[Paper Review] Thermal versus entropic Mpemba effect in molecular gases with nonlinear drag

Alberto Megías, Andrés Santos+1 more|arXiv (Cornell University)|Jan 31, 2022

Advanced Thermodynamics and Statistical Mechanics97 references30 citations

TL;DR

This paper investigates the thermal and entropic Mpemba effects in a molecular gas with nonlinear drag using kinetic theory and numerical simulations. It demonstrates that the thermal and entropic Mpemba effects are not mutually equivalent—crossing of temperature and relative entropy curves can occur independently—highlighting the need to redefine the thermal Mpemba effect in terms of local equilibrium distribution relaxation, with an extended Sonine approximation accurately capturing simulation results.

ABSTRACT

Loosely speaking, the Mpemba effect appears when hotter systems cool sooner or, in a more abstract way, when systems further from equilibrium relax faster. In this paper, we investigate the Mpemba effect in a molecular gas with nonlinear drag, both analytically (by employing the tools of kinetic theory) and numerically (direct simulation Monte Carlo of the kinetic equation and event-driven molecular dynamics). The analysis is carried out via two alternative routes, recently considered in the literature: first, the kinetic or thermal route, in which the Mpemba effect is characterized by the crossing of the evolution curves of the kinetic temperature (average kinetic energy), and, second, the stochastic thermodynamics or entropic route, in which the Mpemba effect is characterized by the crossing of the distance to equilibrium in probability space. In general, a nonmutual correspondence between the thermal and entropic Mpemba effects is found, i.e., there may appear the thermal effect without its entropic counterpart or vice versa. Furthermore, a nontrivial overshoot with respect to equilibrium of the thermal relaxation makes it necessary to revise the usual definition of the thermal Mpemba effect, which is shown to be better described in terms of the relaxation of the local equilibrium distribution. Our theoretical framework, which involves an extended Sonine approximation in which not only the excess kurtosis but also the sixth cumulant is retained, gives an excellent account of the behavior observed in simulations.

Motivation & Objective

• To examine the interplay between thermal and entropic Mpemba effects in a molecular gas with nonlinear drag.
• To assess whether the thermal and entropic definitions of the Mpemba effect are equivalent or distinct in nonequilibrium relaxation.
• To challenge the conventional definition of the thermal Mpemba effect based on kinetic temperature crossing, proposing instead a redefinition via local equilibrium distribution.
• To validate the theoretical framework using direct simulation Monte Carlo and event-driven molecular dynamics.
• To quantify memory effects and nonexponential relaxation due to nonlinear drag in the system.

Proposed method

• Uses the Enskog–Fokker–Planck equation (EFPE) to model the velocity distribution function (VDF) with nonlinear drag and binary elastic collisions.
• Applies an extended Sonine approximation retaining both excess kurtosis and sixth cumulant to describe non-Gaussian VDFs.
• Employs direct simulation Monte Carlo (DSMC) and event-driven molecular dynamics (EDMD) for numerical validation.
• Defines the thermal Mpemba effect via crossing of kinetic temperature evolution curves.
• Defines the entropic Mpemba effect via crossing of Kullback–Leibler divergence (KLD) from equilibrium.
• Compares the two routes using KLD as a distance-to-equilibrium measure in probability space.

Experimental results

Research questions

• RQ1 Do the thermal and entropic Mpemba effects always co-occur in systems with nonlinear drag?
• RQ2 Is there a fundamental mismatch between the thermal and entropic definitions of the Mpemba effect?
• RQ3 Can the conventional thermal Mpemba effect definition be improved by considering relaxation of the local equilibrium distribution?
• RQ4 How do memory effects and non-Maxwellian velocity distributions influence the Mpemba effect?
• RQ5 To what extent does the extended Sonine approximation accurately describe the system's relaxation dynamics?

Key findings

• The thermal and entropic Mpemba effects are not mutually equivalent; one can occur without the other.
• The conventional definition of the thermal Mpemba effect based on kinetic temperature crossing is insufficient due to overshoot behavior in relaxation.
• The relaxation of the local equilibrium distribution provides a better characterization of the thermal Mpemba effect than kinetic temperature alone.
• The extended Sonine approximation, including the sixth cumulant, accurately describes the non-Gaussian velocity distributions and matches simulation results.
• Nonlinear drag induces algebraic, nonexponential relaxation with strong memory effects, disrupting standard exponential cooling assumptions.
• The Kullback–Leibler divergence (KLD) serves as a robust measure of distance to equilibrium, enabling a consistent entropic definition of the Mpemba effect.