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 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 ↑ ∞.

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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, and is not restricted to the examples given in this paper.