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One-Dimensional Lattice Dynamics of the Diffusion-Limited Reaction A+A→A+S: Transient Behavior

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Abstract

We use a Boolean cellular automaton model to describe the diffusion-limited dynamics of the irreversible reaction A+A→A+S on a 1D lattice. We derive a set of equations for the dynamics of the empty interval probabilities from which explicit expressions for the particle concentration and the two-point correlation can be obtained. It is shown that the long-time dynamics is in agreement with the off-lattice solution. The early-time behavior, however, predicts a slower decay of the concentration.

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REFERENCES

  1. D. ben Avraham, M. Burschka, and C. Doering, J. Stat. Phys. 60:695 (1990).

    Google Scholar 

  2. E. Abad, G. Nicolis, and P. Grosfils, Nonlinear reactive systems viewed as stochastic dynamical systems, unpublished.

  3. V. Privman, Phys. Rev. E 50:50 (1994).

    Google Scholar 

  4. See J. L. Spouge, Phys. Rev. Lett. 60:871 (1988) when initial “concentration” p is 1/2; for the case with reverse reaction see J. C. Lin, C. R. Doering, and D. ben-Avraham, Chem. Phys. 146:355 (1990). There exist some exact solutions of the equivalent diffusion-annihilation process A+A→S+S, see, e.g., R. J. Glauber, J. Math. Phys. 4:294 (1963); T. Liggett, Interacting Particle Systems (Springer-Verlag, New York, 1985); S. Redner and K. Kang, Phys. Rev. A 32:435 (1985); A. A. Lushnikov, Phys. Lett. A 120:135 (1987); V. Kuzokov and E. Kotomin, Rep. Prog. 87:1941 (1988); M. Bramson and J. Lebowitz, Phys. Rev. Lett. 61:2397 (1988); D. Balding, P. Clifford, and N. J. B. Green, Phys. Lett. A 126:481 (1988); F. Family and J. G. Amar, J. Stat. Phys. 65:1235 (1991); M. D. Grynberg and Robin B. Stinchcombe, Phys. Rev. Lett. 76:851 (1996) and references therein.

    Google Scholar 

  5. J. Crank, The Mathematics of Diffusion, 2nd ed. reprinted (Clarendon, Oxford, 1976).

    Google Scholar 

  6. I. M. Ryshik and I. S. Gradstein, Tables of Series, Products and Integrals, 2nd ed. (VEB Deutscher Verlag der Wissenschaften, Berlin, 1963).

    Google Scholar 

  7. H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, 2nd ed. (Clarendon, Oxford, 1959).

    Google Scholar 

  8. G. Doetsch, Theorie und Anwendung der Laplace Transformation (Dover Pub., New York, 1943).

    Google Scholar 

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Authors and Affiliations

  1. Center for Nonlinear Phenomena and Complex Systems, Université Libre de Bruxelles, B-1050, Brussels, Belgium

    Enrique Abad & Gregoire Nicolis

  2. Department of Chemistry, State University of New York at Albany, Albany, New York, 12222

    Harry L. Frisch

Authors
  1. Enrique Abad
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  2. Harry L. Frisch
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  3. Gregoire Nicolis
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Abad, E., Frisch, H.L. & Nicolis, G. One-Dimensional Lattice Dynamics of the Diffusion-Limited Reaction A+A→A+S: Transient Behavior. Journal of Statistical Physics 99, 1397–1407 (2000). https://doi.org/10.1023/A:1018649108442

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  • DOI: https://doi.org/10.1023/A:1018649108442

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