Universal scaling of direct and inverse Mpemba effects near criticality
In plain words
Near a phase transition (such as a magnet losing its magnetization at a critical temperature), relaxation becomes slow and governed by a few universal numbers. The question is whether the existence and timing of Mpemba effects there are also fixed by those universal numbers.
Precise statement
Ising-class system with non-conserved order parameter (Glauber or model-A dynamics), bath temperature $T_{\mathrm{b}}$ near the critical temperature $T_{\mathrm{crit}}$, initial equilibrium at $T_{0}$. Using a stated distance from final equilibrium, determine whether the existence of direct $\left(T_{0} > T_{\mathrm{b}}\right)$ and inverse $\left(T_{0} < T_{\mathrm{b}}\right)$ Mpemba effects and the crossing time $t_{x}$ obey scaling forms $t_{x} \sim \left|T_{\mathrm{b}} - T_{\mathrm{crit}}\right|^{-\nu z} F\left(\left(T_{0} - T_{\mathrm{crit}}\right)/\left(T_{\mathrm{b}} - T_{\mathrm{crit}}\right)\right)$ set by the static exponent $\nu$, the dynamic exponent $z$ and the order of the transition, independent of lattice and microscopic rates. The answer is a classification of where each effect occurs, with the scaling function $F$ or a demonstration that none exists.
What would settle it
Large-scale Monte Carlo or renormalization-group computation across several models of one universality class showing a common F and common boundaries for direct and inverse effects.
Status in the literature
Unverified note
A Landau theory predicts Mpemba effects at phase transitions (Holtzman and Raz 2022), and Monte Carlo work on Ising and Potts models reports a scaling picture, set by the initial-state correlation length, that holds across first- and second-order transitions (Chatterjee et al., arXiv:2309.03709); no scaling function $F$ in terms of $\nu$ and $z$ with RG or multi-model confirmation has been established (2026).