Non-additive large deviation function for the particle densities of driven systems in contact

Abstract

We investigate the non-equilibrium large deviation function of the particle densities in two steady-state driven systems exchanging particles at a vanishing rate. We first derive through a systematic multi-scale analysis the coarse-grained master equation satisfied by the distribution of the numbers of particles in each system. Assuming that this distribution takes for large systems a large deviation form, we obtain the equation (similar to a Hamilton–Jacobi equation) satisfied by the large deviation function of the densities. Depending on the systems considered, this equation may satisfy or not the macroscopic detailed balance property, i.e., a time-reversibility property at large deviation level. In the absence of macroscopic detailed balance, the large deviation function can be determined as an expansion close to a solution satisfying macroscopic detailed balance. In this case, the large deviation function is generically non-additive, i.e., it cannot be split as two separate contributions from each system. In addition, the large deviation function can be interpreted as a non-equilibrium free energy, as it satisfies a generalization of the second law of thermodynamics, in the spirit of the Hatano–Sasa relation. Some of the results are illustrated on an exactly solvable driven lattice gas model.

Nonequilibrium grand-canonical ensemble built from a physical particle reservoir

“…In line with our previous works [27][28][29]38], we consider the following general framework of two systems in contact in the weak exchange rate limit, that we call for short weak contact. Our general set-up consists in two stochastic Markovian systems A and B that exchange particles at a low rate as compared to the characteristic frequency of the internal dynamics of each system.…”

“…We are specifically interested in determining the joint distribution P (ρ A , ρ B ) of particles densities ρ A and ρ B . It has been argued in [27,28,38] that in the weak exchange rate limit, the contact dynamics can be conveniently encoded into a coarse-grained exchange rate ϕ(∆N A ; ρ A , ρ B ) with ∆N A = N A −N A the number of exchanged particles during a single transition, and ρ A and ρ B the densities in each system. In the limit of a large total volume V T = V A + V B , the joint stationary distribution P (ρ A , ρ B ) of the number of particles in systems A and B takes the large deviation form…”

“…In this framework, additivity is satisfied when the contact dynamics is factorized between the two systems [27,28]. When the contact dynamics is not factorized, or when the macroscopic detailed balance relation ( 7) is not obeyed, the large deviation function I(ρ A , ρ B ) is generically non-additive [38].…”

“…Having introduced a proper nonequilibrium framework to define chemical potentials for systems in contact, one can try to define a nonequilibrium grand-canonical ensemble and study whether its properties are equivalent to that of the nonequilibrium canonical (fixed particle number) system. The theoretical framework allowing for the definition of nonequilibrium chemical potentials consists in considering two systems in contact, in the weak exchange rate limit [26][27][28]38] (however, note that interesting phenomena also appear for non-vanishing exchange rates [39]). While the two systems have previously been assumed to have comparable sizes, it is of interest to discuss the case when one of the systems is much larger than the other and plays the role of a particle reservoir.…”

“…In Sec. II, we briefly review the framework introduced in [27,28,38] to describe the steady state of driven systems in contact exchanging particles at a vanishing rate. Then in Sec.…”