On thermalization of a system with discrete phase space

We investigate the thermalization of a stochastic system with a discrete phase space, initially at equilibrium at temperature T i and then thermalizing in an environment at temperature T f, considering both cases T i > T f and T i < T f. For the simple case of a system with constant energy gaps, we show that the relation between the time scales of the cooling and heating processes is not univocal and depends on the magnitude of the energy gap itself. Specifically, the eigenvalues of the corresponding stochastic matrix determine the relaxation time scales: for large energy gaps, the cooling process exhibits the shortest relaxation times to equilibrium, whereas for small gaps, the heating process is faster at all scales. To quantify the degree of thermalization, we consider both the Kullback–Leibler divergence and the Fisher information (along with related quantities). In the intermediate-to-long-time regime, both quantities convey the same type of information regarding the rate of thermalization and follow the ordering predicted by the dynamical eigenvalues. We then examine a more complex system with a more intricate stochastic matrix—namely, a 1D Ising model—and confirm the existence of two regimes, one in which cooling becomes faster than heating. Finally, we relate our findings to a previous work in which a harmonic oscillator was used as the working fluid, where the heating process was always found to be faster than the cooling one.