Kullback--Leibler Divergence of a Freely Cooling Granular Gas

TL;DR

This paper investigates the Kullback-Leibler divergence as a Lyapunov functional for the inelastic Boltzmann equation in granular gases, supporting the homogeneous cooling state as the appropriate reference distribution through simulations.

Contribution

It provides evidence that the Kullback-Leibler divergence with the homogeneous cooling state distribution acts as a Lyapunov functional for granular gases, excluding the Maxwellian as a reference.

Findings

Maxwellian distribution is not a suitable reference.

Homogeneous cooling state distribution is reinforced as the correct reference.

KLD reveals non-monotonic dependence on the coefficient of restitution.

Abstract

Finding the proper entropy-like Lyapunov functional associated with the inelastic Boltzmann equation for an isolated freely cooling granular gas is a still unsolved challenge. The original $H$-theorem hypotheses do not fit here and the $H$-functional presents some additional measure problems that are solved by the Kullback--Leibler divergence (KLD) of a reference velocity distribution function from the actual distribution. The right choice of the reference distribution in the KLD is crucial for the latter to qualify or not as a Lyapunov functional, the asymptotic "homogeneous cooling state" (HCS) distribution being a potential candidate. Due to the lack of a formal proof far from the quasielastic limit, the aim of this work is to support this conjecture aided by molecular dynamics simulations of inelastic hard disks and spheres in a wide range of values for the coefficient of restitution ($\alpha$) and for different initial conditions. Our results reject the Maxwellian distribution as a possible reference, whereas they reinforce the HCS one. Moreover, the KLD is used to measure the amount of information lost on using the former rather than the latter, revealing a non-monotonic dependence with $\alpha$.