Macroscopic Thermodynamic Framework for the Mpemba Effect

TL;DR

This paper develops a macroscopic thermodynamic theory based on irreversible thermodynamics to explain the Mpemba effect, revealing how initial conditions and structural evolution influence anomalous cooling behaviors.

Contribution

It introduces a generalized Newton's law incorporating a memory term, providing a universal framework to predict the Mpemba effect in complex systems.

Findings

The coefficient $ ext{M}$ determines the occurrence of the Mpemba effect.

The framework maps phase diagrams for different systems exhibiting the effect.

Provides criteria for anomalous relaxation behaviors based on thermodynamic parameters.

Abstract

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.