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

Filyokov and Karpov [Inzhenerno-Fizicheskii Zhurnal 13, 624 (1967)] 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 to 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 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 $W_{i \to j}$ between configurations $i$ and $j$ : instead of choosing the dynamical rules $W_{i \to 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 [Phys. Rev. Lett. 92, 150601 (2004)] under the name 'Non-equilibrium Counterpart to detailed balance', and with the 'invariant quantities' derived by Baule and Evans [Phys. Rev. Lett. 101, 240601 (2008)], but provide a slightly different perspective via the formulation in terms of an eigenvalue problem.