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Fluctuation Theorem for Nonequilibrium Reactions Journal of Chemical Physics 120 (2004) 8898-8905 Fluctuation theorem for nonequilibrium reactions Pierre Gaspard Center for Nonlinear Phenomena and Complex Systems, Universit´eLibre de Bruxelles, Code Postal 231, Campus Plaine, B-1050 Brussels, Belgium A fluctuation theorem is derived for stochastic nonequilibrium reactions ruled by the chemical master equation. The theorem is expressed in terms of the generating and large-deviation functions characterizing the fluctuations of a quantity which measures the loss of detailed balance out of thermodynamic equilibrium. The relationship to entropy production is established and discussed. The fluctuation theorem is verified in the Schl¨oglmodel of far-from-equilibrium bistability. PACS numbers: 82.20.Uv; 05.70.Ln; 02.50.Ey I. INTRODUCTION Reacting systems can be driven out of equilibrium when in contact with several particle reservoirs or chemiostats generating fluxes of matter across the system. The fluxes are caused by the differences of chemical potentials between the chemiostats. Such an open system may be thought of as a reactor with inlets for reactants and an outlet for the products. In this case, the open system is driven out of equilibrium at the boundaries with the chemiostats. Even if detailed balance is satisfied for all the reactions in the bulk of the reactor, the nonequilibrium boundary conditions will break detailed balance for the reactions establishing the contact with the chemiostats. According to the second law of thermodynamics, the resulting nonequilibrium states are characterized by the production of entropy inside the open system. Above the nanoscale, the reactions taking place in the system can be described in terms of the numbers of molecules of the different species. These numbers undergo random jumps on time scales longer than the time scale of the elementary reacting steps because of the randomness of the elastic and inelastic collisions among the molecules composing the system. The random jumps in the numbers of reacting particles form a stochastic process ruled by a master equation proposed in the seventies by Nicolis and coworkers1–4. This master equation obeys a H-theorem, which leads to a mesoscopic expression for the entropy production in terms of probability distributions5,6. In this theoretical framework, the master equation rules the relaxation of the probability distribution toward an invariant distribution called a nonequilibrium steady state even if the open system is driven far from equilibrium in a regime which would be identified as time-dependent in the macroscopic description. This invariant probability distribution describes the molecular fluctuations around the trajectory of the macroscopic (deterministic) time evolution. In a stroboscopic observation at regular time intervals, the system randomly visit successive states, forming a path in the state space. The probability of such a path can be compared with the probability of the time-reversed path. If the system is at equilibrium, both probabilities are in balance on average since detailed balance holds. In contrast, if the system is driven out of equilibrium, the probabilities are no longer in balance. This nonequilibrium loss of balance manifests itself on average as well as for individual paths. The purpose of the present paper is to show that the ratio of the forward- and backward-path probabilities decays on average according to the entropy production of the nonequilibrium reactions taking place in the open system and that, moreover, the large deviations of this ratio with respect to its average behavior obey a remarkable property called a fluctuation theorem. This result has its origins in recent work on a symmetry property for Markov stochastic processes obtained by Kurchan7, Lebowitz and Spohn8, as well as Maes9, who were inspired by previous work on dynamical chaos in the many-particle systems of statistical mechanics10–16. Here, our goal is to establish a fluctuation theorem for reactions in NESS including far-from-equilibrium situations. This fluctuation theorem is derived in the framework of the master equation by Nicolis and coworkers and is applied to a bistable far-from-equilibrium reaction. The paper is organized as follows. Section II describes the theoretical framework used here for describing nonequi- librium reactions. The fluctuation theorem is obtained in Sec. III. In Sec. IV, the fluctuation theorem is applied to the Schl¨ogltrimolecular model of bistability17,18. Conclusions are drawn in Sec. V 2 II. STOCHASTIC APPROACH TO NONEQUILIBRIUM REACTIONS A. Reaction as a stochastic process We consider a network of reactions ρ = 1, 2, ..., r: a c a c k X i i X j j +ρ X i i X j j ν<ρ A + ν<ρ X ν>ρ A + ν>ρ X (1) k−ρ i=1 j=1 i=1 j=1 j c i a involving the intermediate species {X }j=1 and the species {A }i=1 supplied at constant concentration by reservoirs, also called chemiostats. We impose that the reactions take place in a homogeneously system such as a continuously stirred tank reactor connected to the chemiostats. The system is assumed to be isothermal. A reversed reaction is associated with each reaction so that each process can be balanced by the reversed process. An elementary event of the reaction ρ ∈ {±1, ±2, ..., ±r} changes the integer numbers Xj ∈ N of molecules of the intermediate species Xj by an amount equal to the stoichiometric coefficients j j j j νρ ≡ ν>ρ − ν<ρ = −ν−ρ (2) (j = 1, 2, ..., c). N denotes the set of non-negative integer numbers. We introduce the numbers of molecules of the different species Xj inside a fictitious boundary separating the reactor j c c (i.e., the open system) from the chemiostats. We denote by X = {X }j=1 ∈ N the vector of the numbers of molecules j c of the different species and by ν ρ = {νρ}j=1 the vector of the stoichiometric coefficients. An elementary event of the reaction ρ induces the random jump ρ X−→X + ν ρ (3) with the transition rate Wρ(X|X + ν ρ). On the other hand, an elementary event of the reversed reaction −ρ induces the random jump −ρ X−→X − ν ρ (4) with the transition rate W−ρ(X|X − ν ρ). The random jumps and the waiting times between the jumps can be generated by Gillespie’s algorithm19,20. If the system is in the state X the next reaction ρ to occur is a random variable with probabilities W (X|X + ν ) P = ρ ρ (5) ρ P±r ρ0=±1 Wρ0 (X|X + ν ρ0 ) while the waiting time τ is another random variable which is exponentially distributed with the probability density ±r X p(τ) = κ exp(−κτ) with κ = Wρ0 (X|X + ν ρ0 ) (6) ρ0=±1 The successive jumps are statistically independent. For reactions among diluted species, the transition rates are given by the mass-action law according to1–4 a c j j j j j Y i νi Y X X − 1 X − 2 X − ν<ρ + 1 W (X|X + ν ) = Ω k [A ] <ρ ··· (7) ρ ρ ρ Ω Ω Ω Ω i=1 j=1 (ρ = ±1, ±2, ..., ±r) where Ω is the so-called extensivity parameter which is proportional to the total number of i particles in the open system, while kρ is the reaction coefficient. In the case the reaction ρ involves some species A supplied by a chemiostat, the transition rate is proportional to the product of the concentrations [Ai] of the species Ai from the chemiostats. 3 B. Master equation The master equation ruling the reactive stochastic process described here above is given by1–4 ±r d X P (X; t) = [W (X − ν |X) P (X − ν ; t) − W (X|X − ν ) P (X; t)] (8) dt ρ ρ ρ −ρ ρ ρ=±1 where P (X; t) is the probability that the reactor contains the numbers X of molecules of the intermediate species at time t. In Eq. (8), the sum over the reactions runs over both the direct and reversed reactions ρ = ±1, ±2, ..., ±r. In general, the probabilities P (X; t) relax in time toward stationary values Pst(X), defining some nonequilibrium steady state (NESS) or the state of thermodynamic equilibrium. In a stationary state, detailed balance is satisfied for reaction ρ if Wρ(X − ν ρ|X) Pst(X − ν ρ) = W−ρ(X|X − ν ρ) Pst(X) (9) If the system is closed, detailed balance holds for all the reactions ρ ∈ {±1, ±2, ..., ±r} and the stationary state becomes the equilibrium state Pst(X) = Peq(X). If the system is open and in contact with out-of-equilibrium chemiostats, there is no detailed balance for the reactions involving the chemiostats. The loss of detailed balance is a consequence of the nonequilibrium boundary conditions imposed at the contacts of the reactor with the chemiostats. C. Entropy production The entropy of the open system in a state described by the probabilities P (X; t) is given by X X S(t) = S0(X)P (X; t) − P (X; t) ln P (X; t) (10) X X in units where Boltzmann’s constant takes the unit value, kB = 1. The first term of Eq. (10) is the average of the entropy S0(X) of a system with the fixed numbers X of molecules due to the disorder in the degrees of freedom other than the numbers X themselves. The second term of Eq. (10) is the contribution to entropy due to the probability distribution of these numbers X of molecules, i.e., due to the disorder in the probability distribution of the numbers X of molecules. Nicolis and coworkers6 have proved the following H-theorem according to which the time variation of the entropy splits as dS d S d S = e + i (11) dt dt dt deS diS into the entropy flow dt and the entropy production dt .
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