Extension of Haff's cooling law in granular flows

The total energy E(t) in a fluid of inelastic particles is dissipated through inelastic collisions. When such systems are prepared in a homogeneous initial state and evolve undriven, E(t) decays initially as t(-2) similar to exp[-2 epsilon tau] (known as Haff's law), where tau is the average number of collisions suffered by a particle within time t, and epsilon = 1 - alpha(2) measures the degree of inelasticity, with ct the coefficient of normal restitution. This decay law is extended for large times to E(t) similar to tau(-d/2) in d dimensions, far into the nonlinear clustering regime. The theoretical predictions are quantitatively confirmed by computer simulations, and hold for small to moderate inelasticities with 0.6 < alpha < 1.