3 0 M ay 2 00 5 Theorem on the Distribution of Short - Time Single Particle Displacements with Physical Applications

The distribution of the initial short-time displacements of a single particle is considered for a class of classical systems of particles under rather general conditions. This class of systems contains canonical equilibrium of a multi-component Hamiltonian system as a special case. We prove that for this class of systems the nth order cumulant of the initial short-time displacements behaves as the 2n-th power of time for all n > 2, rather than exhibiting a general nth power scaling. This has direct applications to the initial short-time behavior of the Van Hove self-correlation function, to its non-equilibrium generalizations the Green's functions for mass transport, and to the non-Gaussian parameters used in supercooled liquids and glasses. Moreover , in the context of the Green's functions this theorem is expected to be relevant for mass transport at (sub)picosecond time scales.