Advertisement

Journal of Statistical Physics

, Volume 80, Issue 3–4, pp 517–543 | Cite as

Ballistic annihilation and deterministic surface growth

  • Vladimir Belitsky
  • Pablo A. Ferrari
Articles

Abstract

A model of deterministic surface growth studied by Krug and Spohn, a model of the annihilating reactionA+B→inert studied by Elskens and Frisch, a one-dimensional three-color cyclic cellular automaton studied by Fisch, and a particular automaton that has the number 184 in the classification of Wolfram can be studied via a cellular automaton with stochastic initial data called ballistic annihilation. This automaton is defined by the following rules: At timet=0, one particle is put at each integer point of ℝ. To each particle, a velocity is assigned in such a way that it may be either +1 or −1 with probabilities 1/2, independent of the velocities of the other particles. As time goes on, each particle moves along ℝ at the velocity assigned to it and annihilates when it collides with another particle. In the present paper we compute the distribution of this automaton for each timet ∈ ℕ. We then use this result to obtain the hydrodynamic limit for the surface profile from the model of deterministic surface growth mentioned above. We also show the relation of this limit process to the process which we call moving local minimum of Brownian motion. The latter is the processB x min ,x ∈ ℝ, defined byB x min ≔min{B y ;x−1≤yx+1} for everyx ∈ ℝ, whereB x ,x ∈ ℝ, is the standard Brownian motion withB0=0.

Key Words

Cellular automaton deterministic model of surface growth ballistic annihilation three-color cyclic cellular automaton annihilating two-species reaction hydrodynmic limit moving local minimum of Brownian motion 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    P. Billingsley,Convergence of Probability Measures (Wiley, New York, 1968).Google Scholar
  2. 2.
    E. Ben-Naim, S. Redner, and F. Leyvraz, Decay kinetics of ballistic annihilation,Phys. Rev. Lett. 70:1890–1893 (1993).Google Scholar
  3. 3.
    Y. Elskens and H. L. Frisch, Annihilation kinetics in the one-dimensional ideal gas,Phys. Rev. A 31(6):3812–3816 (1985).Google Scholar
  4. 4.
    W. Feller,An Introduction to Probability Theory and Its Application, Vol. I (Wiley, New York, 1964).Google Scholar
  5. 5.
    R. Fisch, Clustering in the one-dimensional three-color cyclic cellular automaton,Ann. Prob. 20(3):1528–1548 (1992).Google Scholar
  6. 6.
    B. V. Gnedenko and A. N. Kolmogorov,Limit Distributions for Sums of Independent Random Variables (Addison-Wesley, Reading, Massachusetts, 1954).Google Scholar
  7. 7.
    I. Karatzas and S. E. Shreve,Brownian Motion and Stochastic Calculus (Springer-Verlag, Berlin, 1991).Google Scholar
  8. 8.
    J. Krug and H. Spohn, Universility classes for deterministic surface growth,Phys. Rev. A 38:4271–4283 (1988).Google Scholar
  9. 9.
    J. Neveu and J. Pitman, Renewal property of the extrema and tree property of the excursion of a one-dimensional Brownian motion, inSéminare de Probabilités XXIII, J. Azéma, P. A. Meyer, and M. Yor, eds. (Springer-Verlag, Berlin, 1989).Google Scholar
  10. 10.
    J. Neveu and J. Pitman, The branching process in a Brownian excursion, inSéminare de Probabilités XXIII, (Springer-Verlag, 1989).Google Scholar
  11. 11.
    S. Wolfram, Statistical mechanics of cellular automata,Rev. Mod. Phys. 55:601 (1983).Google Scholar

Copyright information

© Plenum Publishing Corporation 1995

Authors and Affiliations

  • Vladimir Belitsky
    • 1
  • Pablo A. Ferrari
    • 1
  1. 1.Instituto de Matematica e EstatísticaUniversidade de São PauloSão PauloBrazil

Personalised recommendations