Diffusion of Tagged Particle in an Exclusion Process

TL;DR

This paper investigates the diffusion behavior of a tagged particle in an exclusion process under external forces, deriving general equations and revealing behaviors different from classical sub-diffusion, applicable to various diffusion types.

Contribution

It introduces a general framework for analyzing tagged particle diffusion under external forces, extending beyond classical models and linking to order statistics and non-Gaussian diffusion.

Findings

Derived general equations for particle distribution and mean square displacement.

Identified new diffusion behaviors differing from classical sub-diffusion.

Connected results to order statistics and non-Gaussian diffusion models.

Abstract

We study the diffusion of tagged hard core interacting particles under the influence of an external force field. Using the Jepsen line we map this many particle problem onto a single particle one. We obtain general equations for the distribution and the mean square displacement $<(x_T)^2>$ of the tagged center particle valid for rather general external force fields and initial conditions. A wide range of physical behaviors emerge which are very different than the classical single file sub-diffusion $<(x_T)^2 > \sim t^{1/2}$ found for uniformly distributed particles in an infinite space and in the absence of force fields. For symmetric initial conditions and potential fields we find $<(x_T)^2 > = {{\cal R} (1 - {\cal R})\over 2 N {\it r} ^2} $ where $2 N$ is the (large) number of particles in the system, ${\cal R}$ is a single particle reflection coefficient obtained from the single particle Green function and initial conditions, and $r$ its derivative. We show that this equation is related to the mathematical theory of order statistics and it can be used to find $<(x_T)^2>$ even when the motion between collision events is not Brownian (e.g. it might be ballistic, or anomalous diffusion). As an example we derive the Percus relation for non Gaussian diffusion.