Growth of circular crystals in a circular region

Abstract

We compute the mean and standard deviation of the surface coverage of a circular region that is partially covered by N circular crystals (discs). It is assumed that the centres of crystals are randomly and independently distributed, with uniform probability inside the region and zero probability outside it. The results are presented graphically as functions of the number of crystals and their dimensionless radii. We also compute the mean and standard deviation of the fraction of the total peripheral length of the crystals which is lying inside the region and which is not covered by other crystals. Results are given for cases up to and including N= 64.