Necessary and sufficient conditions for the Markovian Mpemba effect
In plain words
In simple models where a system hops randomly between a few states, a hotter start can lose the overlap with the slowest-decaying pattern and so cool faster. The task is to state exactly which energy landscapes allow this.
Precise statement
Continuous-time Markov jump process on n states with energies $E_i$ and barriers $B_{ij}$, Arrhenius rates $W_{ij} = \operatorname{exp}(-(B_{ij} - E_j)/T_b)$ obeying detailed balance at bath temperature $T_b$, initial condition the Boltzmann distribution $\pi(T_0)$. Expand $p(t) - \pi(T_b) = \operatorname{sum}_{k \ge 2} a_k(T_0) v_k \operatorname{exp}(-\lambda_k t)$ with $0 < \lambda_2 < \lambda_3 \le ...$; a weak direct Mpemba effect is $\mid a_2(T_h)\mid < \mid a_2(T_c)\mid$ for some $T_h > T_c > T_b$, a strong one is $a_2(T_h) = 0$, and an inverse effect is $\mid a_2(T_c)\mid < \mid a_2(T_h)\mid$ for some $T_c < T_h < T_b$. Find necessary and sufficient conditions on $\{E_i, B_{ij}, T_b\}$ for the weak and strong, direct and inverse effects, for general n, stated as explicit inequalities or sign conditions (for example on the graph of barriers) that do not require computing the eigenvectors of the generator.
What would settle it
A theorem characterizing the parameter sets $\{E_i, B_{ij}, T_b\}$ for which $a_2(T_0)$ is nonmonotonic or vanishes, valid for all $n$.
Status in the literature
Unverified note
Sufficient conditions via $a_{2}$ (Lu and Raz 2017) and a topological count of zeros of $a_{2}$ (Klich, Raz, Hirschberg and Vucelja 2019) exist; a March 2026 preprint gives only necessary conditions for $n = 3$ (Avitan, Factor and Gelbwaser-Klimovsky, arXiv:2603.04567).