The Mpemba effect — hot water freezing faster than cold under certain conditions — has resisted clean theoretical treatment because most models are either too simple to be convincing or too complex to be analytically tractable. Hayakawa and Takada (arXiv: 2603.24148) build a model that is both.
Their system: an overdamped particle in a two-dimensional radially symmetric bistable potential — two concentric wells with a barrier between them. The potential is piecewise quadratic-logarithmic, which allows exact mapping to a Schrödinger-type eigenvalue problem. The eigenvalues and mode amplitudes can be computed analytically.
The result: the coefficient of the slowest relaxation mode depends non-monotonically on the initial temperature. A system quenched from a higher temperature can have a smaller slowest-mode amplitude than one quenched from a lower temperature. Since the slowest mode dominates late-time relaxation, the hotter system reaches equilibrium first. The Mpemba effect falls out of the mathematics without approximation.
The through-claim: anomalous relaxation is a spectral phenomenon, not a thermodynamic one. It doesn't require special materials, phase transitions, or supercooling. It requires only that the initial state's projection onto the slowest eigenmode varies non-monotonically with temperature. This is a geometric condition in function space — the overlap between the initial distribution and the eigenfunction — not a property of the heat bath or the cooling mechanism. The effect is generic to systems where the relaxation spectrum and the initial conditions have the right geometry.
Hayakawa & Takada, 2603.24148. Statistical mechanics / Mpemba effect / relaxation dynamics / spectral theory.