Velocity-correlation distributions in thermal and granular hard-core gases

Collision statistics of hard-core systems (thermal and dissipative) is investigated through the velocity-correlation distributions after n collisions of a tagged hard-core particle: These quantities provide information on the velocity correlations for a given number of collisions. We obtain exact results for arbitrary dimension for the velocity-correlation distribution after the first collision as well as for the velocity-correlation function after an infinite number of collisions. For Gaussian velocity distributions, we show that the decay of the first-collision velocity-correlation distribution for negative argument is always exponential in any dimension; the decay rate is then a function of the mass and the coefficient of restitution. For granular gases, where deviations from Gaussian are relevant, expressions including Sonine corrections are also derived for the velocity-correlation distribution and a comparison with a direct simulation Monte Carlo (DSMC) shows accurate agreement with theoretical results. We emphasize that these quantities can be easily obtained in simulations and most likely also in experiments: therefore they could be an efficient probe of the local environment and of the degree of inelasticity of the collisions.