Kinetics of Diffusion-Controlled Annihilation with Sparse Initial Conditions
| Autor: | Ben-Naim, E., Krapivsky, P. L. |
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| Rok vydání: | 2016 |
| Předmět: | |
| Zdroj: | J. Phys. A 49, 504005 (2016) |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1088/1751-8113/49/50/504005 |
| Popis: | 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, $\Delta\leq 2$; 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 $\Delta=2$. Comment: 5 pages, 4 figures |
| Databáze: | arXiv |
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