Working Paper Article Version 1 This version is not peer-reviewed

Kullback–Leibler Divergence of a Freely Cooling Granular Gas

Version 1 : Received: 7 October 2020 / Approved: 8 October 2020 / Online: 8 October 2020 (10:43:55 CEST)

A peer-reviewed article of this Preprint also exists.

Megías, A.; Santos, A. Kullback–Leibler Divergence of a Freely Cooling Granular Gas. Entropy 2020, 22, 1308. Megías, A.; Santos, A. Kullback–Leibler Divergence of a Freely Cooling Granular Gas. Entropy 2020, 22, 1308.


Finding the proper entropy functional associated with the inelastic Boltzmann equation for a granular gas is a yet 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 “homogeneous cooling state” (HCS) distribution of the freely cooling system being a potential candidate. Due to the lack of a formal proof, 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 (α). Our results reject the Maxwellian distribution as a possible reference, whereas reinforce the HCS one. Moreover, the KLD is used to measure the amount of information lost on using the former rather than the latter, and reveals a nonmonotonic dependence with α. Additionally, a Maxwell-demon-like velocity-inversion experiment highlights the microscopic irreversibility of the granular gas dynamics.


Kullback–Leibler divergence; granular gases; kinetic theory; molecular dynamics


Physical Sciences, Acoustics

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