Non-equilibrium steady states with a spatial Markov structure
Frank Redig, Berend van Tol
TL;DR
This work analyzes non-equilibrium steady states (NESS) of boundary-driven chains through representations as mixtures of equilibrium product measures with a spatial Markov hidden parameter. It shows that, under natural scaling and translation invariance, the entire family of n-site mixture densities is determined by a single-site density and reduces to ordered Dirichlet distributions. The authors develop a two-sided Markov framework, prove uniqueness and provide a recursive construction for all n, and establish that Dirichlet-type mixtures are the only invariant solutions under the stated symmetries. The results connect the microscopic Markov structure of hidden parameters to macroscopic non-equilibrium features, such as additivity principles and long-range correlations, in a rigorous setting.
Abstract
We investigate the structure of non-equilibrium steady states (NESS) for a class of exactly solvable models in the setting of a chain with left and right reservoirs. Inspired by recent results on the harmonic model, we focus on models in which the NESS is a mixture of equilibrium product measures, and where the probability measure which describes the mixture has a spatial Markovian property. We completely characterize the structure of such mixture measures, and show that under natural scaling and translation invariance properties, the only possible mixture measures are coinciding with the Dirichlet process found earlier in the context of the harmonic model.