Trees and forests for nonequilibrium purposes: an introduction to graphical representations

TL;DR

This paper introduces graphical tree-based methods to analyze nonequilibrium Markov processes, providing new formulas for stationary distributions, response, and quasi-potentials with applications in thermal physics.

Contribution

It develops a novel graphical framework using trees and forests to represent nonequilibrium stationary states and related quantities, extending classical formulas with new interpretability.

Findings

Derived a tree-ensemble representation of stationary distributions

Established response formulas in large driving regimes

Connected graph-theoretic structures to physical quantities like heat and work

Abstract

Using local detailed balance we rewrite the Kirchhoff formula for stationary distribution of Markov jump processes in terms of a physically interpretable tree-ensemble. We use that arborification of path-space integration to derive a McLennan-tree characterization close to equilibrium, as well as to obtain response formula for the stationary distribution in the asymptotic regime of large driving. Graphical expressions of currents and of traffic are obtained, allowing the study of various asymptotic regimes. Finally, we present how the matrix-forest theorem gives a representation of quasi-potentials, as used e.g. for computing excess work and heat in nonequilibrium thermal physics. A variety of examples illustrate and explain the graph elements and constructions.