Escape and finite-size scaling in diffusion-controlled annihilation

Ben-Naim, Eli; Krapivsky, Paul L.

doi:10.1088/1751-8113/49/50/504004

Title: Escape and finite-size scaling in diffusion-controlled annihilation

Abstract

In this paper, we study diffusion-controlled single-species annihilation with a finite number of particles. In this reaction-diffusion process, each particle undergoes ordinary diffusion, and when two particles meet, they annihilate. We focus on spatial dimensions d>2 where a finite number of particles typically survive the annihilation process. Using scaling techniques we investigate the average number of surviving particles, M, as a function of the initial number of particles, N. In three dimensions, for instance, we find the scaling law M ~ N^1/3 in the asymptotic regime N»1. We show that two time scales govern the reaction kinetics: the diffusion time scale, T ~ N^2/3, and the escape time scale, τ ~ N^4/3. The vast majority of annihilation events occur on the diffusion time scale, while no annihilation events occur beyond the escape time scale.

Authors:

^[1]; Krapivsky, Paul L. ^[2]

Los Alamos National Lab. (LANL), Los Alamos, NM (United States). Theoretical Division and Center for Nonlinear Studies
Boston Univ., MA (United States). Department 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 Laboratory Directed Research and Development (LDRD) Program

OSTI Identifier:: 1337106

Report Number(s):: LA-UR-16-23613
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:: 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; Mathematics

Citation Formats


                    Ben-Naim, Eli, and Krapivsky, Paul L. Escape and finite-size scaling in diffusion-controlled annihilation.  United States: N. p., 2016. 
Web.  doi:10.1088/1751-8113/49/50/504004.

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                    Ben-Naim, Eli, & Krapivsky, Paul L. Escape and finite-size scaling in diffusion-controlled annihilation.  United States.  https://doi.org/10.1088/1751-8113/49/50/504004

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                    Ben-Naim, Eli, and Krapivsky, Paul L. Fri .  
"Escape and finite-size scaling in diffusion-controlled annihilation".  United States.  https://doi.org/10.1088/1751-8113/49/50/504004.  https://www.osti.gov/servlets/purl/1337106.

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@article{osti_1337106,

  title        = {Escape and finite-size scaling in diffusion-controlled annihilation},

  author       = {Ben-Naim, Eli and Krapivsky, Paul L.},

  abstractNote = {In this paper, we study diffusion-controlled single-species annihilation with a finite number of particles. In this reaction-diffusion process, each particle undergoes ordinary diffusion, and when two particles meet, they annihilate. We focus on spatial dimensions d>2 where a finite number of particles typically survive the annihilation process. Using scaling techniques we investigate the average number of surviving particles, M, as a function of the initial number of particles, N. In three dimensions, for instance, we find the scaling law M ~ N1/3 in the asymptotic regime N»1. We show that two time scales govern the reaction kinetics: the diffusion time scale, T ~ N2/3, and the escape time scale, τ ~ N4/3. The vast majority of annihilation events occur on the diffusion time scale, while no annihilation events occur beyond the escape time scale.},

  doi          = {10.1088/1751-8113/49/50/504004},

  journal      = {Journal of Physics. A, Mathematical and Theoretical},

  number       = 50,

  volume       = 49,

  place        = {United States},

  year         = {Fri Dec 16 00:00:00 EST 2016},

  month        = {Fri Dec 16 00:00:00 EST 2016}

}

Copy to clipboard

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Scientific Reports, Vol. 7, Issue 1
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Title: Escape and finite-size scaling in diffusion-controlled annihilation

Abstract

Citation Formats

Stochastic Problems in Physics and Astronomy journal, January 1943

Pattern Formation and Dynamics in Nonequilibrium Systems book, January 2009

Front propagation into unstable states journal, November 2003

Applications of field-theoretic renormalization group methods to reaction–diffusion problems journal, April 2005

Role of density fluctuations in bimolecular reaction kinetics journal, February 1978

Asymptotics for interacting particle systems onZ d journal, June 1980

Kinetics of diffusion‐controlled processes in dense polymer systems. I. Nonentangled regimes journal, March 1982

Diffusion-limited reactions in one dimension journal, May 1983

Particle–antiparticle annihilation in diffusive motion journal, March 1983

Diffusion-controlled annihilation in the presence of particle sources: Exact results in one dimension journal, October 1985

Fluctuation-dominated kinetics in diffusion-controlled reactions journal, July 1985

Exact solutions for a diffusion-reaction process in one dimension journal, March 1988

Exact solution for a steady-state aggregation model in one dimension journal, April 1989

Statics and dynamics of a diffusion-limited reaction: Anomalous kinetics, nonequilibrium self-ordering, and a dynamic transition journal, September 1990

Diffusion-limited exciton fusion reaction in one-dimensional tetramethylammonium manganese trichloride (TMMC) journal, April 1993

One-dimensional diffusion-limited relaxation of photoexcitations in suspensions of single-walled carbon nanotubes journal, July 2006

Measurement of a Reaction-Diffusion Crossover in Exciton-Exciton Recombination inside Carbon Nanotubes Using Femtosecond Optical Absorption journal, November 2013

Alternating kinetics of annihilating random walks near a free interface journal, March 1998

Finite-size scaling studies of one-dimensional reaction-diffusion systems. Part I. Analytical results journal, March 1995

Scaling Theory for Finite-Size Effects in the Critical Region journal, June 1972

Finite-size scaling at first-order phase transitions journal, August 1984

A Guide to First-Passage Processes book, January 2001

Solution of the spatial neutral model yields new bounds on the Amazonian species richness journal, February 2017

Stochastic Problems in Physics and Astronomy
journal, January 1943

Pattern Formation and Dynamics in Nonequilibrium Systems
book, January 2009

Front propagation into unstable states
journal, November 2003

Applications of field-theoretic renormalization group methods to reaction–diffusion problems
journal, April 2005

Role of density fluctuations in bimolecular reaction kinetics
journal, February 1978

Asymptotics for interacting particle systems onZ d
journal, June 1980

Kinetics of diffusion‐controlled processes in dense polymer systems. I. Nonentangled regimes
journal, March 1982

Diffusion-limited reactions in one dimension
journal, May 1983

Particle–antiparticle annihilation in diffusive motion
journal, March 1983

Diffusion-controlled annihilation in the presence of particle sources: Exact results in one dimension
journal, October 1985

Fluctuation-dominated kinetics in diffusion-controlled reactions
journal, July 1985

Exact solutions for a diffusion-reaction process in one dimension
journal, March 1988

Exact solution for a steady-state aggregation model in one dimension
journal, April 1989

Statics and dynamics of a diffusion-limited reaction: Anomalous kinetics, nonequilibrium self-ordering, and a dynamic transition
journal, September 1990

Diffusion-limited exciton fusion reaction in one-dimensional tetramethylammonium manganese trichloride (TMMC)
journal, April 1993

One-dimensional diffusion-limited relaxation of photoexcitations in suspensions of single-walled carbon nanotubes
journal, July 2006

Measurement of a Reaction-Diffusion Crossover in Exciton-Exciton Recombination inside Carbon Nanotubes Using Femtosecond Optical Absorption
journal, November 2013

Alternating kinetics of annihilating random walks near a free interface
journal, March 1998

Finite-size scaling studies of one-dimensional reaction-diffusion systems. Part I. Analytical results
journal, March 1995

Scaling Theory for Finite-Size Effects in the Critical Region
journal, June 1972

Finite-size scaling at first-order phase transitions
journal, August 1984

A Guide to First-Passage Processes
book, January 2001

Solution of the spatial neutral model yields new bounds on the Amazonian species richness
journal, February 2017