Nonequilibrium Microcanonical and Canonical Ensembles and Their Equivalence

Abstract

Generalizations of the microcanonical and canonical ensembles for paths of Markov processes have been proposed recently to describe the statistical properties of nonequilibrium systems driven in steady states. Here, we propose a theory of these ensembles that unifies and generalizes earlier results and show how it is fundamentally related to the large deviation properties of nonequilibrium systems. Using this theory, we provide conditions for the equivalence of nonequilibrium ensembles, generalizing those found for equilibrium systems, construct driven physical processes that generate these ensembles, and rederive in a simple way known and new product rules for their transition rates. A nonequilibrium diffusion model is used to illustrate these results.

Classical stochastic dynamics and continuous matrix product states: gauge transformations, conditioned and driven processes, and equivalence of trajectory ensembles

“…As we explain below, this approach enables to derive in a compact and concise way many (known) properties of trajectory ensembles associated to dynamical large-deviations [3][4][5][6] and non-equilibrium fluctuations [7,8].…”

Numerical comparison of a constrained path ensemble and a driven quasisteady state

“…(2), this extra constraint on the (time-reversal-symmetric) potential energy density only has the effect of confining Eq. (2) to a set of microstates of equal potential energy density [5].…”

“…Such an ensemble is appealing in that it shares many features of an equilibrium ensemble, and admits elegant techniques for investigation of it properties, both in the case where the constrained dynamical quantity is antisymmetric under time reversal (a flux) [4,5,9], and where it is symmetric (a "dynamical activity") [4,5,[10][11][12][13][14][15][16]. However, it remains unclear whether such ensembles are realized in practise, i.e.…”

“…(2) no longer holds for all microstates i, but continues to hold for all microstates of equal potential energy [5].…”

“…A class of far-fromequilibrium system, that shares the Hamiltonian of an equilibrium system, can be defined by a subset of the equilibrium ensemble of phase-space trajectories, conditioned by a finite flux [2][3][4][5]. That is to say, those members of the equilibrium ensemble of systems, which exhibit a given flux during some time interval, are defined as belonging to the non-equilibrium constrained-flux ensemble.…”