Approximate symmetry laws for percolation in complex systems: Percolation in polydisperse composites

The concept of so-called global symmetry of percolation models is discussed and extended to multicolored models. An integral equation is obtained, which determines the partial percolation probabilities Pa for sites of color a. This equation is applied to a polydisperse particulate composite: a mixture of conducting (of relative fraction xm) and nonconducting spheres with distributions of sizes nm(R) and ni(R), respectively. We find the probability PR for a conducting particle of radius R to belong to the percolation cluster as a function of xm and a functional of nm(R') and ni(R'). The percolation threshold x is shown to decrease with increasing dispersion Δ of particle sizes. A simple law x=1/(3[1+(Δ/4)]) is obtained in the range of moderate dispersions. The theory is applicable also to a mixture of electronic and ionic conductors