Universal cluster size distribution in a system of randomly spaced particles

Abstract: The distribution function of particles over clusters is proposed for a system of identical intersecting spheres, the centres of which are uniformly distributed in space. Consideration is based on the concept of the rank number of clusters, where the rank is assigned to clusters according to the cluster sizes. Distribution is universal in the sense that it does not depend on boundary conditions and is valid for infinite medium. The form of the distribution function is determined by only one parameter, equal to the ratio of the sphere radius (`interaction radius') to the average distance between the centres of the spheres. This parameter plays also a role of the order parameter. It is revealed under what conditions the universal distribution behaves like well known log-normal distribution. Applications of the proposed distribution to some realistic physical situations, which are close to the conditions of the gas condensation to liquid, are considered.