Trees and Forests for Nonequilibrium Purposes: An Introduction to Graphical Representations

Using local detailed balance we rewrite the Kirchhoff formula for stationary distributions 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.

Automated summary

In this paper, we use the method of local detailed balance to rewrite the Kirchhoff formula for stationary distributions of Markov jump processes. We introduce a physically interpretable tree-ensemble to describe the concept. By arborification of path-space integration, we derive a McLennan-tree characterization close to equilibrium and obtain response formula for the stationary distribution in the asymptotic regime of large driving. Graphical expressions of currents and traffic are obtained, allowing the study of various asymptotic regimes. We also demonstrate how the matrix-forest theorem provides a representation of quasi-potentials, which are used in computing excess work and heat in nonequilibrium thermal physics. Various examples are presented to illustrate and explain the graph elements and constructions.