# Gibbsian stationary non equilibrium states

## TL;DR

This paper investigates the structure of stationary non-equilibrium states in interacting particle systems using geometric and algebraic methods, providing new ways to construct models with fixed invariant measures.

## Contribution

It introduces two geometric constructions for analyzing non-reversible transition rates and applies a discrete Hodge decomposition to characterize stationary states.

## Key findings

- Decomposition of currents into gradient, circulation, and harmonic parts.
- Construction of models with prescribed invariant measures using divergence free flows.
- A novel, constructive approach to understanding non-equilibrium stationary states.

## Abstract

We study the structure of stationary non equilibrium states for interacting particle systems from a microscopic viewpoint. In particular we discuss two different discrete geometric constructions. We apply both of them to determine non reversible transition rates corresponding to a fixed invariant measure. The first one uses the equivalence of this problem with the construction of divergence free flows on the transition graph. Since divergence free flows are characterized by cyclic decompositions we can generate families of models from elementary cycles on the configuration space. The second construction is a functional discrete Hodge decomposition for translational covariant discrete vector fields. According to this, for example, the instantaneous current of any interacting particle system on a finite torus can be canonically decomposed in a gradient part, a circulation term and an harmonic component. All the three components are associated to functions on the configuration space. This decomposition is unique and constructive. The stationary condition can be interpreted as an orthogonality condition with respect to an harmonic discrete vector field and we use this decomposition to construct models having a fixed invariant measure.