Diffusion-limited reactions and mortal random walkers in confined geometries

TL;DR

This paper investigates the encounter probabilities of mortal random walkers in confined geometries, providing exact solutions and analyzing the effects of lattice shape and size, with implications for diffusion-reaction processes.

Contribution

It offers an exact expression for encounter probability of mortal walkers in confined 2D geometries and explores the impact of lattice shape and size on first-passage behavior.

Findings

Exact encounter probability expression derived

Logarithmic corrections in the continuum limit

Lattice shape influences encounter probability and transition to 1D behavior

Abstract

Motivated by the diffusion-reaction kinetics on interstellar dust grains, we study a first-passage problem of mortal random walkers in a confined two-dimensional geometry. We provide an exact expression for the encounter probability of two walkers, which is evaluated in limiting cases and checked against extensive kinetic Monte Carlo simulations. We analyze the continuum limit which is approached very slowly, with corrections that vanish logarithmically with the lattice size. We then examine the influence of the shape of the lattice on the first-passage probability, where we focus on the aspect ratio dependence: Distorting the lattice always reduces the encounter probability of two walkers and can exhibit a crossover to the behavior of a genuinely one-dimensional random walk. The nature of this transition is also explained qualitatively.