Back
 JAMP  Vol.4 No.8 , August 2016
A Study on Numerical Calculation Method of Small Cluster Density in Percolation Model
Abstract:
Percolation theory deals with the numbers and properties of the clusters formed in the different occupation probability. In this Paper, we study the calculation method of small clusters. We calcu-lated the small cluster density of 1, 2 and 3 in the percolation model with the exact method and the numerical method. The results of the two methods are very close, which can be verified by each other. We find that the cluster density of all three kinds of small clusters reaches the highest value when the occupation probability is between 0.1 and 0.2. It is very difficult to get the analytical formula for the exact method when the cluster area is relatively large (such as the area is more than 50), so we can get the density value of the cluster by numerical method. We find that the time required calculating the cluster density is proportional to the percolation area, which is indepen-dent of the cluster size and the occupation probability.
Cite this paper: Wang, X. and Gao, J. (2016) A Study on Numerical Calculation Method of Small Cluster Density in Percolation Model. Journal of Applied Mathematics and Physics, 4, 1507-1512. doi: 10.4236/jamp.2016.48159.
References

[1]   Broadbent, S. and Hammersley, J. (1957) Percolation Processes I. Crystals and Mazes. Proceedings of the Cambridge Philosophical Society, 53, 629-641.

[2]   Newman, C.M. and Schulmang, L.S. (1981) Number and Density of Percolating Clusters. J. Phys. A: Math. Gen., 14, 1735-1743.

[3]   Hongler, C. and Smirnov, S. (2009) Critical Percolation: The Expected Number of Clusters in a Rectangle. Probability Theory & Related Fields, 151, 735-756.

[4]   Xiao, L.Q. and Zhou, S.P. (2014) Stochastic Simulation Method and Its Application. Peking University Press, 246-249.

[5]   Christensen, K. and Moloney, N.R. (2005) Complexity and Criticality. 35-36.

[6]   Burks, A.W. Essay on Cellular Automata. University of Illinois Press, Urbana. Yoshio Yuge, H. (1978) Renormalization-Group Approach for Critical Percolation Behavior in Two Dimesions. Phys Rev (B), 18, 1514-1517.

 
 
Top