Nonequilibrium chemical potentials of steady-state lattice gas models in contact: A large-deviation approach

Abstract

We introduce a general framework to describe the stationary state of two driven systems exchanging particles or mass through a contact, in a slow exchange limit. The definition of chemical potentials for the systems in contact requires that the large-deviations function describing the repartition of mass between the two systems is additive, in the sense of being a sum of contributions from each system. We show that this additivity property does not hold for an arbitrary contact dynamics, but is satisfied on condition that a macroscopic detailed balance condition holds at contact, and that the coarse-grained contact dynamics satisfies a factorization property. However, the nonequilibrium chemical potentials of the systems in contact keep track of the contact dynamics, and thus do not obey an equation of state. These nonequilibrium chemical potentials can be related either to the equilibrium chemical potential, or to the nonequilibrium chemical potential of the isolated systems. Results are applied both to an exactly solvable driven lattice gas model, and to the Katz-Lebowitz-Spohn model using a numerical procedure to evaluate the chemical potential. The breaking of the additivity property is also illustrated on the exactly solvable model. arXiv:1909.00432v2 [cond-mat.stat-mech] 2 Dec 2019with (∆N A ) c referring to the sum over configurations C A , C B that, respectively, contain N (C A ) = ρ A V A + ∆N A and N (C B ) = ρ B V B − ∆N A particles (the second sum being exactly the same with ∆N A = 0).

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

“…A more detailed discussion of this point, in particular regarding the scaling assumptions underlying Eq. ( 16), can be found in [22]. For later convenience, we repeat here the expression of the coarse-grained transition rates ϕ(ρ A , ∆N A ), using explicit notations:…”

“…A more detailed discussion on the derivation of this Hamilton-Jacobi equation starting from the coarse-grained master equation ( 14) can be found in [22].…”

“…The large deviations function I(ρ A |ρ) is a convenient tool to define non-equilibrium chemical potentials [15,16,17,18,19,20,21,22] As mentioned in the introduction, the definition of a non-equilibrium chemical potential relies on an additivity condition on the large deviations function, which reads [15,16,20,22]…”

“…A sufficient condition for the additivity property (38) to hold has been found in [20,22]. It relies both on the macroscopic detailed balance condition (21),…”

“…Such an approach actually relies on a additivity property of the large deviations function describing density fluctuations in subsystems [15,16,17,18,19]. Recent works have more explicitly focused on the properties of the non-equilibrium chemical potential, which turns out to generically depend on the contact dynamics and to lack an equation of state [20,21,22]. The validity of the additivity condition has been traced back to both a macroscopic detailed balance condition at contact and a factorization of the contact dynamics [20,22].…”