Extinction and Survival in Two-Species Annihilation
J.G. Amar, E. Ben-Naim, S.M. Davis, P.L. Krapivsky
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 {\em 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 {\em populations} are equal, these two quantities are
comparable $M_+\sim M_-\sim N^{1/3}$.
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