Reaction-Diffusion Process Driven by a Localized Source: First Passage Properties

TL;DR

This paper analyzes a reaction-diffusion process with immobile and diffusing atoms, focusing on the first passage properties and how the survival probability of diffusing atoms evolves over time in different dimensions.

Contribution

It provides a detailed analysis of the asymptotic behavior of survival probabilities in a reaction-diffusion system with a localized source across various dimensions.

Findings

Survival probability saturates in 1D.

Vanishes algebraically in 2D with a flux-dependent exponent.

Exhibits stretched exponential decay in 3D.

Abstract

We study a reaction-diffusion process that involves two species of atoms, immobile and diffusing. We assume that initially only immobile atoms, uniformly distributed throughout the entire space, are present. Diffusing atoms are injected at the origin by a source which is turned on at time t=0. When a diffusing atom collides with an immobile atom, the two atoms form an immobile stable molecule. The region occupied by molecules is asymptotically spherical with radius growing as t^{1/d} in d>=2 dimensions. We investigate the survival probability that a diffusing atom has not become a part of a molecule during the time interval t after its injection and the probability density of such a particle. We show that asymptotically the survival probability (i) saturates in one dimension, (ii) vanishes algebraically with time in two dimensions (with exponent being a function of the dimensionless flux and determined as a zero of a confluent hypergeometric function), and (iii) exhibits a stretched exponential decay in three dimensions.