The Mpemba Index and anomalous relaxation

Mpemba effect is a counter-intuitive relaxation phenomenon that occurs when, upon coupling to a cold thermal reservoir, a system prepared at a hot temperature relaxes to equilibrium faster than a system prepared at an intermediate temperature. Here we study the possibility of the Mpemba effect for a large class of thermal relaxation processes. We show that an enhanced Mpemba effect is possible for some choices of the initial temperatures, decreasing considerably the cooling times, and introduce a topological index that identifies the existence of such enhancement. We estimate the probability of the Mpemba index for a class of systems with random barriers and energies and show that it is non-vanishing.