On the Superposition of Probabilities

Abstract

Superposition methods—as used in many lattice diffusion calculations to estimate joint probabilities from limited information on lower order correlations—are here reviewed in the general inhomogeneous context. The Kirkwood-Kikuchi-Barker (KKB) superposition principle is shown to be a special case of the application of combinatorial Möbius function techniques over a Boolean lattice of partitions into clusters, or motifs which uses a logarithmic/exponential device. This method is conceptually distinct from the approach utilizing the linear algebra of probabilities over an equal a priori probability phase space. It is shown that the probabilities produced by KKB superposition in general fail to satisfy the basic rules of conservation of probability; and a new algorithm is given for embedding the Möbius formalism into probability space. The general problem of superposition of probabilities is formulated in terms of linear convex analysis. The concept of marginal coordinate systems generalizing both the Möbius formalism and the idea of conditional probabilities is introduced and pidgeonhole cluster coordinates developed. The linear convex formalism additionally permits the use of interior point methods for the determination of the permissible polytope in probability space and provides an exact representation of the configurational entropy which incorporating all correlations in finite structures. It is shown that knowledge of the permissible polytope in probability space can restrict the class of permissible structures so severely that entropy maximization is unnecessary; and it is suggested that this is particularly true in the case of ordered structures. The equilibrium configurational free energy computation is discussed in the light of linear convex theory and the ergodic hypothesis.