A Nernst heat theorem for nonequilibrium jump processes

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.