Theorem on the distribution of short time single particle displacements

The distribution of the initial very short-time displacements of a single particle is considered for a class of classical systems with Gaussian initial velocity distributions and arbitrary initial particle positions. A very brief sketch is given of a rather intricate and lengthy proof that for this class of systems the nth order cumulants behave as t2n for all n>2, rather than as tn. We also briefly discuss some physical consequences for liquids.