Dynamics of heterogeneous hard spheres in a file

TL;DR

This paper investigates the dynamics of heterogeneous hard spheres in a quasi-one-dimensional channel, deriving a scaling law for mean square displacement and analyzing the Gaussian nature of the tagged particle's position.

Contribution

It introduces a novel approximation for the N-particle PDF in a heterogeneous system and derives a new scaling law for particle displacement.

Findings

Mean square displacement scales as t^((1-γ)/(2c-γ))

Tagged particle position follows a Gaussian distribution

Provides a generalized framework for heterogeneous particle dynamics

Abstract

Normal dynamics in a quasi-one-dimensional channel of length L (\to\infty) of N hard spheres are analyzed. The spheres are heterogeneous: each has a diffusion coefficient D that is drawn from a probability density function (PDF), W D^(-{\gamma}), for small D, where 0\leq{\gamma}<1. The initial spheres' density {\rho} is non-uniform and scales with the distance (from the origin) l as, {\rho} l^(-a), 0\leqa\leq1. An approximation for the N-particle PDF for this problem is derived. From this solution, scaling law analysis and numerical simulations, we show here that the mean square displacement for a particle in such a system obeys, <r^2>~t^(1-{\gamma})/(2c-{\gamma}), where c=1/(1+a). The PDF of the tagged particle is Gaussian in position. Generalizations of these results are considered.