Non-equilibrium Steady States with a Spatial Markov Structure

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 Large deviations and additivity principle for the open harmonic process, (2023), (JSP 191(1):10, 2024). 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 in Carinci Gioia, Franceschini Chiara, Frassek Rouven, Giardinà Cristian, Redig Frank. Large deviations and additivity principle for the open harmonic process, (2023), in the context of the harmonic model.