On the Algebraic Structure of the Cluster Expansion in Statistical Mechanics

• Published in: Journal of mathematical physics 1965-08, Vol.6 (8), p.1179-1188
• Main Authors: Arf, Cahït, Ïmre, Kaya, Özïzmïr, Ercüment
• Format: Article
• Language: English
• ISSN: 0022-2488; 1089-7658
• DOI: 10.1063/1.1704757

The structure of cluster expansion which is widely used in statistical mechanics is studied from an algebraic point of view. In doing this, a commutative algebra is constructed which is generated by partitions of a finite set by regarding them as operators which divide the set into disjoint parts. Physically, these operators correspond to operations which remove interaction among certain clusters of particles. It is shown that the cluster expansion stems from the relation between two basis sets of this algebra; the first set is the set of all partitions and the second is the set of pairwise orthogonal minimal idempotents. This property enables one to demonstrate the equivalence of the product versus cluster properties of the distribution and correlation functions, respectively, in general terms. This is done by constructing a simple representation space for the partition algebra corresponding to the distribution functions. A second application of the partition algebra is considered in the case when correlations are well‐ordered with respect to the interaction strength λ, so that to a given order in λ the distribution functions are not all independent and can be expressed in terms of a finite irreducible set of functions involving smaller numbers of particles. The combinatorial problem of calculating the expansion coefficients is carried out explicitly using a graded representation space with respect to the order in λ. It is concluded that the partition algebra can be used as a mathematical tool in handling problems involving cluster expansion.