Energy decay of freely cooling granular gases in three dimensions

Freely cooling granular gases, wherein a dilute system of macroscopic particles with uncorrelated initial velocities lose energy through inelastic collisions, have been extensively studied both as a simple model for granular systems as well as a nonequilibrium system showing nontrivial coarsening at late times. As the system cools, inelasticity induces clustering, making the system inhomogeneous. While the form of energy decay ($E(t)\sim t^{-\theta}$) in the initial homogeneous regime is well established by Haff's law ($\theta=2$), the energy decay in the clustered regime is still unresolved in higher dimensions. Within mean field theory, $\theta=2 d/(d+2)$ (where $d$ is the spatial dimension), while a correspondence to Burgers equation implies an exponent $\theta= 2/3 (d=1), d/2 (d>1)$. In one and two dimensions, the two formulae predict the same exponents. By performing extensive event driven molecular dynamics simulations, we show that in three dimensions, the energy decays asymptotically with a power $\approx 1.2$, for all coefficients of restitution $r<1$, consistent with the mean field exponent. However, we argue that the mean field arguments fail due to non local interactions between mass clusters.