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Analytical Description of the Diffusion in a Cellular Automaton with the Margolus Neighbourhood in Terms of the Two-Dimensional Markov Chain

Author

Listed:
  • Anton E. Kulagin

    (Division for Electronic Engineering, Tomsk Polytechnic University, 30 Lenina av., 634050 Tomsk, Russia
    These authors contributed equally to this work.)

  • Alexander V. Shapovalov

    (Department of Theoretical Physics, Tomsk State University, 1 Novosobornaya Sq., 634050 Tomsk, Russia
    Laboratory for Theoretical Cosmology, International Centre of Gravity and Cosmos, Tomsk State University of Control Systems and Radioelectronics, 40 Lenina av., 634050 Tomsk, Russia
    These authors contributed equally to this work.)

Abstract

The one-parameter two-dimensional cellular automaton with the Margolus neighbourhood is analyzed based on considering the projection of the stochastic movements of a single particle. Introducing the auxiliary random variable associated with the direction of the movement, we reduce the problem under consideration to the study of a two-dimensional Markov chain. The master equation for the probability distribution is derived and solved exactly using the probability-generating function method. The probability distribution is expressed analytically in terms of Jacobi polynomials. The moments of the obtained solution allowed us to derive the exact analytical formula for the parametric dependence of the diffusion coefficient in the two-dimensional cellular automaton with the Margolus neighbourhood. Our analytic results agree with earlier empirical results of other authors and refine them. The results are of interest for the modelling two-dimensional diffusion using cellular automata especially for the multicomponent problem.

Suggested Citation

  • Anton E. Kulagin & Alexander V. Shapovalov, 2023. "Analytical Description of the Diffusion in a Cellular Automaton with the Margolus Neighbourhood in Terms of the Two-Dimensional Markov Chain," Mathematics, MDPI, vol. 11(3), pages 1-18, January.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:3:p:584-:d:1043899
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    References listed on IDEAS

    as
    1. Bastien Chopard & Laurent Frachebourg & Michel Droz, 1994. "Multiparticle Lattice Gas Automata For Reaction Diffusion Systems," International Journal of Modern Physics C (IJMPC), World Scientific Publishing Co. Pte. Ltd., vol. 5(01), pages 47-63.
    2. Wai Ki Ching & Eric S. Fung & Michael K. Ng, 2004. "Higher‐order Markov chain models for categorical data sequences," Naval Research Logistics (NRL), John Wiley & Sons, vol. 51(4), pages 557-574, June.
    3. Alexander V. Shapovalov & Anton E. Kulagin, 2021. "Semiclassical Approach to the Nonlocal Kinetic Model of Metal Vapor Active Media," Mathematics, MDPI, vol. 9(23), pages 1-17, November.
    4. Hari Mohan Srivastava, 2022. "Some Families of Generating Functions Associated with Orthogonal Polynomials and Other Higher Transcendental Functions," Mathematics, MDPI, vol. 10(20), pages 1-23, October.
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