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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:
ORCiD logo [1];  [2]
  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:
Research Org.:
Los Alamos National Lab. (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

Citation Formats

Ben-Naim, Eli, and Krapivsky, Paul. Kinetics of diffusion-controlled annihilation with sparse initial conditions. United States: N. p., 2016. Web. https://doi.org/10.1088/1751-8113/49/50/504005.
Ben-Naim, Eli, & Krapivsky, Paul. Kinetics of diffusion-controlled annihilation with sparse initial conditions. United States. https://doi.org/10.1088/1751-8113/49/50/504005
Ben-Naim, Eli, and Krapivsky, Paul. Fri . "Kinetics of diffusion-controlled annihilation with sparse initial conditions". United States. https://doi.org/10.1088/1751-8113/49/50/504005. https://www.osti.gov/servlets/purl/1337109.
@article{osti_1337109,
title = {Kinetics of diffusion-controlled annihilation with sparse initial conditions},
author = {Ben-Naim, Eli and Krapivsky, Paul},
abstractNote = {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.},
doi = {10.1088/1751-8113/49/50/504005},
journal = {Journal of Physics. A, Mathematical and Theoretical},
number = 50,
volume = 49,
place = {United States},
year = {2016},
month = {12}
}

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