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.

We study a class of stochastic models of mass transport on discrete vertex set V. For these models, a one-parameter family of homogeneous product measures ⊗i∈Vνθ is reversible. We prove that the set of mixtures of inhomogeneous product measures with equilibrium marginals, i.e., the set of measures of the form (Formula presented.) is left invariant by the dynamics in the course of time, and the “mixing measure” Ξ evolves according to a Markov process which we then call “the hidden parameter model”. This generalizes results from De Masi et al. (Preprint arXiv:2310.01672, 2023) to a larger class of models and on more general graphs. The class of models includes discrete and continuous generalized KMP models, as well as discrete and continuous harmonic models. The results imply that in all these models, the non-equilibrium steady state of their reservoir driven version is a mixture of product measures where the mixing measure is in turn the stationary state of the corresponding “hidden parameter model”. For the boundary-driven harmonic models on the chain {1,…,N} with nearest neighbor edges, we recover that the stationary measure of the hidden parameter model is the joint distribution of the ordered Dirichlet distribution (cf. Carinci et al., Preprint arXiv:2307.14975, 2023), with a purely probabilistic proof based on a spatial Markov property of the hidden parameter model.