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

Abstract

Using local detailed balance we rewrite the Kirchhoff formula for stationary distributions of Markov jump processes in terms of a physically interpretable treeensemble. We use that arborification of path-space integration to derive a McLennantree 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 dynamical activity 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. For pedagogical purposes, we add simple examples to illustrate and explain the various graphical elements.

The Sun within: active processes from two-temperature models

“…where M = y m(y), T is the set of all spanning trees, T η is a spanning tree rooted at η (all edges are directed toward η) and m(T η ) is the weight of the directed spanning tree T η ; see [53,54].…”

“…We prove the uniformization (III.5) by using a graphical representation of the stationary probability law ρ s . In general, the stationary solution of the Master Equation for a Markov jump process can be represented by the Kirchhoff formula where M = ∑ y m ( y ), 𝒯 is the set of all spanning trees, 𝒯 η is a spanning tree rooted at η (all edges are directed toward η ) and m (𝒯 η ) is the weight of the directed spanning tree 𝒯 η ; see [53, 54].…”

Exact computation of heat capacities for active particles on a graph

“…Another reason for our choice of excess heat, is that it avoids discussions related to entropy and the second law. For further discussion on this issue, see [14,15].…”

“…The matrix-tree theorem states that the solution to (15) may be expressed in terms of weights of the spanning trees of that graph [17][18][19]. The Kirchhoff formula, making use of the matrix-tree theorem, gives a way to represent the solution to equation (15) in terms of spanning tree weights By a tree, we mean a connected cycle-less component of a graph, and a spanning tree is a tree including all vertices of the graph. A tree can be defined by giving all of its constituent edges.…”

“…Thus, to find ρ(i), we simply need to find the sets of all spanning trees rooted in i for all i ; a simple example is shown in figure 2, for more examples and details see [20]. These trees can be generated algorithmically, yet the computational complexity increases sharply with the number of states.…”