Capture of particles undergoing discrete random walks

TL;DR

This paper derives the probability of particles undergoing discrete random walks being captured by an adsorbing sphere in 3D, revealing universal behaviors and implications for simulations of reactions and aggregation.

Contribution

It provides a novel analytical expression for capture probability of discrete random walk particles near an adsorbing sphere, including universal asymptotic behaviors.

Findings

Capture probability formula involving jump distribution Fourier transform

Non-zero survival probability for particles starting on the sphere surface

Universal behavior depending only on sigma/R ratio

Abstract

It is shown that particles undergoing discrete-time jumps in 3D, starting at a distance r0 from the center of an adsorbing sphere of radius R, are captured with probability (R - c sigma)/r0 for r0 much greater than R, where c is related to the Fourier transform of the scaled jump distribution and sigma is the distribution's root-mean square jump length. For particles starting on the surface of the sphere, the asymptotic survival probability is non-zero (in contrast to the case of Brownian diffusion) and has a universal behavior sigma/(R sqrt(6)) depending only upon sigma/R. These results have applications to computer simulations of reaction and aggregation.