MULTIFRACTALITY OF SNOWFLAKES

Abstract

The results of the extensive numerical simulations of the Dielectric Breakdown Model (DBM) with noise reduction on the hexagonal lattice are presented. Seventy-five clusters grown under different boundary conditions consisting of 16 000 particles on the lattice 1001×1001 were generated. The simulations were done for the noise reduction parameter s equal to 200 and two values of the parameter η, namely for η=0.5 and 1. For the latter case, two boundary conditions were considered: the DBM and DLA b.c. Such a growth model leads to the formation of the fractal objects resembling real snowflakes. The growth probabilities were stored at the five stages of the clusters growth: at the masses of 1000, 2000, 4000, 8000 and 16 000 particles. Multifractal analysis was performed and obtained results are presented. The comparison of two methods: the histogram and moments methods, is provided. We discover that for the η=0.5 parameter, there is a phase transition, while for η=1, there appears to be no phase transition. Besides the usual growth probability measure, the measure given by the noise reduction counters is considered and multifractality of it is presented.