Concomitant clusters structure creating by Hammersley--Leath--Alexandrowichz algorithm for~percolation cluster generating

D. V. Alekseev ; G. A. Kazunina

Prikladnaâ diskretnaâ matematika, no. 1 (2020), pp. 117-127.

Résumé

A three-dimensional cellular automaton implementing the simulation of growth of a percolation cluster on a simple cubic lattice according to the Hammersley–Leath–Alexandrowichz algorithm was built for the first time introducing into consideration the concomitant cluster structure formed out of cells excluded from the growth process. The concomitant cluster structure is modelled in a wide interval of perimeter germination probability values $0{,}3117$ on a $100 \times 100 \times 100$ lattice and analyzed by using the functions of distribution of number and mass of clusters of the accompanying structure by size. As a result of the computational experiment, there were obtained dependencies on the probability of perimeter germination for such basic characteristics of the cluster structure as the mass of the main cluster; the mass of the maximum cluster of concomitant structure; the total mass of the concomitant structure; mean-square radii of the main cluster and the maximum cluster of the concomitant structure; the number of clusters of concomitant structure; mass ratio of the maximum cluster of the concomitant structure to the mass of the main cluster. It has been established that in the interval of germination probability $0{,}3117$ in the concomitant structure, the dominant cluster is formed with the mean-square radius close to the mean-square radius of the main cluster. With a further increase in probability of germination, the size of the dominant cluster decreases sharply, and at $P\leq 0{,}67$ its decay is observed.

Détail

Keywords: cellular automaton, kinetic growth models, Hammersley–Leath–Alexandrowichz algorithm.

Bibliographie

[1] Feder J., Fractals, Springer, N.Y., 1988, 258 pp. | MR

[2] Guld Kh., Tobochnik Ya., Computer Modeling in Physics, v. 2, Mir Publ., M., 1990, 390 pp. (in Russian)

[3] Bandman O. L., “Discrete models of physical-chemical processes”, Prikladnaya Diskretnaya Matematika, 2009, no. 3(5), 33–49 (in Russian)

[4] Bandman O. L., “A discrete stochastic model of water permeation through a porous substance: parallel implementation peculiarities”, Numer. Analys. Appl., 11:1, 4–15 | DOI | MR

[5] Alekseev D. V., Kazunina G. A., “Modeling the of damage accumulation of probabilistic cellular automata”, Fizika Tverdogo Tela, 48:2 (2006), 255–261 (in Russian)

[6] Alekseev D. V., Kazunina G. A., Cherednichenko A. V., “Cellular automaton simulation of the fracture process for brittle materials”, Prikladnaya Diskretnaya Matematika, 2015, no. 2(28), 103–118 (in Russian)

[7] Lobanov A. I., “Model of cellular automata”, Computer Research and Modeling, 2:3 (2010), 273–293 (in Russian) | MR

[8] Matyushkin I. V., Zapletina M. A., “Review on the subject of cellular automata on the basis of modern Russian publications”, Computer Research and Modeling, 11:1 (2019), 9–57 (in Russian) | MR

[9] Tarasevich Yu. Yu., Percolation: theory, applications, algorithms, Librokom Publ., M., 2018, 112 pp. (in Russian)

[10] Moskalev P. V., “Analysis of the percolation cluster structure”, J. Tehnicheskoi Fiziki, 79:6 (2009), 2–7 (in Russian)

[11] Moskalev P. V., “The structure of site percolation models on three-dimensional square lattices”, Computer Research and Modeling, 5:4 (2013), 607–622 (in Russian) | MR

[12] Alexandrowicz Z., “Critically branched chains and percolation clusters”, Phys. Lett. A, 80:4 (1980), 284–286 | DOI | MR

[13] Leath P., “Cluster size and boundary distribution near percolation threshold”, Phys. Rev. B, 14:11 (1976), 5046–5055 | DOI

[14] Alekseev D. V., Kazunina G. A., “Concomitant clusters structure creating by percolation cluster generating Hammersley–Leath–Alexandrowichz algorithm”, Aktualnie Napravleniya Nauchnih Issledovanii XXI Veka: Teorya i Praktika, 2018, no. 6(42), 18–20 (in Russian)

[15] Alekseev D. V., An Introduction to Computer Simulation Physical Systems: Using Microsoft Visual Basic, LENAND Publ., M., 2019, 272 pp. (in Russian)