Scaled-particle theory and the short distance behavior of continuum percolation

Abstract

In this paper, we use results from geometric probability theory to constrain the behavior of continuum percolation models. Specifically, we consider the random percolation of spheres, in which particles are distributed at random with density ρ̄, each pair being considered connected if its separation is less than a distance a. For this model we prove a zero-separation theorem, which gives the first three terms in a Taylor series expansion around zero separation of the two-point connectedness function. These expressions are then used in a closure for the Born–Green equations of percolation. The result is an approximate equation of state, or formula for the mean number of clusters 〈nc〉 as a function of density, which is quite accurate at moderate densities. Using the relation between continuum percolation and the continuum Potts model, we develop two different forms of scaled particle theory for continuum percolation. These theories are then combined with the zero-separation theorems to give several approximate equations of state for percolation, i.e., formulas for the mean number of clusters 〈nc〉 as a function of density ρ̄. Finally, we use an argument from geometric probability theory to provide a closure of the Born–Green hierarchy for percolation. The equation of state resulting from this procedure is exact to third order in a density expansion.