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

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## Abstract

**:**

## 1. Introduction

## 2. Diffusion in the Margolus Cellular Automaton

## 3. One-Dimensional Movement of a Single Particle

## 4. Master Equation

## 5. Probability Generating Function

**Theorem 1.**

## 6. Discussion of the Results

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Proof of Theorem 1

**Lemma A1.**

**Proof.**

**Corollary A1.**

**Proof.**

## References

- Wolfram, S. A New Kind of Science; Wolfram Media: Champaign, IL, USA, 2002. [Google Scholar]
- Palchaudhuri, A.; Anand, D.; Dhar, A. FPGA fabric conscious architecture design and automation of speed-area efficient Margolus neighborhood based cellular automata with variegated scan path insertion. J. Parallel Distrib. Comput.
**2022**, 167, 50–63. [Google Scholar] [CrossRef] - Cicuttin, A.; De Micco, L.; Crespo, M.; Antonelli, M.; Garcia, L.; Florian, W. Physical implementation of asynchronous cellular automata networks: Mathematical models and preliminary experimental results. Nonlinear Dyn.
**2021**, 105, 2431–2452. [Google Scholar] [CrossRef] - Cagigas-Muñiz, D.; Diaz-del Rio, F.; Sevillano-Ramos, J.; Guisado-Lizar, J.L. Efficient simulation execution of cellular automata on GPU. Simul. Model. Pract. Theory
**2022**, 118, 102519. [Google Scholar] [CrossRef] - Matolygin, A.; Shalyapina, N.; Gromov, M.; Torgaev, S. Tensor approach to software implementation of cellular automata model of diffusion. J. Phys.: Conf. Ser.
**2020**, 1680, 012035. [Google Scholar] [CrossRef] - Toffoli, T.; Margolus, N. Cellular Automata Machines: A New Environment for Modeling; MIT Press: Cambridge, MA, USA, 1987; p. 276. [Google Scholar]
- Kireeva, A.; Sabelfeld, K.K.; Kireev, S. Synchronous multi-particle cellular automaton model of diffusion with self-annihilation. In Parallel Computing Technologies; Lecture Notes in Computer Science (Including Subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Springer: Cham, Switzerland, 2019; Volume 11657, pp. 345–359. [Google Scholar] [CrossRef]
- Fick, A. On liquid diffusion. J Membr Sci. J. Membr. Sci.
**1995**, 100, 33–38. [Google Scholar] [CrossRef] - Paul, A.; Laurila, T.; Vuorinen, V.; Divinski, S. Fick’s Laws of Diffusion. In Thermodynamics, Diffusion and the Kirkendall Effect in Solids; Springer: Cham, Switzerland, 2014; pp. 115–139. [Google Scholar] [CrossRef]
- Shapovalov, A.; Kulagin, A. Semiclassical approach to the nonlocal kinetic model of metal vapor active media. Mathematics
**2021**, 9, 2995. [Google Scholar] [CrossRef] - Shapovalov, A.; Kulagin, A.; Siniukov, S. Family of Asymptotic Solutions to the Two-Dimensional Kinetic Equation with a Nonlocal Cubic Nonlinearity. Symmetry
**2022**, 14, 577. [Google Scholar] [CrossRef] - Odintsov, S. Editorial for Feature Papers 2021–2022. Symmetry
**2022**, 15, 32. [Google Scholar] [CrossRef] - Mickens, R. Nonstandard finite difference schemes for reaction--diffusion equations having linear advection. Numer. Methods Partial Differ. Equations
**2000**, 16, 361–364. [Google Scholar] [CrossRef] - Pankov, P.; Zheentaeva, Z.; Shirinov, T. Asymptotic reduction of solution space dimension for dynamic systems. TWMS J. Pure Appl. Math.
**2021**, 12, 243–253. [Google Scholar] - Shokri, A. The multistep multiderivative methods for the numerical solution of first order initial value problems. TWMS J. Pure Appl. Math.
**2016**, 7, 88–97. [Google Scholar] - Rachinskaya, M.; Fedotkin, M. Research of a multidimensional Markov Chain as a model for the class of queueing systems controlled by a threshold priority algorithm. Reliab. Theory Appl.
**2018**, 13, 47–58. [Google Scholar] - Ching, W.K.; Fung, E.S.; Ng, M.K. Higher-order Markov chain models for categorical data sequences. Nav. Res. Logist.
**2004**, 51, 557–574. [Google Scholar] [CrossRef] - Morzfeldi, M.; Tong, X.T.; Marzouk, Y.M. Localization for MCMC: Sampling high-dimensional posterior distributions with local structure. J. Comput. Phys.
**2019**, 380, 1–28. [Google Scholar] [CrossRef][Green Version] - Ching, W.K.; Ng, M.K.; Fung, E.S. Higher-order multivariate Markov chains and their applications. Linear Algebra Its Appl.
**2008**, 428, 492–507. [Google Scholar] [CrossRef][Green Version] - Bandman, O.L. Comparative study of cellular-automata diffusion models. Parallel Comput. Technol.
**1999**, 1662, 395–409. [Google Scholar] [CrossRef] - Malinetskii, G.G.; Stepantsov, M.E. Modeling of diffusion processes by cellular automata with Margolus neighborhood. Comput. Math. Math. Phys.
**1998**, 38, 973–975. [Google Scholar] - Bandman, O.L. Invariants of cellular automata models for reaction-diffusion processes. Appl. Discret. Math.
**2012**, 3, 108–120. (In Russian) [Google Scholar] [CrossRef] - Shalyapina, N.; Gromov, M.; Matolygin, A.; Torgaev, S. Empirical dependence of the probability of blocks rotations on the diffusion coefficient in a cellular automaton with a Margolus neighbourhood. J. Phys.: Conf. Ser.
**2021**, 2140, 012031. [Google Scholar] [CrossRef] - Nelson, R. Probability, Stochastic Processes, and Queueing Theory; Springer: New York, NY, USA, 1995. [Google Scholar] [CrossRef]
- Gluzman, S. Padé and Post-Padé Approximations for Critical Phenomena. Symmetry
**2020**, 12, 1600. [Google Scholar] [CrossRef] - Medvedev, Y. Multi-particle cellular-automata models for diffusion simulation. In Methods and Tools of Parallel Programming Multicomputers; Lecture Notes in Computer Science; Springer: Berlin/Heidelberg, Germany, 2010; Volume 6083, pp. 204–211. [Google Scholar] [CrossRef]
- Bandman, O.L. Computation properties of spatial dynamics simulation by probabilistic cellular automata. Future Gener. Comput. Syst.
**2005**, 21, 633–643. [Google Scholar] [CrossRef] - Chopard, B.; Frachebourg, L.; Droz, M. Multiparticle lattice gas automata for reaction diffusion systems. Int. J. Mod. Phys. C
**1994**, 5, 47–63. [Google Scholar] [CrossRef] - Bateman, H.; Erdélyi, A. Higher Transcendental Functions; McGraw-Hill: New York, NY, USA, 1953; Volume 1. [Google Scholar]
- Srivastava, H.M. A Survey of Some Recent Developments on Higher Transcendental Functions of Analytic Number Theory and Applied Mathematics. Symmetry
**2021**, 13, 2294. [Google Scholar] [CrossRef] - Srivastava, H.M. Some Families of Generating Functions Associated with Orthogonal Polynomials and Other Higher Transcendental Functions. Mathematics
**2022**, 10, 3730. [Google Scholar] [CrossRef] - Bateman, H.; Erdélyi, A. Higher Transcendental Functions; McGraw-Hill: New York, NY, USA, 1953; Volume 2. [Google Scholar]

**Figure 1.**Odd and even partitions of the cell grid. Solid lines divide the cell grid into blocks of the odd partition and dashed lines divide the cell grid into blocks of the even partition. Arrows demonstrate the rotation rule.

**Figure 2.**Examples of possible movements of a particle in the MCA. The black dot is for the particle movement at an odd time step (the rotation rule applies for blocks of the odd partition) and the white dot is for the particle movement at an even time step (the rotation rule applies for block of the even partition).

**Figure 3.**The plot of the probability distribution ${P}_{t}\left(x\right)$ along with the plot of the normal probability distribution function ${f}_{t}\left(x\right)$ for $p=\frac{1}{3}$.

**Figure 4.**The plot of the probability distribution ${P}_{t}\left(x\right)$ along with the plot of the normal probability distribution ${f}_{t}\left(x\right)$ for $p=\frac{1}{2}$.

**Figure 5.**The plot of the probability distribution ${P}_{t}\left(x\right)$ along with the plot of the normal probability distribution ${f}_{t}\left(x\right)$ for $p=\frac{3}{4}$.

**Figure 6.**The time plot of the dispersion of the ${X}_{t}$ for two types of the MCA. It illustrates the macroscopic behaviour of these MCA at small times. The parameters in (

**a**) correspond to MCA with the diffusion coefficient of $0.25$, and the parameters in (

**b**) correspond to MCA with the diffusion coefficient of $0.125$.

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**MDPI and ACS Style**

Kulagin, A.E.; Shapovalov, A.V.
Analytical Description of the Diffusion in a Cellular Automaton with the Margolus Neighbourhood in Terms of the Two-Dimensional Markov Chain. *Mathematics* **2023**, *11*, 584.
https://doi.org/10.3390/math11030584

**AMA Style**

Kulagin AE, Shapovalov AV.
Analytical Description of the Diffusion in a Cellular Automaton with the Margolus Neighbourhood in Terms of the Two-Dimensional Markov Chain. *Mathematics*. 2023; 11(3):584.
https://doi.org/10.3390/math11030584

**Chicago/Turabian Style**

Kulagin, Anton E., and 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* 11, no. 3: 584.
https://doi.org/10.3390/math11030584