Extinction and Survival in Two-Species Annihilation

TL;DR

This paper analyzes the stochastic diffusion-controlled annihilation of two particle species in three dimensions, revealing how initial differences and concentrations influence survival outcomes and scaling behaviors.

Contribution

It introduces a detailed analysis of the critical initial difference and scaling laws for surviving particles in two-species annihilation in three dimensions.

Findings

Critical difference scales as N^{1/3} for large N.

Survivor counts scale as N^{1/2} and N^{1/6} depending on initial conditions.

Equal initial concentrations lead to comparable survivor numbers scaling as N^{1/3}.

Abstract

We study diffusion-controlled two-species annihilation with a finite number of particles. In this stochastic process, particles move diffusively, and when two particles of opposite type come into contact, the two annihilate. We focus on the behavior in three spatial dimensions and for initial conditions where particles are confined to a compact domain. Generally, one species outnumbers the other, and we find that the difference between the number of majority and minority species, which is a conserved quantity, controls the behavior. When the number difference exceeds a critical value, the minority becomes extinct and a finite number of majority particles survive, while below this critical difference, a finite number of particles of both species survive. The critical difference $\Delta_c$ grows algebraically with the total initial number of particles $N$, and when $N\gg 1$, the critical difference scales as $\Delta_c\sim N^{1/3}$. Furthermore, when the initial concentrations of the two species are equal, the average number of surviving majority and minority particles, $M_+$ and $M_-$, exhibit two distinct scaling behaviors, $M_+\sim N^{1/2}$ and $M_-\sim N^{1/6}$. In contrast, when the initial populations are equal, these two quantities are comparable $M_+\sim M_-\sim N^{1/3}$.