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

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 α) 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(c). The behavior of F(c) in the domain of thermal velocities (c ∼ 1) can be characterized by the two first non-trivial coefficients (a2 and a3) 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 α-dependence of a2 and a3, to report new computer simulations results of a2 and a3 for two-dimensional systems, and to investigate the possibility of proposing theoretical estimates of a2 and a3 with an optimal compromise between simplicity and accuracy.