Escape and Finite-Size Scaling in Diffusion-Controlled Annihilation

TL;DR

This paper investigates the behavior of finite particle systems undergoing diffusion-controlled annihilation in dimensions greater than two, revealing scaling laws for surviving particles and characteristic time scales.

Contribution

It introduces a scaling framework for understanding finite-size effects and identifies key time scales governing the annihilation process in higher dimensions.

Findings

Surviving particles scale as M ~ N^{1/3} in 3D.

Two main time scales: diffusion time T ~ N^{2/3} and escape time τ ~ N^{4/3}.

Most annihilations occur during the diffusion time scale.

Abstract

We study diffusion-controlled single-species annihilation with a finite number of particles. In this reaction-diffusion process, each particle undergoes ordinary diffusion, and when two particles meet, they annihilate. We focus on spatial dimensions $d>2$ where a finite number of particles typically survive the annihilation process. Using the rate equation approach and scaling techniques we investigate the average number of surviving particles, $M$, as a function of the initial number of particles, $N$. In three dimensions, for instance, we find the scaling law $M\sim N^{1/3}$ in the asymptotic regime $N\gg 1$. We show that two time scales govern the reaction kinetics: the diffusion time scale, $T\sim N^{2/3}$, and the escape time scale, $\tau\sim N^{4/3}$. The vast majority of annihilation events occur on the diffusion time scale, while no annihilation events occur beyond the escape time scale.