Drazin-Inverse and heat capacity for driven random walks on the ring

TL;DR

This paper introduces a novel method combining algebraic and graph-theoretical techniques to exactly compute the nonequilibrium heat capacity of driven Markov jump processes on a ring, including the diffusion limit.

Contribution

It presents a new graphical representation of the Drazin-inverse of the generator, enabling precise calculations of excess heat in nonequilibrium Markov processes.

Findings

Exact computation of nonequilibrium heat capacity

Effective analysis of diffusion limit as N approaches infinity

Novel use of graph-theoretical methods in Markov process analysis

Abstract

We apply single and double tree-like representations of Markov jump processes on $\mathbb{Z}_N$ for obtaining their nonequilibrium heat capacity and for taking the diffusion limit $N\uparrow \infty$. The main tool is a graphical representation of the Drazin-inverse of the backward generator. In that way, the combination of algebraic and graph-theoretical approaches enables an exact computation of an important nonequilibrium quantity (excess heat) for Markov jump processes.