  An asymmetry in thermal relaxation toward equilibrium has been uncovered for
Langevin systems near stable minima [Phys. Rev. Lett. 125, 110602 (2020)]. It
is conjectured that given the same degree of nonequilibrium of the initial
states, relaxation from a lower-temperate state (warming) is always faster than
that from a higher-temperature state (cooling). Here we elucidate this
conjecture for discrete-state systems undergoing Markovian dynamics. We
rigorously prove that warming is always faster than cooling for arbitrary
two-state systems, whereas for systems with more than two distinct energy
levels, the conjecture is, in general, no longer valid. Moreover, for systems
whose energy levels are degenerate to two energy states, we find that there
exist critical thresholds of the energy gap. Depending on the magnitude of this
energy gap, warming can be definitely faster or slower than cooling,
irrespective of the transition rates between states. Our results clarify the
conjecture for discrete-state systems and reveal several hidden features
inherent in thermal relaxation.
