Exciting Hard Spheres

We investigate the collision cascade that is generated by a single moving incident particle on a static hard-sphere gas. We argue that the number of moving particles at time t grows as t^{xi} and the number collisions up to time t grows as t^{eta}, with xi=2d/(d+2) and eta=2(d+1)/(d+2) and d the spatial dimension. These growth laws are the same as those from a hydrodynamic theory for the shock wave emanating from an explosion. Our predictions are verified by molecular dynamics simulations in d=1 and 2. For a particle incident on a static gas in a half-space, the resulting backsplatter ultimately contains almost all the initial energy.