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Title: Extinction and survival in two-species annihilation

Abstract

In this paper, we study diffusion-controlled two-species annihilation with a finite number of particles. In this stochastic process, particles move diffusively, and when two particles of opposite type come into contact, the two annihilate. We focus on the behavior in three spatial dimensions and for initial conditions where particles are confined to a compact domain. Generally, one species outnumbers the other, and we find that the difference between the number of majority and minority species, which is a conserved quantity, controls the behavior. When the number difference exceeds a critical value, the minority becomes extinct and a finite number of majority particles survive, while below this critical difference, a finite number of particles of both species survive. The critical difference $${\mathrm{{\Delta}}}_{c}$$ grows algebraically with the total initial number of particles N, and when $$N{\gg}1$$, the critical difference scales as $${\mathrm{{\Delta}}}_{c}{\sim}{N}^{1/3}$$. Furthermore, when the initial concentrations of the two species are equal, the average number of surviving majority and minority particles $${M}_{+}$$ and $${M}_{{-}}$$, exhibit two distinct scaling behaviors, $${M}_{+}{\sim}{N}^{1/2}$$ and $${M}_{{-}}{\sim}{N}^{1/6}$$. Finally, in contrast, when the initial populations are equal, these two quantities are comparable $${M}_{+}{\sim}{M}_{{-}}{\sim}{N}^{1/3}$$.

Authors:
 [1];  [2];  [1];  [3]
  1. Univ. of Toledo, OH (United States). Dept. of Physics and Astronomy
  2. Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
  3. Boston Univ., MA (United States). Dept. of Physics
Publication Date:
Research Org.:
Los Alamos National Lab. (LANL), Los Alamos, NM (United States); Univ. of Toledo, OH (United States)
Sponsoring Org.:
USDOE; National Science Foundation (NSF)
OSTI Identifier:
1435527
Report Number(s):
LA-UR-17-30128
Journal ID: ISSN 2470-0045; TRN: US1900066
Grant/Contract Number:  
AC52-06NA25396; DMR-1410840; PHY-1262810
Resource Type:
Accepted Manuscript
Journal Name:
Physical Review E
Additional Journal Information:
Journal Volume: 97; Journal Issue: 2; Journal ID: ISSN 2470-0045
Publisher:
American Physical Society (APS)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; Brownian motion; irreversible processes; nonequilibrium statistical mechanics; stochastic processes

Citation Formats

Amar, J. G., Ben-Naim, E., Davis, S. M., and Krapivsky, P. L. Extinction and survival in two-species annihilation. United States: N. p., 2018. Web. doi:10.1103/PhysRevE.97.022112.
Amar, J. G., Ben-Naim, E., Davis, S. M., & Krapivsky, P. L. Extinction and survival in two-species annihilation. United States. doi:10.1103/PhysRevE.97.022112.
Amar, J. G., Ben-Naim, E., Davis, S. M., and Krapivsky, P. L. Fri . "Extinction and survival in two-species annihilation". United States. doi:10.1103/PhysRevE.97.022112. https://www.osti.gov/servlets/purl/1435527.
@article{osti_1435527,
title = {Extinction and survival in two-species annihilation},
author = {Amar, J. G. and Ben-Naim, E. and Davis, S. M. and Krapivsky, P. L.},
abstractNote = {In this paper, we study diffusion-controlled two-species annihilation with a finite number of particles. In this stochastic process, particles move diffusively, and when two particles of opposite type come into contact, the two annihilate. We focus on the behavior in three spatial dimensions and for initial conditions where particles are confined to a compact domain. Generally, one species outnumbers the other, and we find that the difference between the number of majority and minority species, which is a conserved quantity, controls the behavior. When the number difference exceeds a critical value, the minority becomes extinct and a finite number of majority particles survive, while below this critical difference, a finite number of particles of both species survive. The critical difference ${\mathrm{{\Delta}}}_{c}$ grows algebraically with the total initial number of particles N, and when $N{\gg}1$, the critical difference scales as ${\mathrm{{\Delta}}}_{c}{\sim}{N}^{1/3}$. Furthermore, when the initial concentrations of the two species are equal, the average number of surviving majority and minority particles ${M}_{+}$ and ${M}_{{-}}$, exhibit two distinct scaling behaviors, ${M}_{+}{\sim}{N}^{1/2}$ and ${M}_{{-}}{\sim}{N}^{1/6}$. Finally, in contrast, when the initial populations are equal, these two quantities are comparable ${M}_{+}{\sim}{M}_{{-}}{\sim}{N}^{1/3}$.},
doi = {10.1103/PhysRevE.97.022112},
journal = {Physical Review E},
number = 2,
volume = 97,
place = {United States},
year = {2018},
month = {2}
}

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