Mpemba effect in molecular gases under nonlinear drag

TL;DR

This paper investigates the Mpemba effect in a molecular gas with nonlinear drag, showing how initial temperature differences influence relaxation dynamics and providing a linearized theoretical framework for understanding this phenomenon.

Contribution

The study introduces a kinetic theory model for the Mpemba effect in molecular gases with nonlinear drag, including a linearized theory that predicts crossover times and temperature differences.

Findings

Mpemba effect occurs in early relaxation stages with close initial temperatures.

Linearized theory semi-quantitatively predicts crossover time and temperature difference.

The model highlights the role of excess kurtosis in the effect.

Abstract

We look into the Mpemba effect---the initially hotter sample cools sooner---in a molecular gas with nonlinear viscous drag. Specifically, the gas particles interact among them via elastic collisions and also with a background fluid at equilibrium. Thus, within the framework of kinetic theory, our gas is described by an Enskog--Fokker--Planck equation. The analysis is carried out in the first Sonine approximation, in which the evolution of the temperature is coupled to that of the excess kurtosis. This coupling leads to the emergence of the Mpemba effect, which is observed in an early stage of the relaxation and when the initial temperatures of the two samples are close enough. This allows for the development of a simple theory, linearizing the temperature evolution around a reference temperature---namely the initial temperature closer to the asymptotic equilibrium value. The linear theory provides a semiquantitative description of the effect, including expressions for the crossover time and the maximum temperature difference. We also discuss the limitations of our linearized theory.