n-point probability functions for a lattice model of heterogeneous media

The macroscopic properties of two-phase random heterogeneous media depend upon the infinite set of n-point probability functions S1,...,Sn. The quantity Sn(x1,...,xn) gives the probability of finding n points with positions x1,...,xn all in one of the phases. We derive a series representation of Sn for a finite-sized D-dimensional lattice model of heterogeneous media. By performing certain averages over the Sn, we then obtain explicit expressions for translationally invariant and rotationally invariant n-point probabilities. Computer simulations are carried out for low-order n-point probability functions. The theoretical and simulation results are found to be in excellent agreement.