Large Mpemba-like effect in a gas of inelastic rough hard spheres

Abstract

We report the emergence of a giant Mpemba effect in the uniformly heated gas of inelastic rough hard spheres: The initially hotter sample may cool sooner than the colder one, even when the initial temperatures differ by more than one order of magnitude. In order to understand this behavior, it suffices to consider the simplest Maxwellian approximation for the velocity distribution in a kinetic approach. The largeness of the effect stems from the fact that the rotational and translational temperatures, which obey two coupled evolution equations, are comparable. Our theoretical predictions agree very well with molecular dynamics and direct simulation Monte Carlo data.Let us consider two beakers of water at different temperatures. Mpemba and Osborne showed that the initially hotter sample cools sooner under certain conditions [1], i.e., the curve giving the time evolution of its temperature crosses that of the initially cooler sample and stays below it for longer times. This is called the Mpemba memory effect, which is known since antiquity in cultures for which water in the form of ice and snow is common [2]. Later, the Mpemba effect has been clearly identified in different physical systems [3][4][5][6][7], although there is still some debate about its existence in water [8].From a physical point of view, one would like to answer how different the initial preparation of two samples of the system under study must be so that the Mpemba effect arises. This is the main-currently unresolved in general-question, although there has been some recent progress in this respect [5,6]. Lu and Raz [5] analyzed the Mpemba effect in a generic Markovian system by monitoring the relaxation of an entropy-like variable that measures the distance to the steady state. This makes it possible to define and investigate Mpemba-like effects in systems for which there is not an obvious definition of a nonequilibrium temperature, but makes the comparison with the usual experimental setup described above difficult.A different approach was carried out by some of us in the study of the Mpemba effect for a granular fluid of smooth hard spheres [6]. Therein, the granular temperature-basically the average kinetic energy per particle-is the physical quantity monitored to investigate the Mpemba effect. In the smooth-sphere case, the angular velocities play no role since there is no energy transfer between the translational and the rotational degrees of freedom, and the kinetic energy is thus purely * prados@us.es translational. We showed that the Mpemba effect stems from the coupling of the granular temperature and the kurtosis, which measures the deviation of the velocity distribution function from the Maxwellian shape at the lowest order. More specifically, it is the difference between the initial values of the kurtosis of the two samples that controls the appearance of the Mpemba effect.In the granular fluid of smooth hard spheres, the kurtosis is typically small. On the one hand, this facilitates the theoretical analysis, because it makes it possi...

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