# Escape and Finite-Size Scaling in Diffusion-Controlled Annihilation

Source: [Escape and Finite-Size Scaling in Diffusion-Controlled Annihilation](<https://www.sota2.com/research/paper/68fa34b41e209131ce89b734>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2025-10-24T20:50:53.202Z

Authors: E. Ben-Naim, P.L. Krapivsky

Published: 2016-05-23T23:32:25.000Z

## 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.

## Recorded benchmark results

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## Original sources and implementation

[Original paper](<https://arxiv.org/abs/1605.07238>)

[PDF](<https://arxiv.org/pdf/1605.07238.pdf>)