Escape and finite-size scaling in diffusion-controlled annihilation

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 where a finite number of particles typically survive the annihilation process. Using 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 in the asymptotic regime . We show that two time scales govern the reaction kinetics: the diffusion time scale, , and the escape time scale, . The vast majority of annihilation events occur on the diffusion time scale, while no annihilation events occur beyond the escape time scale.