Cluster size distribution in percolation theory and fractal Cantor dust.
Results of numerical simulation of cluster size distribution in the site percolation problem are presented. These results disagree with the theoretical data obtained on the basis of the standard drop model of finite cluster structure, in particular, they give a different value of exponent zeta (lnn{s} approximately -s{zeta}). Therefore, a more precise fractal model for describing the structure of clusters in a percolation system is proposed. The consideration is based on the solution of a kinetic equation for the number of finite clusters. In the framework of the proposed approach (fractal model together with kinetic equation), a correct value of exponent zeta is obtained and an explanation is given to the dependence of this exponent on the fraction of occupied sites p, which was revealed by numerical simulations. Additionally, a relation is established between the characteristics of cluster size distribution and fractal dimension of the Cantor dust constructed on the percolation cluster.
Grinchuk
Pavel
P
A. V. Luikov Heat and Mass Transfer Institute, National Academy of Sciences of Belarus, 15 Petrus Brovka Str., Minsk 220072, Belarus. gps@hmti.ac.by