Escape and Finite-Size Scaling in Diffusion-Controlled Annihilation

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 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 similar to N-1/3 in the asymptotic regime N >> 1. We show that two time scales govern the reaction kinetics: the diffusion time scale, T similar to N-2/3, and the escape time scale, tau similar to 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.