Kinetics of diffusion-controlled annihilation with sparse initial conditions

Journal Article · 15 December 2016 · Journal of Physics. A, Mathematical and Theoretical

DOI: https://doi.org/10.1088/1751-8113/49/50/504005 · OSTI ID:1337109

Ben-Naim, Eli  ^[1]; Krapivsky, Paul ^[2]

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1. Los Alamos National Lab. (LANL), Los Alamos, NM (United States). Theoretical Division and Center for Nonlinear Studies
2. Boston Univ., MA (United States). Dept. of Physics

Here, 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 also focus on sparse initial conditions where particles occupy a subspace of dimension δ that is embedded in a larger space of dimension d. Furthermore, we find that the co-dimension Δ = d - δ governs the behavior. All particles disappear when the co-dimension is sufficiently small, Δ ≤ 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, δ = 1, we find inverse logarithmic decay, $$S\sim {(\mathrm{ln}t)}^{-1}$$, in three dimensions, and a modified power-law decay, $$S\sim (\mathrm{ln}t){t}^{-1/2}$$, in two dimensions. In general, the survival probability decays algebraically when Δ < 2, and there is an inverse logarithmic decay at the critical co-dimension Δ = 2.

• Research Organization: Los Alamos National Laboratory (LANL)
• Sponsoring Organization: USDOE
• Grant/Contract Number: AC52-06NA25396
• OSTI ID: 1337109
• Report Number(s): LA-UR-16-25627
• Journal Information: Journal of Physics. A, Mathematical and Theoretical, Journal Name: Journal of Physics. A, Mathematical and Theoretical Journal Issue: 50 Vol. 49; ISSN 1751-8113
• Publisher: IOP PublishingCopyright Statement
• Country of Publication: United States
• Language: English