Exact Short Time Dynamics for Steeply Repulsive Potentials

TL;DR

This paper derives exact short-time dynamics for particles interacting via steeply repulsive power-law potentials, revealing a universal scaling function that connects soft sphere behavior to hard sphere limits and validating results with molecular dynamics simulations.

Contribution

It provides an exact analytical calculation of short-time correlation functions for steeply repulsive potentials, establishing a universal scaling law and linking soft sphere dynamics to hard sphere limits.

Findings

Universal scaling function $S(\tau)$ describes short-time dynamics.

As $\nu \to \infty$, correlation functions approach delta functions.

Results agree with molecular dynamics simulations for steep potentials.

Abstract

The autocorrelation functions for the force on a particle, the velocity of a particle, and the transverse momentum flux are studied for the power law potential $v(r)=\epsilon (\sigma /r)^{\nu}$ (soft spheres). The latter two correlation functions characterize the Green-Kubo expressions for the self-diffusion coefficient and shear viscosity. The short time dynamics is calculated exactly as a function of $\nu $. The dynamics is characterized by a universal scaling function $S(\tau)$, where $\tau =t/\tau_{\nu}$ and $ \tau _{\nu}$ is the mean time to traverse the core of the potential divided by $\nu $. In the limit of asymptotically large $\nu $ this scaling function leads to delta function in time contributions in the correlation functions for the force and momentum flux. It is shown that this singular limit agrees with the special Green-Kubo representation for hard sphere transport coefficients. The domain of the scaling law is investigated by comparison with recent results from molecular dynamics simulation for this potential.