Escape and finite-size scaling in diffusion-controlled annihilation. (22nd November 2016)

• Record Type: Journal Article
• Authors: Ben-Naim, E; Krapivsky, P L
• Is Part Of: Journal of physics. Volume 49:Number 50(2016)
• Publication Date: 2016-11-22
• Subjects: reaction kinetics -- finite-size scaling -- reaction-diffusion processes -- stochastic processes; Mathematical physics -- Periodicals; Statistical physics -- Periodicals; Quantum theory -- Periodicals; Matter -- Properties -- Periodicals; 530.105
• DOI: 10.1088/1751-8113/49/50/504004
• Languages: English
• ISSNs: 1751-8113

Abstract

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 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 ∼ N 1 / 3 in the asymptotic regime N ≫ 1 . We show that two time scales govern the reaction kinetics: the diffusion time scale, T ∼ N 2 / 3, and the escape time scale, τ ∼ 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.