Non-equilibrium steady states: Maximization of the Shannon entropy associated with the distribution of dynamical trajectories in the presence of constraints

Abstract

Filyokov and Karpov (1967 Inzh.-Fiz.Zh. 13 624) have proposed a theory of non-equilibrium steady states in direct analogy with the theory of equilibrium states: the principle is to maximize the Shannon entropy associated with the probability distribution of dynamical trajectories in the presence of constraints, including the macroscopic current of interest, via the method of Lagrange multipliers. This maximization leads directly to the generalized Gibbs distribution for the probability distribution of dynamical trajectories, and to some fluctuation relation of the integrated current. The simplest stochastic dynamics where these ideas can be applied are discrete-time Markov chains, defined by transition probabilities Wi→j between configurations i and j: instead of choosing the dynamical rules Wi→j a priori, one determines the transition probabilities and the associate stationary state that maximize the entropy of dynamical trajectories with the other physical constraints that one wishes to impose. We give a self-contained and unified presentation of this type of approach, both for discrete-time Markov chains and for continuous-time master equations. The obtained results are in full agreement with the Bayesian approach introduced by Evans (2004 Phys.Rev.Lett.92150601) under the name 'Non-equilibrium Counterpart to detailed balance', and with the 'invariant quantities' derived by Baule and Evans (2008 Phys.Rev.Lett.101 240601), but provide a slightly different perspective via the formulation in terms of an eigenvalue problem. © 2011 IOP Publishing Ltd and SISSA.