Title: Kinetics of diffusion-controlled annihilation with sparse initial conditions

Abstract

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.

Authors:

Ben-Naim, Eli  ^[1];

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Krapivsky, Paul ^[2]

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+ Show Author Affiliations

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

Publication Date:

Fri Dec 16 00:00:00 EST 2016

Research Org.:

Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)

Sponsoring Org.:

USDOE

OSTI Identifier:

1337109

Report Number(s):

LA-UR-16-25627
Journal ID: ISSN 1751-8113

Grant/Contract Number:  

AC52-06NA25396

Resource Type:

Accepted Manuscript

Journal Name:

Journal of Physics. A, Mathematical and Theoretical

Additional Journal Information:

Journal Volume: 49; Journal Issue: 50; Journal ID: ISSN 1751-8113

Publisher:

IOP Publishing

Country of Publication:

United States

Language:

English

Subject:

97 MATHEMATICS AND COMPUTING; Mathematics