PAPER: Classical statistical mechanics, equilibrium and non-equilibrium Generalized time evolution of the homogeneous cooling state of a granular gas with positive and negative coefficient of normal restitution

摘要

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 ? [?1, 1], which is 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 intermediate times. At the long-time limit, the relevant information of the HCS is given by a scaling distribution function ?_s(c), where the time dependence occurs through a dimensionless velocity c. For α ? ?0.75, ?_s remains close to the Gaussian distribution in the thermal region, its cumulants and exponential tails being well described by the first Sonine approximation. In contrast, for α ? ?0.75, the distribution function becomes multimodal, its maxima located at c≠0, and its observable tails algebraic. The latter is a consequence of an unbalanced relaxation–dissipation competition, and is analytically demonstrated for α ? ?1, thanks to a reduction of the Boltzmann equation to a Fokker–Plank-like equation. Finally, a generalized scaling solution to the Boltzmann equation is also found ?(c, β). Apart from the time dependence occurring through the dimensionless velocity, ?(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 ?(c, β) describes the time evolution of the HCS for almost all times. The relevance of the new scaling is also discussed.