  Numerical experiments of the statistical evolution of an ensemble of
non-interacting particles in a time-dependent billiard with inelastic
collisions, reveals the existence of three statistical regimes for the
evolution of the speeds ensemble, namely, diffusion plateau, normal
growth/exponential decay and stagnation. These regimes are linked numerically
to the transition from Gauss-like to Boltzmann-like speed distributions.
Further, the different evolution regimes are obtained analytically through
velocity-space diffusion analysis. From these calculations the asymptotic root
mean square of speed, initial plateau, and the growth/decay rates for
intermediate number of collisions are determined in terms of the system
parameters. The analytical calculations match the numerical experiments and
point to a dynamical mechanism for thermalization, where inelastic collisions
and a high-dimensional phase space lead to a bounded diffusion in the velocity
space towards a stationary distribution function with a kind of reservoir
temperature determined by the boundary oscillation amplitude and the
restitution coefficient.
