Kinetics of Diffusion-Controlled Annihilation with Sparse Initial Conditions

Abstract

We study diffusion-controlled single-species annihilation with sparse initial conditions. In this random process, particles undergo Brownian motion, and when two particles meet, both disappear. We focus on sparse initial conditions where particles occupy a subspace of dimension $\delta$ that is embedded in a larger space of dimension $d$. We find that the co-dimension $\Delta =d-\delta$ governs the behavior. All particles disappear when the co-dimension is sufficiently small, $\Delta \le 2$; otherwise, a finite fraction of particles indefinitely survive. We establish the asymptotic behavior of the probability $S(t)$ that a test particle survives until time $t$. When the subspace is a line, $\delta =1$, we find inverse logarithmic decay, $S\sim (\ln t{)}^{-1}$, in three dimensions, and a modified power-law decay, $S\sim (\ln t)t^{-1/2}$, in two dimensions. In general, the survival probability decays algebraically when $\Delta <2$, and there is an inverse logarithmic decay at the critical co-dimension $\Delta =2$.

arXiv comment5 pages, 4 figures

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Cited by in corpus (2)

• Extinction and Survival in Two-Species Annihilation
• Annihilation of single-species charged particles based on the Dyson gas dynamics