# 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.**

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**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