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 from 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 for Markovian dynamics and show that under proper conditions, exponentially faster relaxations are possible. These special relaxations, which we named ``the strong Mpemba effect'', can be classified by a topological index. 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 thermodynamical limit. Thus, we expect that the strong Mpemba effect can be observed experimentally in a wide variety of systems. We analyze the different types of Mpemba relaxations in the mean field anti-ferromagnet Ising model and show surprisingly rich Mpemba phase diagram.