Persistent memory for a Brownian walker in a random array of obstacles

TL;DR

This paper demonstrates that particles undergoing Brownian motion in a fixed array of obstacles exhibit long-time correlations in their mean-square displacement, with power-law tails in the velocity autocorrelation function due to repeated encounters with the same obstacle.

Contribution

The authors provide an analytic solution for a tracer in a dilute planar Lorentz gas showing persistent correlations caused by quenched disorder, applicable across different microdynamics.

Findings

Long-time correlations emerge in Brownian motion in obstacle arrays.

Power-law tails govern the velocity autocorrelation function.

Quenched disorder induces persistent correlations regardless of microdynamics.

Abstract

We show that for particles performing Brownian motion in a frozen array of scatterers long-time correlations emerge in the mean-square displacement. Defining the velocity autocorrelation function (VACF) via the second time-derivative of the mean-square displacement, power-law tails govern the long-time dynamics similar to the case of ballistic motion. The physical origin of the persistent memory is due to repeated encounters with the same obstacle which occurs naturally in Brownian dynamics without involving other scattering centers. This observation suggests that in this case the VACF exhibits these anomalies already at first order in the scattering density. Here we provide an analytic solution for the dynamics of a tracer for a dilute planar Lorentz gas and compare our results to computer simulations. Our result support the idea that quenched disorder provides a generic mechanism for persistent correlations irrespective of the microdynamics of the tracer particle.