Macroscopic Mpemba Effect from Cumulative-Heat-Enhanced Relaxation

The counterintuitive Mpemba effect, wherein a hotter system cools faster, critically lacks a universal macroscopic theory. Here, starting from linear irreversible thermodynamics, we formulate a generalized Newton's cooling law for the system-reservoir temperature difference ΔT, given by dΔT/dt = -\[γ₀ + mathcal{M}Q(t)\]\[ΔT - mathcal{I}Q(t)\], where γ₀ is the bare relaxation rate, and the cumulative heat exchange Q(t) explicitly encodes initial-state memory. The coefficients mathcal{M} and mathcal{I}, arising from the interplay between heat flux and structural evolution, govern diverse anomalous relaxation behaviors. Specifically, mathcal{M} > 0 $mathcal{M} < 0$ induces the (inverse) Mpemba effect, while mathcal{I} imposes a non-vanishing asymptotic ΔT, predicting incomplete thermalization. Our findings capture the full spectrum of memory-dependent relaxation, bridging kinetic speedup with structural freezing in complex systems.