The Metastable Mpemba Effect Corresponds to a Non-monotonic Temperature Dependence of Extractable Work

The Mpemba effect refers to systems whose thermal relaxation time is a non-monotonic function of the initial temperature. Thus, a system that is initially hot cools to a bath temperature more quickly than the same system, initially warm. In the special case where the system dynamics can be described by a double-well potential with metastable and stable states, dynamics occurs in two stages: a fast relaxation to local equilibrium followed by a slow equilibration of populations in each coarse-grained state. We have recently observed the Mpemba effect experimentally in such a setting, for a colloidal particle immersed in water. Here, we show that this metastable Mpemba effect arises from a non-monotonic temperature dependence of the maximum amount of work that can be extracted from the local-equilibrium state at the end of Stage 1.