Direction-Cut Method and Two-Dimensional Bond Percolation in Basic Archimedes Lattices Addressed on Square Latis

Öz

Percolation theory is a statistical approach that aims to understand and identify events that occur during phase transitions on networks. In this study, with the developed direction-cut method, it has been shown that percolation threshold and critical exponents can be determined with Monte Carlo-based simulation on two-dimensional grids that are the most basic and common application of percolation theory. By considering the neighbors of any site on the lattice, the method of defining the bond potential of this site with an array is based on cutting the related directions from the array so that the established bonds are not repeated and the infeasible bonds are not established. In addition, the algorithm, which allows examining the cluster behavior with the cluster system based on reference numbers, enables the detection of universal behaviors and critical exponents. In order to test the method, percolation simulations have been performed in three different Archimedean lattices addressed on square grids. In two-dimensional bond percolation, on a 1000 × 1000 grid, the percolation thresholds for triangular, square, and honeycomb lattices, exact values of which are approximately 0.3473, 0.5 and 0.6527 respectively, have been found as 0.3469 ± 0.0016 for the triangular lattice, 0.4992 ± 0.0022 for the square lattice, and 0.6510 ± 0.0027 for the honeycomb lattice. Also, critical exponents such as β, γ, ν, and fractal dimension D, values of which are universal in two-dimensions and the exact values are 5/36 (~0.1389), 43/18 (~2.3889), 4/3 (~1.3333), and 91/48 (~1.8958), respectively, have been found as 0.1389, 0.1386, and 0.1390 for β; 2.3886, 2.3272, and 2.3275 for γ; 1.3326, 1.3392, and 1.3168 for ν; and finally 1.8801, 1.8729, and 1.8932 for D in triangular, square and honeycomb lattices, respectively. The results of the simulations which are quite convenient for all three different lattices showed that the direction-cut method is a strong candidate to be an effective and relatively easy algorithm for percolation simulations targeted to different lattice types and potentially in different dimensions.

Anahtar Kelimeler

• Percolation Theory
• Percolation Threshold
• Critical Exponents
• Direction-Cut Method

Yön-Kesme Yöntemi ve Kare Izgarada Adreslenmiş Temel Arşimet Latislerinde İki-Boyutlu Bağ Perkolasyonu

Öz

Perkolasyon teorisi, ağ yapıları üzerinde faz geçişleri sırasında gerçekleşen olayları anlamayı ve tanımlamayı hedefleyen istatistiksel bir yaklaşımdır. Yapılan bu çalışmada, geliştirilen yön-kesme yöntemi tanıtılarak, perkolasyon teorisinin en temel ve yaygın uygulaması olan iki-boyutlu ızgaralar üzerinde, Monte Carlo temelli bir benzetimde, perkolasyon eşiği ve kritik üstellerin tespit edilebileceği gösterilmiştir. Latis üzerindeki herhangi bir düğüm noktasına ait komşuları gözeterek, bu noktaya ait bağ potansiyelini bir dizi ile tanımlayan yöntem, kurulmuş bağların tekrarlanmaması ve tanımlı olmayan bağların kurulmaması için ilgili yönlerin kesilerek diziden çıkartılmasına dayanmaktadır. Ayrıca, referans numaralarına dayalı kümelenme sistemiyle, küme davranışlarını incelemeye olanak sağlayan algoritma, bu yöntemle evrensel davranışların ve kritik üstellerin de tespit edilmesini sağlamaktadır. Çalışmada, yöntemin sınanması adına, kare ızgara üzerine adreslenen üç farklı temel Arşimet latislerinde, perkolasyon benzetimleri yapılmıştır. İki-boyutta bağ perkolasyonu için, üçgen, kare ve bal peteği latislerinde, gerçek değerleri sırasıyla yaklaşık olarak 0,3473, 0,5 ve 0,6527 olan perkolasyon eşikleri, 1000×1000 boyutlarına sahip bir ızgara üzerinde üçgen latis için 0,3469±0,0016, kare latis için 0,4992±0,0022 ve bal peteği latis için 0,6510±0,0027 olarak tespit edilmiştir. Yine değerleri iki-boyutta evrensel olan kritik üsteller β, γ, ν ve fraktal boyut D, gerçek değerleri sırasıyla 5/36 (~0,1389), 43/18 (~2,3889), 4/3 (~1,3333) ve 91/48 (~1,8958) olmak üzere, üçgen, kare ve bal peteği latislerinde sırasıyla, β değeri 0,1389, 0,1386 ve 0,1390, γ değeri 2,3886, 2,3272 ve 2,3275, ν değeri 1,3326, 1,3392 ve 1,3168 ve son olarak D değeri 1,8801, 1,8729 ve 1,8932 şeklide tespit edilmiştir. Her üç farklı latis için de oldukça uygun sonuçların elde edildiği benzetimler ışığında gösterilmiştir ki yön-kesme yöntemi farklı latis tiplerinde ve potansiyelde farklı boyutlarda, gerçekleştirilmesi hedeflenen perkolasyon benzetimlerine etkin ve nispeten kolay bir algoritma olmaya aday durumdadır.

Anahtar Kelimeler

• Perkolasyon Eşiği
• Perkolasyon Teorisi
• Kritik Üsteller
• Yön-Kesme Yöntemi

Kaynakça

1. Araújo, N., Grassberger, P., Kahng, B., Schrenk, K. J. ve Ziff, R. M. (2014). Recent advances and open challenges in percolation. European Physical Journal: Special Topics, C. 223, ss. 2307–2321. Springer Verlag.
2. Broadbent, S. R. ve Hammersley, J. M. (1957). Percolation processes. Mathematical Proceedings of the Cambridge Philosophical Society, 53(3), 629–641.
3. Cook, A., Blom, H. A. P., Lillo, F., Mantegna, R. N., Miccichè, S., Rivas, D., Vazquez, R. ve Zanin, M. (2015). Applying complexity science to air traffic management. Journal of Air Transport Management, 42, 149–158.
4. Feldman, D. (2008). Polymer history. Designed Monomers and Polymers, 11(1), 1–15.
5. Fernandez-Anez, N., Christensen, K. ve Rein, G. (2017). Two-dimensional model of smouldering combustion using multi-layer cellular automaton: The role of ignition location and direction of airflow. Fire Safety Journal, 91, 243–251.
6. Fisher, M. E. (1961). Critical Probabilities for Cluster Size and Percolation Problems. Journal of Mathematical Physics, 2(4), 620–627.
7. Fisher, M. E. ve Essam, J. W. (1961). Some Cluster Size and Percolation Problems. Journal of Mathematical Physics, 2(4), 609–619.
8. Flory, P. J. (1941a). Molecular Size Distribution in Three Dimensional Polymers. I. Gelation. Journal of the American Chemical Society, 63(11), 3083–3090.

9. Flory, P. J. (1941b). Molecular Size Distribution in Three Dimensional Polymers. II. Trifunctional Branching Units. Journal of the American Chemical Society, 63(11), 3091–3096.
10. Flory, P. J. (1941c). Molecular Size Distribution in Three Dimensional Polymers. III. Tetrafunctional Branching Units. Journal of the American Chemical Society, 63(11), 3096–3100.
11. Hoshen, J. ve Kopelman, R. (1976). Percolation and cluster distribution. I. Cluster multiple labeling technique and critical concentration algorithm. Physical Review B, 14(8), 3438–3445.
12. Hoshen, J., Kopelman, R. ve Monberg, E. M. (1978). Percolation and cluster distribution. II. layers, variable-range interactions, and exciton cluster model. Journal of Statistical Physics, 19(3), 219–242.
13. Kaynan, O., Yıldız, A., Bozkurt, Y. E., Yenigun, E. O. ve Cebeci, H. (2020). Electrically Conductive High–Performance Thermoplastic Filaments for Fused Filament Fabrication. Composite Structures, 237, 111930.
14. Li, J., Ma, P. C., Chow, W. S., To, C. K., Tang, B. Z. ve Kim, J.-K. (2007). Correlations between Percolation Threshold, Dispersion State, and Aspect Ratio of Carbon Nanotubes. Advanced Functional Materials, 17(16), 3207–3215.
15. Newman, M. E. J. ve Ziff, R. M. (2001). Fast Monte Carlo algorithm for site or bond percolation. Physical Review E, 64(1), 016706.
16. Parviainen, R. (2007). Estimation of bond percolation thresholds on the Archimedean lattices. Journal of Physics A: Mathematical and Theoretical, 40(31), 9253–9258.
17. Saberi, A. A. (2015). Recent advances in percolation theory and its applications. Physics Reports, 578, 1–32.
18. Solomon, S., Weisbuch, G., de Arcangelis, L., Jan, N. ve Stauffer, D. (2000). Social percolation models. Physica A: Statistical Mechanics and its Applications, 277(1–2), 239–247.
19. Stauffer, D. (1979). Scaling theory of percolation clusters. Physics Reports, 54(1), 1–74.
20. Stauffer, D. ve Aharony, A. (1985). Introduction to Percolation Theory. Introduction to Percolation Theory. Abingdon, UK: Taylor & Francis.
21. Stauffer, D., Coniglio, A. ve Adam, M. (1982). Gelation and critical phenomena. Polymer Networks (C. 44, ss. 103–158). Berlin, Heidelberg: Springer Berlin Heidelberg.
22. Suding, P. N. ve Ziff, R. M. (1999). Site percolation thresholds for Archimedean lattices. Physical Review E, 60(1), 275–283.
23. Sykes, M. F. ve Essam, J. W. (1963). Some Exact Critical Percolation Probabilities for Bond and Site Problems in Two Dimensions. Physical Review Letters, 10(1), 3–4.
24. Sykes, M. F. ve Essam, J. W. (1964). Exact Critical Percolation Probabilities for Site and Bond Problems in Two Dimensions. Journal of Mathematical Physics, 5(8), 1117–1127.
25. Trompeta, A.-F., Koumoulos, E., Stavropoulos, S., Velmachos, T., Psarras, G. ve Charitidis, C. (2019). Assessing the Critical Multifunctionality Threshold for Optimal Electrical, Thermal, and Nanomechanical Properties of Carbon Nanotubes/Epoxy Nanocomposites for Aerospace Applications. Aerospace, 6(1), 7.
26. Tüzel, E., Özmetin, M. S., Yýlmaz, Y. ve Pekcan, Ö. (2000). A new critical point and time dependence of bond formation probability in sol–gel transition: a Monte Carlo study in two dimension. European Polymer Journal, 36(4), 727–733.
27. Vogel, E. E., Lebrecht, W. ve Valdés, J. F. (2010). Bond percolation for homogeneous two-dimensional lattices. Physica A: Statistical Mechanics and its Applications, 389(8), 1512–1520.
28. Winterfeld, P. H., Scriven, L. E. ve Davis, H. T. (1981). Percolation and conductivity of random two-dimensional composites. Journal of Physics C: Solid State Physics, 14(17), 2361–2376.
29. Zeng, G., Li, D., Guo, S., Gao, L., Gao, Z., Stanley, H. E. ve Havlin, S. (2019). Switch between critical percolation modes in city traffic dynamics. Proceedings of the National Academy of Sciences, 116(1), 23–28.
30. Zerko, S., Polanowski, P. ve Sikorski, A. (2012). Percolation in two-dimensional systems containing cyclic chains. Soft Matter, 8(4), 973–979.