Stochastic interacting particle systems out of equilibrium

Abstract

This paper provides an introduction to some stochastic models of lattice gases out of equilibrium and a discussion of results of various kinds obtained in recent years. Although these models are different in their microscopic features, a unified picture is emerging at the macroscopic level, applicable, in our view, to real phenomena where diffusion is the dominating physical mechanism. We rely mainly on an approach developed by the authors based on the study of dynamical large fluctuations in stationary states of open systems. The outcome of this approach is a theory connecting the non equilibrium thermodynamics to the transport coefficients via a variational principle. This leads ultimately to a functional derivative equation of Hamilton-Jacobi type for the non equilibrium free energy in which local thermodynamic variables are the independent arguments. In the first part of the paper we give a detailed introduction to the microscopic dynamics considered, while the second part, devoted to the macroscopic properties, illustrates many consequences of the Hamilton-Jacobi equation. In both parts several novelties are included.

Clausius Inequality and Optimality of Quasistatic Transformations for Nonequilibrium Stationary States

“…It has been very powerful in studying concrete microscopic models but can be used also as a phenomenological theory. It has led to several new interesting predictions [2][3][4][5][6][7].…”

“…We thus consider an open system in contact with boundary reservoirs, characterized by their chemical potential λ, and under the action of an external field E. We denote by Λ ⊂ R d the bounded region occupied by the system, by x the macroscopic space coordinates and by t the macroscopic time. With respect to our previous work [1,3,4], we here consider the case in which λ and E depend explicitly on the time t, driving the system from a nonequilibrium state to another one. The macroscopic dynamics is given by the hydrodynamic equation for the density which satisfies the following general assumptions based on the notion of local equilibrium.…”

“…Consider a system with time independent driving and let (û(t),(t)), t ∈ [T 1 , T 2 ] be a pair density-current satisfying the continuity equation ∂ tû + ∇ · = 0. According to the basic principles of the macroscopic fluctuation theory [1,3,4], the probability of observing this path is given, up to a prefactor, by exp − ε −d β I [T1,T2] (û,) where ε is the scaling parameter, i.e., the ratio between the microscopic length scale (say the typical intermolecular distance) and the macroscopic one, β = 1/κT (here T is the temperature and κ is Boltzmann's constant), and the action functional I has the form…”

Entropy production in the nonequilibrium steady states of interacting many-body systems

“…On the one hand large deviation functions form the basis of a modern approach to equilibrium statistical mechanics [1][2][3], on the other hand they are increasingly recognized of being of fundamental interest for the characterization of non-equilibrium systems where they are sometimes considered to play a role similar to that played by the free energy at equilibrium. Important examples are given by the large deviations of the steady-state currents [4,5] in systems that are far from equilibrium.…”

The cellular basis of cell sorting kinetics

“…There exist already several mathematical results in this direction, for instance Bertini, De Sole, Gabrielli, Jona-Lasinio and Landim [2007], but for a detailed application to the cell sorting model these results must be developed further.…”