The second and third Sonine coefficients of a freely cooling granular gas revisited

TL;DR

This paper reviews and updates the understanding of Sonine coefficients in granular gases, presents new simulation data for 2D systems, and explores theoretical estimates balancing simplicity and accuracy.

Contribution

It provides a comprehensive review, new simulation results for 2D systems, and proposes improved theoretical estimates for Sonine coefficients in granular gases.

Findings

New simulation data for 2D systems

Analysis of $oldsymbol{a_2}$ and $oldsymbol{a_3}$ dependence on restitution coefficient

Proposed theoretical estimates balancing simplicity and accuracy

Abstract

In its simplest statistical-mechanical description, a granular fluid can be modeled as composed of smooth inelastic hard spheres (with a constant coefficient of normal restitution $\alpha$) whose velocity distribution function obeys the Enskog-Boltzmann equation. The basic state of a granular fluid is the homogeneous cooling state, characterized by a homogeneous, isotropic, and stationary distribution of scaled velocities, $F(\mathbf{c})$. The behavior of $F(\mathbf{c})$ in the domain of thermal velocities ($c\sim 1$) can be characterized by the two first non-trivial coefficients ($a_2$ and $a_3$) of an expansion in Sonine polynomials. The main goals of this paper are to review some of the previous efforts made to estimate (and measure in computer simulations) the $\alpha$-dependence of $a_2$ and $a_3$, to report new computer simulations results of $a_2$ and $a_3$ for two-dimensional systems, and to investigate the possibility of proposing theoretical estimates of $a_2$ and $a_3$ with an optimal compromise between simplicity and accuracy.