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.

We give a graphical representation for the Drazin-inverse of the backward generator L for biased random walkers on Z N . From the matrix-forest theorem an algorithm is constructed for computing the pseudo-potential V , solution of LV = f for any function f on Z N having zero stationary expectation. As an application to thermal physics, we give the nonequilibrium heat capacity as function of bias and temperature. Finally, we discuss the diffusion limit N ↑ ∞.

The notion of a nonequilibrium heat capacity is important for bio-energetics and for calorimetry of active materials more generally. It centers around the notion of excess heat or excess work dissipated during a quasistatic relaxation between different nonequilibrium conditions. We give exact results for active random walks moving in an energy landscape on a graph, based on calculations employing the matrix-tree and matrix-forest theorems. That graphical method applies to any Markov jump process under the physical condition of local detailed balance.

We generalize a version of the Third Law of Thermodynamics to nonequilibrium processes described as Markov jump dynamics on a finite connected graph. In addition to the stationary heat flux, after a small change in an external parameter, an excess heat flows into the thermal bath at temperature T during relaxation. We state conditions under which that excess heat and the nonequilibrium heat capacity vanish as T ↓ 0. The interpretation is that typical relaxation paths towards dominant states do not dramatically differ from each other in their dissipated heat.

We discuss via general arguments and examples when and why the steady nonequilibrium heat capacity vanishes with temperature. The framework is that of Markov jump processes on finite connected graphs where the condition of local detailed balance allows to identify the heat fluxes, and where the discreteness more easily enables sufficient nondegeneracy of the stationary distribution at absolute zero, as under equilibrium. However, for the nonequilibrium extension of the Third Law of Thermodynamics, a dynamic condition is needed as well: the low-temperature dynamical activity and accessibility of the dominant state must remain sufficiently high so that relaxation times do not start to dramatically differ between different initial states. It suffices that the relaxation times do not exceed the dissipation time.