Velocity-correlation distributions in granular systems

We investigate the velocity-correlation distributions after $n$ collisions of a tagged particle undergoing binary collisions. Analytical expressions are obtained in any dimension for the velocity-correlation distribution after the first-collision as well as for the velocity-correlation function after an infinite number of collisions, in the limit of Gaussian velocity distributions. It appears that the decay of the first-collision velocity-correlation distribution for negative argument is exponential in any dimension with a coefficient that depends on the mass and on the coefficient of restitution. We also obtained the velocity-correlation distribution when the velocity distributions are not Gaussian: by inserting Sonine corrections of the velocity distributions, we derive the corrections to the velocity-correlation distribution which agree perfectly with a DSMC (Direct Simulation Monte Carlo) simulation. We emphasize that these new quantities can be easily obtained in simulations and likely in experiments: they could be an efficient probe of the local environment and of the degree of inelasticity of the collisions.