Abstract
In this article, we review the problem of reaction annihilation \(A+A \rightarrow \emptyset \) on a real lattice in one dimension, where A particles move ballistically in one direction with a discrete set of possible velocities. We first discuss the case of pure ballistic annihilation, that is a model in which each particle moves simultaneously at constant speed. We then review ballistic annihilation with superimposed diffusion in one dimension. This model consists of diffusing particles each of which diffuses with a fixed bias, which can be either positive or negative with probability 1/2, and annihilate upon contact. When the initial concentration of left- and right-moving particles is same, the concentration c(t) decays as \(t^{-1/2}\) with time, for pure ballistic annihilation. However, when the diffusion is superimposed decay is faster and the concentration \(c(t) \sim t^{-3/4}\). We also discuss the nearest-neighbor distance distribution as well as crossover behavior.
Graphical Abstract
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data Availability Statement
This manuscript has no associated data or the data will not be deposited. [Authors’ comment: The manuscript contains several theoretical works and does not contain any empirical data.]
References
For a review of diffusion-controlled annihilation, see S. Redner, Nonequilibrium Statistical Mechanics in One Dimension ed V. Privman, Cambridge: Cambridge University Press (1996)
P.L. Krapivsky, S. Redner, E. Ben-Naim, A kinetic view of statistical physics (Cambridge University Press, Cambridge, 2010)
J.L. Spouge, Exact solutions for a diffusion-reaction process in one dimension. Phys. Rev. Lett. 60, 871 (1988)
F. Leyvraz, N. Jan, Critical dynamics for one-dimensional models. J. Phys. A 19, 603–605 (1986)
J.L. Spouge, Exact solutions for a diffusion-reaction process in one dimension: 11. Spatial distributions. J. Phys. A 21, 4183 (1988)
S. Biswas and M. M. Saavedra Contreras, “Zero-temperature ordering dynamics in a two-dimensional biaxial next-nearest-neighbor Ising model” Phys. Rev. E 100, 042129 (2019)
I. Ispolatov, P.L. Krapivsky, S. Redner, War: The dynamics of vicious civilizations. Phys. Rev. E 54, 1274 (1996)
S. Biswas, P. Sen, Model of binary opinion dynamics: Coarsening and effect of disorder. Phys. Rev. E 80, 027101 (2009)
S. Biswas, P. Sen, and P. Ray, “Opinion dynamics model with domain size dependent dynamics: novel features and new universality class” J. Phys.: Conf. Series 297, 012003 (2011)
Y. Elskens, H.L. Frisch, Annihilation kinetics in the one-dimensional ideal gas. Phys. Rev. A 31, 3812 (1985)
E. Ben-Naim, S. Redner, P.L. Krapivsky, Two scales in asynchronous ballistic annihilation. J. Phys. A 29, L561 (1996)
William Feller, An Introduction to Probability Theory and Its Applications, John Wiley and Sons Ltd; 3rd edition (January 1, 1968)
P.L. Krapivsky, S. Redner, F. Leyvraz, Ballistic annihilation kinetics: the case of discrete velocity distributions. Phys. Rev. E 51, 3977 (1995)
J. Piasecki, Ballistic annihilation in a one-dimensional fluid. Phys. Rev. E 51(6), 5535–5540 (1995)
M. Droz, P.-A. Rey, L. Frachebourg, J. Piasecki, Ballistic-annihilation kinetics for a multivelocity one-dimensional ideal gas. Phys. Rev. E 51(6), 5541–5548 (1995)
S. Harris, An introduction to the theory of the Boltzmann equation, Courier Corporation (2004)
E. ben-Naim, S. Redner y F. Leyvraz,“Decay kinetics of ballistic annihilation” Phys. Rev. Lett. 70 1890 (1993)
S. Biswas, H. Larralde, F. Leyvraz, “Ballistic annihilation with superimposed diffusion in one dimension”Phys. Rev. E 93, 022136 (2016)
P.A. Alemany, Novel decay laws for the one-dimensional reaction-diffusion model as consequence of initial distributions. J. Phys. A 30, 3299 (1997)
G.H. Weiss, Aspects and applications of the random walk Random Materials and Processes; ed H (E. Stanley and E, Guyon, 1994)
B.D. Hughes, Random Walks and Random Environments, vol. 1 (Clarendon, Oxford, 1995)
C.R. Doering and D. Ben-Avraham, D. “Interparticle distribution functions and rate equations for diffusion–limited reactions”. Phys. Rev. A, 38 (6) 3035 (1988)
Yu. Jiang, F. Leyvraz, Kinetics of two-species ballistic annihilation. Phys. Rev. E 50, 608 (1994)
M.J.E. Richardson, Exact solution of two-species ballistic annihilation with general pair-reaction probability. J. Stat. Phys. 89, 777 (1997)
J. Masoliver, G.H. Weiss, TelegrapherÕs equations with variable propagation speeds. Phys. Rev. E 49, 3852 (1994)
S.K. Foong, S. Kanno, Properties of the telegrapher’s random process with or without a trap. Stochastic Processes Appl. 53, 147 (1994)
G.H. Weiss, Aspects and applications of the random walk (North-Holland, Amsterdam, 1994)
Acknowledgements
Financial support from the project of CONACyT Ciencia de Frontera 2019, Number 10872, is gratefully acknowledged. FL acknowledges the financial support from the project of CONACyT Ciencias Basicas, Number 254515.
Author information
Authors and Affiliations
Contributions
All authors contributed equally to the paper.
Corresponding author
Additional information
The original online version of this article was revised to delete a typesetting comment that has been left in the article: “query Please provide an explanation as to why there is no data or why the data will not be deposited. Your explanation will be displayed as ‘Authors’ comment’.”.
Rights and permissions
About this article
Cite this article
Biswas, S., Leyvraz, F. Ballistic annihilation in one dimension: a critical review. Eur. Phys. J. B 94, 240 (2021). https://doi.org/10.1140/epjb/s10051-021-00258-w
Received:
Accepted:
Published:
DOI: https://doi.org/10.1140/epjb/s10051-021-00258-w