Anomalous diffusion in infinite horizon billiards

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〉∼t^γ_q (up to factors of lnt). The time exponent, γ_q, is piecewise linear and equal to q/2 for q<2 and q-1 for q>2. 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.