Integral equations for continuum percolation

Abstract

Continuum percolation is studied by means of integral equations for the connectedness function which can be derived from similar equations for the correlation function of fluids with the aid of a diagrammatic expansion. In particular we examine a refinement (which we call PY-d(2)) of the Percus–evick approximation first proposed by Stell and a HNC-type approximation for the connectedness function. These equations are applied to the system of randomly centered spheres, where they are solved numerically. The value of the critical density and the way in which the mean clusters size diverges are given much more accurate by the PY-d(2) approximation than by the PY approximation, while the critical exponents remain unchanged. We find that the HNC approximation does not possess a critical point where the mean clusters size diverges and that the long range behavior of the connectedness bridge function has a more important role in the percolation problem than in the thermal critical point.