Kinetics of diffusioncontrolled annihilation with sparse initial conditions
Here, we study diffusioncontrolled singlespecies 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 codimension Δ = d  δ governs the behavior. All particles disappear when the codimension 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 powerlaw 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 codimension Δ = 2.
 Authors:

^{[1]}
;
^{[2]}
 Los Alamos National Lab. (LANL), Los Alamos, NM (United States). Theoretical Division and Center for Nonlinear Studies
 Boston Univ., MA (United States). Dept. of Physics
 Publication Date:
 Report Number(s):
 LAUR1625627
Journal ID: ISSN 17518113
 Grant/Contract Number:
 AC5206NA25396
 Type:
 Accepted Manuscript
 Journal Name:
 Journal of Physics. A, Mathematical and Theoretical
 Additional Journal Information:
 Journal Volume: 49; Journal Issue: 50; Journal ID: ISSN 17518113
 Publisher:
 IOP Publishing
 Research Org:
 Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
 Sponsoring Org:
 USDOE
 Country of Publication:
 United States
 Language:
 English
 Subject:
 97 MATHEMATICS AND COMPUTING; Mathematics
 OSTI Identifier:
 1337109
BenNaim, Eli, and Krapivsky, Paul. Kinetics of diffusioncontrolled annihilation with sparse initial conditions. United States: N. p.,
Web. doi:10.1088/17518113/49/50/504005.
BenNaim, Eli, & Krapivsky, Paul. Kinetics of diffusioncontrolled annihilation with sparse initial conditions. United States. doi:10.1088/17518113/49/50/504005.
BenNaim, Eli, and Krapivsky, Paul. 2016.
"Kinetics of diffusioncontrolled annihilation with sparse initial conditions". United States.
doi:10.1088/17518113/49/50/504005. https://www.osti.gov/servlets/purl/1337109.
@article{osti_1337109,
title = {Kinetics of diffusioncontrolled annihilation with sparse initial conditions},
author = {BenNaim, Eli and Krapivsky, Paul},
abstractNote = {Here, we study diffusioncontrolled singlespecies 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 codimension Δ = d  δ governs the behavior. All particles disappear when the codimension 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 powerlaw 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 codimension Δ = 2.},
doi = {10.1088/17518113/49/50/504005},
journal = {Journal of Physics. A, Mathematical and Theoretical},
number = 50,
volume = 49,
place = {United States},
year = {2016},
month = {12}
}