Network representations of non-equilibrium steady states: Cycle decompositions, symmetries and dominant paths

TL;DR

This paper introduces an iterative cycle decomposition method for non-equilibrium steady states of Markov processes, revealing symmetries and dominant paths, with applications to models like TASEP and mass transit dynamics.

Contribution

It presents a novel cycle decomposition approach that satisfies detailed balance and expresses observables as cycle averages, advancing understanding of non-equilibrium steady states.

Findings

Cycle decomposition captures steady state fluxes effectively.

Symmetries reduce the number of cycles needed.

Changes in dominant paths cause structural shifts in cycles.

Abstract

Non-equilibrium steady states (NESS) of Markov processes give rise to non-trivial cyclic probability fluxes. Cycle decompositions of the steady state offer an effective description of such fluxes. Here, we present an iterative cycle decomposition exhibiting a natural dynamics on the space of cycles that satisfies detailed balance. Expectation values of observables can be expressed as cycle "averages", resembling the cycle representation of expectation values in dynamical systems. We illustrate our approach in terms of an analogy to a simple model of mass transit dynamics. Symmetries are reflected in our approach by a reduction of the minimal number of cycles needed in the decomposition. These features are demonstrated by discussing a variant of an asymmetric exclusion process (TASEP). Intriguingly, a continuous change of dominant flow paths in the network results in a change of the structure of cycles as well as in discontinuous jumps in cycle weights.