The Mpemba index and anomalous relaxation

TL;DR

The paper investigates the Mpemba effect, revealing a stronger version with exponentially faster relaxation times, characterized by the Mpemba index, and demonstrates its occurrence in various systems through theoretical and model analysis.

Contribution

It introduces the concept of the strong Mpemba effect and the Mpemba index, analyzing their topological and probabilistic properties in Markovian systems and models.

Findings

Strong Mpemba effect leads to exponentially faster relaxation.

The Mpemba index's parity is a topological property.

The effect persists in the thermodynamic limit and is observable in diverse systems.

Abstract

The Mpemba effect is a counter-intuitive relaxation phenomenon, where a system prepared at a hot temperature cools down faster than an identical system initiated 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, polymers, and driven granular gases. We analyze the Mpemba effect in Markovian dynamics and discover that a stronger version of the effect often exists for a carefully chosen set of initial temperatures. In this \emph{strong Mpemba effect}, the relaxation time jumps to a smaller value leading to exponentially faster equilibration dynamics. The number of such special initial temperatures defines the \emph{Mpemba index}, whose parity is a topological property of the system. To demonstrate these concepts, we first analyze the different types of Mpemba relaxations in the mean field anti-ferromagnet Ising model, which demonstrates a surprisingly rich Mpemba phase diagram. Moreover, we show that the strong effect survives the thermodynamic limit and that it 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. Using the parity of the Mpemba index, we then study the occurrence of the strong Mpemba effect in a large class of thermal quench processes and show that it happens with non-zero probability even in the thermodynamic limit. This is done by introducing the \emph{isotropic} model for which we obtain analytical lower bound estimates for the probability of the strong Mpemba effects. Consequently, we expect that such exponentially faster relaxations can be observed experimentally in a wide variety of systems.