Breakdown of the Sonine expansion for the velocity distribution of Granular Gases

TL;DR

This paper investigates the limitations of the Sonine polynomial expansion in describing the velocity distribution of granular gases, revealing a breakdown at higher inelasticities due to tail overpopulation.

Contribution

The study derives an analytical expression for the third Sonine coefficient and demonstrates the expansion's breakdown at higher dissipation levels.

Findings

The third Sonine coefficient a_3 is comparable to a_2, contrary to common assumptions.

Higher-order coefficients grow rapidly with increasing inelasticity.

The Sonine expansion fails at high dissipation due to tail overpopulation.

Abstract

The velocity distribution of a granular gas is analyzed in terms of the Sonine polynomials expansion. We derive an analytical expression for the third Sonine coefficient a_3. In contrast to frequently used assumptions this coefficient is of the same order of magnitude as the second Sonine coefficient a_2. For small inelasticity the theoretical result is in good agreement with numerical simulations. The next-order Sonine coefficients a_4, a_5 and a_6 are determined numerically. While these coefficients are negligible for small dissipation, their magnitude grows rapidly with increasing inelasticity for 0< epsilon < 0.6. We conclude that this behavior of the Sonine coefficients manifests the break down of the Sonine polynomial expansion caused by the increasing impact of the overpopulated high-energy tail of the distribution function.