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

Abstract

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.

v4=final version

References in corpus (16)

• The large deviation approach to statistical mechanics
• Non equilibrium steady states: fluctuations and large deviations of the density and of the current
• Dynamic first-order phase transition in kinetically constrained models of glasses
• Fluctuation theorems for stochastic dynamics
• First-order dynamical phase transition in models of glasses: an approach based on ensembles of histories
• Probability currents as principal characteristics in the statistical mechanics of non-equilibrium steady states
• Construction of a Coordinate Bethe Ansatz for the asymmetric simple exclusion process with open boundaries
• Entanglement in the XX spin chain with an energy current
• Invariant quantities in shear flow
• A selection of nonequilibrium issues
• Thermodynamic formalism and large deviation functions in continuous time Markov dynamics
• Reflection positivity and phase transitions in lattice spin models
• Properties of a non-equilibrium heat bath
• On and beyond entropy production: the case of Markov jump processes
• Slow relaxation, dynamic transitions and extreme value statistics in disordered systems
• The various facets of random walk entropy

Cited by in corpus (9)

• Trajectory phase transitions, Lee-Yang zeros, and high-order cumulants in full counting statistics
• Maximum Caliber: a general variational principle for dynamical systems
• A derivation of the master equation from path entropy maximization
• Revisiting the Ruelle thermodynamic formalism for Markov trajectories with application to the glassy phase of random trap models
• Generalized optimal paths and weight distributions revealed through the large deviations of random walks on networks
• Modeling of biomolecular machines in non-equilibrium steady states
• Inverse problem in the conditioning of Markov processes on trajectory observables : what canonical conditionings can connect two given Markov generators ?
• Statistical optimization for passive scalar transport: maximum entropy production vs maximum Kolmogorov-Sinay entropy
• Microcanonical ensemble out of equilibrium