Cluster Variation for Spatially Distributed Heterogeneous Systems

Fundamentals of cluster variation (CV) are developed for locally heterogeneous spatially distributed systems. The theory is based on the principles of homogeneous CV in which all variants of the location of the basis cluster on a heterogeneous lattice are additionally considered when it is translated over the system. The structure of the statistical sum of homogeneous CV is shown to remain upon moving to a heterogeneous or homogeneous spatially distributed lattice. However, cofactors of the statistical sum, which previously corresponded to homogeneous clusters, must now consider all arrangements of heterogeneous sites inside each cluster. The general approach is to use a layered structure of the transitional region with variable density between vapor and fluid on a planar square lattice. Explicit expressions for a heterogeneous statistical sum of the transitional region are given on the basis of a 3 × 3 cluster. Using a 2 × 2 cluster, it is shown how an explicit equation for the equilibrium particle distribution in the transitional region can be obtained from the heterogeneous statistical sum. A gradual increase in the size of the m × n basis cluster in the transitional region converges to the exact solution.