Theorem on the Distribution of Short Time Single Particle Displacements

TL;DR

This paper proves that in certain classical systems, the short-time displacement distribution of a particle exhibits cumulants scaling as t^{2n} for n>2, impacting the understanding of particle dynamics in liquids.

Contribution

It provides a rigorous proof that higher-order cumulants scale as t^{2n} in specific classical systems, revealing new insights into short-time particle behavior.

Findings

Cumulants scale as t^{2n} for n>2 in the studied systems.

The proof involves complex mathematical derivations.

Implications for physical properties of liquids are discussed.

Abstract

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 t^{2n} for all n>2, rather than as t^{n}. We also briefly discuss some physical consequences for liquids.