Drazin-inverse and heat capacity for driven random walkers on the ring

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