Macroscopic Thermodynamic Framework for the Mpemba Effect

The counterintuitive Mpemba effect, wherein a hotter system cools faster, critically lacks a general macroscopic theory. Here, starting from linear irreversible thermodynamics, we formulate a generalized Newton's cooling law, $\mathrm{d}T/\mathrm{d}t = -[\gamma_0 + \mathcal{M}Q(t)](T-T_r)$, for a system at temperature $T$ relaxing in a thermal reservoir at $T_r$, where the bare relaxation rate $\gamma_0$ is modified by an initial-state memory term, $Q(t) \propto T(0)-T_r$. Arising from the interplay between heat flux and structural evolution, the coefficient $\mathcal{M}$ governs anomalous relaxation behaviors, where $\mathcal{M} > 0$ ($\mathcal{M} < 0$) induces the (inverse) Mpemba effect. This universal thermodynamic framework maps out phase diagram to provide general criteria for the Mpemba effect in complex systems, offering a macroscopic picture that bridges disparate microscopic approaches.