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

• Published 2005

#### 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 a3. In contrast to frequently used assumptions this coefficient is of the same order of magnitude as the second Sonine coefficient a2. For small inelasticity the theoretical result is in good agreement with numerical simulations. The next-order Sonine coefficients a4, a5 and a6 are determined numerically. While these coefficients are negligible for small dissipation, their magnitude grows rapidly with increasing inelasticity for 0 < ε . 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. Introduction. – The velocity distribution function of granular gases deviates from the Maxwell distribution, as first described by Goldshtein and Shapiro [1]. This deviation depends on the coefficient of restitution ε, which quantifies the loss of energy for a collision of two particles i and j: ~v ′ i = ~vi − 1 + ε 2 [(~vi − ~vj) · ~e ]~e , ~v ′ j = ~vj + 1 + ε 2 [(~vi − ~vj) · ~e ]~e . (1) Here ~v ′ i and ~v ′ j stand for the post-collisional velocities where the unit vector of the relative particle position at the collision instant is ~e ≡ (~ri − ~rj) / |~ri − ~rj |. The deviation from the Maxwell distribution may be described by a Sonine polynomials expansion [1–3]. This expansion is applicable to the main part of the velocity distribution, excluding the high-energy tails, which is known to be exponentially overpopulated [4]. So far it was silently accepted that the Sonine expansion is a converging series while the exponential tail does not noticeably contribute to the coefficients of this expansion. This assumption was supported by Direct Simulation Monte Carlo (DSMC) of the Boltzmann equation [5], as well as by Molecular Dynamics simulations of Granular Gases [6]. Up to now,

### Cite this paper

@inproceedings{Brilliantov2005BreakdownOT, title={Breakdown of the Sonine expansion for the velocity distribution of Granular Gases}, author={Nikolai V. Brilliantov and Thorsten P{\"{o}schel}, year={2005} }