Annihilating random walks in one-dimensional disordered media

TL;DR

This paper derives exact expressions for particle concentration in one-dimensional disordered media undergoing annihilation, revealing asymptotic behaviors and duality relations, with some models challenging traditional diffusion approaches.

Contribution

It provides a novel exact solution for particle density in disordered media and explores duality relations, advancing understanding of diffusion-limited reactions in complex systems.

Findings

Exact expression for particle concentration in disordered media.

Asymptotic relation between particle density and single-walker probabilities.

Failure of Smoluchovsky approach in certain random barrier systems.

Abstract

We study diffusion-limited pair annihilation $A+A\to 0$ on one-dimensional lattices with inhomogeneous nearest neighbour hopping in the limit of infinite reaction rate. We obtain a simple exact expression for the particle concentration $\rho_k(t)$ of the many-particle system in terms of the conditional probabilities $P(m;t|l;0)$ for a single random walker in a dual medium. For some disordered systems with an initially randomly filled lattice this leads asymptotically to $\bar{\rho(t)}=\bar{P(0;2t|0;0)}$ for the disorder-averaged particle density. We also obtain interesting exact relations for single-particle conditional probabilities in random media related by duality, such as random-barrier and random-trap systems. For some specific random barrier systems the Smoluchovsky approach to diffusion-limited annihilation turns out to fail.