Theorem on the Distribution of Short-Time Particle Displacements with Physical Applications

TL;DR

This paper proves that for a broad class of classical systems, the cumulants of initial short-time particle displacements scale as the 2n-th power of time, impacting understanding of particle dynamics in various physical contexts.

Contribution

It establishes a general theorem on the scaling behavior of cumulants of short-time displacements, extending previous results to more general initial conditions and dynamics.

Findings

Cumulants scale as the 2n-th power of time for n>2

Results apply to Van Hove functions and Green's functions

Implications for supercooled liquids and glasses

Abstract

The distribution of the initial short-time displacements of particles is considered for a class of classical systems under rather general conditions on the dynamics and with Gaussian initial velocity distributions, while the positions could have an arbitrary distribution. This class of systems contains canonical equilibrium of a Hamiltonian system as a special case. We prove that for this class of systems the nth order cumulants of the initial short-time displacements behave as the 2n-th power of time for all n>2, rather than exhibiting an 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.