Deviation from Maxwell Distribution in Granular Gases with Constant Restitution Coefficient

TL;DR

This paper investigates deviations from Maxwellian velocity distributions in granular gases during homogeneous cooling, revealing three possible solutions for the distribution's correction term and identifying the stable one.

Contribution

It extends previous linear analyses by exploring nonlinear solutions for the velocity distribution deviation in granular gases, identifying the stable solution among three roots.

Findings

Three roots for the coefficient $a_2$ in the Sonine expansion were found.

Only one of these roots corresponds to a stable scaling solution.

The stable solution closely matches previous linear approximation results.

Abstract

We analyze the velocity distribution function of force-free granular gases in the regime of homogeneous cooling when deviations from the Maxwellian distribution may be accounted only by leading term in the Sonine polynomial expansion. These are quantified by the magnitude of the coefficient $a_2$ of the second term of the expansion. In our study we go beyond the linear approximation for $a_2$ and observe that there are three different values (three roots) for this coefficient which correspond to a scaling solution to the Boltzmann equation. The stability analysis performed showed, however, that among these three roots only one corresponds to a stable scaling solution. This is very close to $a_2$, obtained in previous studies in a linear with respect to $a_2$ approximation.