A possible classification of nonequilibrium steady states

TL;DR

This paper introduces a graph-theoretic classification of nonequilibrium steady states based on stationary probabilities and probability currents, revealing the diversity of transition rates that produce identical steady states and their implications for entropy production.

Contribution

It provides a novel classification framework for nonequilibrium steady states using graph theory, linking stationary distributions and currents, and analyzing transition rate choices.

Findings

Stationary probabilities represented as directed labelled trees.

Probability currents correspond to loops in the graph.

Multiple transition rate choices can produce the same steady state.

Abstract

We propose a general classification of nonequilibrium steady states in terms of their stationary probability distribution and the associated probability currents. The stationary probabilities can be represented graph-theoretically as directed labelled trees; closing a single loop in such a graph leads to a representation of probability currents. This classification allows us to identify all choices of transition rates, based on a master equation, which generate the same nonequilibrium steady state. We explore the implications of this freedom, e.g., for entropy production.