The Strong Mpemba effect in over-damped Langevin dynamics and connections to Kramer's escape rate

Rapid cooling or heating of a physical system to its environment can lead to unusual thermal relaxation phenomena, which we call anomalous. A prime example of anomalous thermal relaxation is the Mpemba effect. The phenomenon occurs in cooling when a system prepared at a hot temperature overtakes an identical system prepared at a warm temperature and equilibrates faster to the cold environment. A similar effect exists in heating. Comparing two identical physical systems in their relaxation to the environment, we would expect that the system with a smaller mismatch between its own and the environment's temperature will thermalize faster -- yet it is not always the case. The effect was observed in various physical systems, including water, magnetic systems, clathrate hydrates, polymers, and colloidal particle systems. Here study the Mpemba effect in over-damped Langevin dynamics. We build upon our previous results for the case of piecewise-continuous potentials and on the experimental observations of Kumar and Bechhoeffer (Nature, 2020). Our analytical results are within the limit of small diffusion, which is the limit where Kramer's escape rate can be analytically determined. We connect the properties of the potential and Kramer's escape rate with the effect's existence. The continuous case extends and generalizes our previous work.