Escape and Finite-Size Scaling in Diffusion-Controlled Annihilation
E. Ben-Naim and P.L. Krapivsky
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.
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