Universal trade-off between irreversibility and intrinsic timescale in thermal relaxation with applications to thermodynamic inference

We establish a general lower bound for the entropy production rate (EPR) based on the Kullback-Leibler divergence and the logarithmic-Sobolev constant that characterizes the timescale of relaxation. This bound can be considered as an enhanced second law of thermodynamics. When applied to thermal relaxation, it reveals a universal trade-off relation between the dissipation rate and the intrinsic relaxation timescale. From this relation, a thermodynamic upper bound on the relaxation time between two given states emerges, acting as an inverse speed limit over the entire time region. We also obtain a quantum version of this upper bound, which is always tighter than its classical counterpart, incorporating an additional term due to decoherence. Remarkably, we further demonstrate that the trade-off relation remains valid for any generally non-Markovian coarse-grained relaxation dynamics, highlighting its significant applications in thermodynamic inference. This trade-off relation is a new tool in inferring EPRs in molecular dynamics simulations and practical experiments.