Generalized time evolution of the homogeneous cooling state of a granular gas with positive and negative coefficient of normal restitution

Abstract

The homogeneous cooling state (HCS) of a granular gas described by the inelastic Boltzmann equation is reconsidered. As usual, particles are taken as inelastic hard disks or spheres, but now the coefficient of normal restitution $α$ is allowed to take negative values $α\in[-1,1]$, a simple way of modeling more complicated inelastic interactions. The distribution function of the HCS is studied at the long-time limit, as well as for intermediate times. At the long-time limit, the relevant information of the HCS is given by a scaling distribution function $ϕ_s(c)$, where the time dependence occurs through a dimensionless velocity $c$. For $α\gtrsim -0.75$, $ϕ_s$ remains close to the gaussian distribution in the thermal region, its cumulants and exponential tails being well described by the first Sonine approximation. On the contrary, for $α\lesssim -0.75$, the distribution function becomes multimodal, its maxima located at $c\ne 0$, and its observable tails algebraic. The latter is a consequence of an unbalanced relaxation-dissipation competition, and is analytically demonstrated for $α\simeq -1$ thanks to a reduction of the Boltzmann equation to a Fokker-Planck-like equation. Finally, a generalized scaling solution to the Boltzmann equation is also found $ϕ(c,β)$. Apart from the time dependence occurring through the dimensionless velocity, $ϕ(c,β)$ depends on time through a new parameter $β$ measuring the departure of the HCS from its long-time limit. It is shown that $ϕ(c,β)$ describes the time evolution of the HCS for almost all times. The relevance of the new scaling is also discussed.