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 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 $M\sim N^{1\mathrm{/}3}$ in the asymptotic regime $N\gg 1$. We show that two time scales govern the reaction kinetics: the diffusion time scale, $T\sim N^{2\mathrm{/}3}$, 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.