The Mpemba index and anomalous relaxation

The Mpemba effect is a counter-intuitive relaxation phenomenon, in which a system prepared at a hot temperature cools faster than an identical system starting at a cold temperature when both are quenched to an even colder bath. Such non-monotonic relaxations were observed in various systems, including water, magnetic alloys, and driven granular gases. We analyze the Mpemba effect in Markovian dynamics and discover that in addition to the usual effect there also exists the possibility of a stronger version of the effect for a carefully chosen set of initial temperatures. In what we call the \emph{strong Mpemba effect} the relaxation time jumps to a smaller value leading to exponentially faster equilibration dynamics. A topological index can be associated with the number of such special initial temperatures. Using the parity of this index, we study the occurrence of the strong Mpemba effect for a large class of thermal quench processes and show that it happens with non-zero probability even in the thermodynamic limit. We also introduce the \emph{isotropic} model for which we obtain analytical lower bound estimates for the probability of the strong Mpemba effect. Consequently, we expect that such exponentially faster relaxations can be observed experimentally in a wide variety of systems. Furthermore, we analyze the different types of Mpemba relaxations in the mean field anti-ferromagnet Ising model and show surprisingly rich Mpemba phase diagram. Lastly, we show that in the thermodynamic limit of this system the strong Mpemba effect is tightly connected with thermal overshoot -- in the relaxation process, the temperature of the relaxing system can decay non-monotonically as a function of time.