Macroscopic Mpemba Effect from Cumulative-Heat-Enhanced Relaxation

Abstract

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 $\mathrm{d}ΔT/\mathrm{d}t = -[γ_0 + \mathcal{M}Q(t)][ΔT - \mathcal{I}Q(t)]$, where $γ_0$ 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.

Figures (2)

• Figure 1: (a) Schematic of Example I. A system is surrounded by a biphasic reservoir ($T_{\mathrm{r}}$). Heat released by the system drives the phase transition ($\Phi_{2} \to \Phi_{1}$), dynamically enhancing the effective thermal conductance from $\Gamma_2$ (air/ice) to $\Gamma_1$ (water) and satisfying $\mu=\Gamma_1/\Gamma_2>1$. Temperature difference $\Delta \tilde{T}=\Delta T/T_r$ as a function of cooling time $\tilde{t}=\gamma_0 t$ with $\mu=10$ (b), $\mu=1$ (c) and $\mu=0.06$ (d). The red dash-dotted curve, orange dashed curve, and blue solid curve in (b-c) with $n_{0}=0.1$ are plotted with $\Delta T_0/T_r=0.3$, $\Delta T_0/T_r=0.2$, and $\Delta T_0/T_r=0.1$, respectively. While the red dash-dotted curve, orange dashed curve, and blue solid curve in (d) with $n_{0}=0.9$ are plotted with $\Delta T_0/T_r=-0.1$, $\Delta T_0/T_r=-0.2$, and $\Delta T_0/T_r=-0.3$; other parameters are fixed at $\Gamma_{2}/C=5$, $CT_r/\Theta=3$, $C=1$ and $\Theta=100$.
• Figure 2: The occurrence of the ME. (a) The exact cooling time $\tau_{c}$ [Eq. (\ref{['eq:exact_tau']})] as a function of dimensionless temperature difference $\tilde{\Delta T_0}\equiv\Delta T_0/\Delta T_f$ with $\mu=1,1.5,3,12$ (in order from top to bottom). (b) Phase diagram of the cooling process in the parametric space. The ME region is characterized by $\partial\tau_{c}/\partial T_{0}<0$. Otherwise, the process is classified as Normal cooling ($\partial\tau_{c}/\partial T_{0}\geq0$).