Toward relaxation asymmetry: Heating is faster than cooling

An asymmetry in thermal relaxation toward equilibrium has been uncovered for Langevin systems near stable minima Phys. Rev. Lett. 125, 110602 (2020). It has been shown that, given the same degree of nonequilibrium of the initial distributions, relaxation from a lower temperature state (heating) is faster than that from a higher temperature state (cooling). In this paper, we elucidate this relaxation asymmetry for discrete-state Markovian systems described by the master equation. We rigorously prove that heating is faster than cooling for arbitrary two-state systems, whereas for systems with more than two distinct energy levels, the relaxation asymmetry is no longer universal. Furthermore, for systems whose energy levels degenerate into two energy states, we find that there exist critical thresholds of the energy gap. Depending on the magnitude of the energy gap, heating can be faster or slower than cooling, irrespective of the transition rates between states. Our results clarify the relaxation asymmetry for discrete-state systems and reveal several hidden features inherent in thermal relaxation.