Kullback--Leibler Divergence of a Freely Cooling Granular Gas

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 $\alpha$.

arXiv comment22 pages, 10 figures; v2: old Sec. 4 removed, two new figures added, plus some text changes

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Cited by in corpus (5)

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• Kinetic Theory and Memory Effects of Homogeneous Inelastic Granular Gases under Nonlinear Drag
• Relative entropy of freely cooling granular gases. A molecular dynamics study
• Thermal versus entropic Mpemba effect in molecular gases with nonlinear drag
• Translational and rotational non-Gaussianities in homogeneous freely evolving granular gases