0 because of the specific transition rates used in the numerics. In contrast to these discretefeatures, trajectories which make jumps lead to a continuous distribution due to the continuousnature of jump times. Therefore the relative weight of discrete contributions to P (Σ) is reducedwhen t f is increased. We note in passing that while the process we are interested in is stationary,the corresponding reference auxiliary is not stationary since p ss = p ss .The same curves are presented in the right panel of Fig. 2 for t f = 20. A comparison of the twopanels shows that when t f is increased the dominant realizations are pushed further into the tailsof P (Σ). This is easily seen from the amount of overlap between P (Σ) and P ( − Σ). Larger values1of t f will show even less overlap.This reduced amount of overlap has a direct impact of the probability to reliably obtain dom-inant realizations with the correct weights, and hence on the convergence of exponential averageEq. (9). For t f = 5 the dominant region is well sampled by the simulation. In contrast, when t f = 20 it is clear that the dominant region is only partially sampled, and the sampling is rathernoisy. In some spots the simulation of the original process gives weights which are too low or twohigh. The leftmost part of the dominant region was never sampled. As a result we expect that asimulation based calculation of (cid:10) e − Σ (cid:11) will give reasonable results for t f = 5 and somewhat poorresults for t f = 20. Sampling of the latter can be improved by adding more realizations.To check the validity and convergence of the exponential average (9) we have used ensembles of3 × , 3 × and 3 × realizations of the jump process of Sec. II. Here R = 0 .
3, while therest of the transition rates are identical to those used to generate Fig. 2. The results are presentedin Fig. 3. The ensemble was divided into 30 sub-ensembles which were summed separately andused to generated an effective standard deviation measuring the fluctuations between different sub-ensembles. It is clear that good convergence is obtained for small t f where the average is close to 1and the standard deviation is small. When t f is increased the fluctuations between sub-ensemblesbecome noticeable. When t f is increased even further, and the dominant region is pushed furtherinto the tail of P (Σ), the dominant region is typically under-sampled and the numerical simulationreturns an average value which is substantially smaller than 1. Occasionally, this region is oversampled and then sometimes the simulation returns a value which can be larger than 1. This isthe hallmark of poorly converged exponential averages of this type.An estimate for the number of realizations needed for convergence was derived by Jarzynski[35]. For the jump process studied here it is given by N ∗ ≈ e Σ typ , (15)where Σ typ is the typical value of Σ in the auxiliary dynamics. This is an approximate criterion,and we will further simplify it by estimating Σ typ as if the auxiliary dynamics is at steady state.As was mentioned earlier this is not true since the auxiliary dynamics exhibits transient relaxationtowards its steady state. We nevertheless make this approximation and obtainΣ typ ≈ [ q ssrev + R ( p ss (2) − p ss (4))] t f ≃ . t f . (16)By substituting Eq.(16) in Eq. (15) one can calculate the value t ∗ f = ln N/ . t f < t ∗ f . The three2 t f < e − Σ > N=3x10 N=3x10 N=3x10 FIG. 3: (Color online) Numerical estimate of the exponential average (9), obtained from N = 3 × (squares), 3 × (circles) and 3 × (triangles) realizations as a function of t f . The error bars areestimates of the standard deviation obtained by dividing the ensemble into 30 sub-ensembles and summingeach one separately. The thick vertical lines depict the approximate criterion for convergence, derived fromEqs. (15) and (16), for N = 3 × (red, dashed), 3 × (solid, black) and 3 × (dashed-dotted, blue). vertical lines (dashed, solid, and dash-dot) in Fig. 3 represent t ∗ f for N = 3 × , 3 × and 3 × ,respectively. Since the estimate is approximate, and we further ignored the possible contribution oftransients, we only expect this estimate to work qualitatively. Indeed all lines are roughly locatedin the transition region between times where the exponential average converges and the regionwhere it does not. Overall the numerical results presented in Fig. 3 qualitatively agree with ourexisting understanding of the difficulties in numerical estimation of exponential averages. V. SUMMARY
The stochastic coarse-grained dynamics of small systems in contact with external thermal reser-voirs exhibit microreversibility which ultimately stems from the time reversal symmetry of the un-derlying deterministic evolution. Nevertheless, there are situations in which it is useful to considermodels which violate microreversibility. We have studied the fluctuations of jump processes in a3system with one or more unidirectional rates. Such systems violate microreversibility and can beviewed as motivated by physical processes such as spontaneous relaxation in quantum systems.The usual formulation of fluctuation theorems can not be used for irreversible systems since itinvolves contributions to the entropy production of the form ln R ji R ij which are not always defined.For such systems one can compare the dynamics to that of an auxiliary system in which theunidirectional transitions are flipped. However, this auxiliary dynamics has no simple physicalinterpretation, and therefore it is of interest to identify measures of fluctuations which can beexpresses only in terms of the physical system. We have shown that it is possible to derive suchan integral fluctuation, Eq. (9), which has a simple and appealing physical interpretation. Therealization dependent quantity, Σ, which appears in the exponent of Eq. (9) is a sum of a welldefined entropy production due to bidirectional transitions and a dynamical term which dependson the residence times in the states connected by the unidirectional transition. It can therefore beviewed as including both thermodynamic and dynamical contributions.The validity of Eq. (9) was checked numerically using simulations of the jump process. Itis well known that exponential averages show poor convergence when dominant realizations areinsufficiently sampled. This was discussed in detail for systems exhibiting microreversibility [35]and our numerical results suggest that the same considerations can be applied also for systems withunidirectional transitions. When the numerical results converge they indeed support the validityof the integral fluctuation theorem. Acknowledgments
We thank Christopher Jarzynski and Massimiliano Esposito for illuminating discussions. SR isgrateful for support from the Israel Science Foundation (grant 924/11) and the US-Israel BinationalScience Foundation (grant 2010363). This work was partially supported by the COST ActionMP1209. UH acknowledges support from the Indian Institute of Science, Bangalore, India. [1] D. J. Evans, E. G. D. Cohen, and G. P. Morriss,
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