IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v11y2023i3p584-d1043899.html
   My bibliography  Save this article

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
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/11/3/584/pdf
    Download Restriction: no

    File URL: https://www.mdpi.com/2227-7390/11/3/584/
    Download Restriction: no
    ---><---

    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.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Hari Mohan Srivastava, 2022. "Higher Transcendental Functions and Their Multi-Disciplinary Applications," Mathematics, MDPI, vol. 10(24), pages 1-3, December.
    2. Yang, Ningkang & Han, Lijin & Xiang, Changle & Liu, Hui & Li, Xunmin, 2021. "An indirect reinforcement learning based real-time energy management strategy via high-order Markov Chain model for a hybrid electric vehicle," Energy, Elsevier, vol. 236(C).
    3. Topuz, Kazim & Urban, Timothy L. & Yildirim, Mehmet B., 2024. "A Markovian score model for evaluating provider performance for continuity of care—An explainable analytics approach," European Journal of Operational Research, Elsevier, vol. 317(2), pages 341-351.
    4. Tie Liu, 2010. "Application of Markov Chains to Analyze and Predict the Time Series," Modern Applied Science, Canadian Center of Science and Education, vol. 4(5), pages 162-162, May.
    5. Anton E. Kulagin & Alexander V. Shapovalov, 2024. "A Semiclassical Approach to the Nonlocal Nonlinear Schrödinger Equation with a Non-Hermitian Term," Mathematics, MDPI, vol. 12(4), pages 1-22, February.
    6. Suryadeepto Nag & Sankarshan Basu & Siddhartha P. Chakrabarty, 2022. "Modeling the Commodity Prices of Base Metals in Indian Commodity Market Using a Higher Order Markovian Approach," Journal of Quantitative Economics, Springer;The Indian Econometric Society (TIES), vol. 20(1), pages 159-171, March.
    7. Flavio Ivo Riedlinger & João Nicolau, 2020. "The Profitability in the FTSE 100 Index: A New Markov Chain Approach," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 27(1), pages 61-81, March.
    8. Chenfeng Xiong & Di Yang & Lei Zhang, 2018. "A High-Order Hidden Markov Model and Its Applications for Dynamic Car Ownership Analysis," Service Science, INFORMS, vol. 52(6), pages 1365-1375, December.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:11:y:2023:i:3:p:584-:d:1043899. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.