Cluster Size Distribution in a System of Randomly Spaced Particles

The distribution function of particles over clusters is proposed for a system of identical intersecting spheres, the centers 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. The distribution function does not depend on boundary conditions and is valid for infinite medium. The form of the distribution is determined by only one parameter, equal to the ratio of the sphere radius (‘interaction radius’) to the average distance between the centers of the spheres. This parameter plays also a role of the order parameter. It is revealed under what conditions the 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.