We consider the long time dependence for the moments of displacement $〈|r{|}^q〉$ of infinite horizon billiards, given a bounded initial distribution of particles. For a variety of billiard models we find $〈|r{|}^q〉\sim t^{{\gamma }_q}$ (up to factors of $\ln t).\mathrm{}$ The time exponent, ${\gamma }_q,$ is piecewise linear and equal to $q/2\mathrm{}$ for $q<\mathrm{}2\mathrm{}$ and $q-1$ for $q>\mathrm{}2.\mathrm{}$ We discuss the lack of dependence of this result on the initial distribution of particles and resolve apparent discrepancies between this time dependence and a prior result. The lack of dependence on initial distribution follows from a remarkable scaling result that we obtain for the time evolution of the distribution function of the angle of a particle’s velocity vector.

• Received 15 October 2002

DOI:https://doi.org/10.1103/PhysRevE.67.021110