Random Walk with a Hop-Over Site: A Novel Approach to Tagged Diffusion and Its Applications

TL;DR

This paper introduces a new model of random walk with a hop-over site on a lattice, providing exact solutions for its distribution and applications to vacancy-mediated diffusion phenomena.

Contribution

It presents a novel approach to tagged diffusion by analyzing a random walk that skips a specific site, with exact probability distributions and applications to disordering processes.

Findings

Exact probability distribution derived for the hop-over walk

Application demonstrated in vacancy-mediated disordering

Provides a simple framework for tagged diffusion analysis

Abstract

We study, on a $d$ dimensional hypercubic lattice, a random walk which is homogeneous except for one site. Instead of visiting this site, the walker hops over it with arbitrary rates. The probability distribution of this walk and the statistics associated with the hop-overs are found exactly. This analysis provides a simple approach to the problem of tagged diffusion, i.e., the movements of a tracer particle due to the diffusion of a vacancy. Applications to vacancy mediated disordering are given through two examples.