Article,
On the Mutual Cancellation of Cluster Integrals in Mayer's Fugacity Series
Proceedings of the Physical Society, 83 (1): 3--16 (1964)
Abstract
Theorems are proved for each of two models representing purely repulsive interactions, the `Gaussian' and the `rigid line'. In both cases we study the fugacity series by starting with the complete diagram of l points connected by ½ l ( l -1) lines. For the Gaussian model, it is proved rigorously that the cluster integral corresponding to any diagram can be expressed as a product , each disjoint set of lines in the complementary diagram contributing one factor . The result for the `rigid-line' model is that any cluster integral can be expressed as a sum , each disjoint set of lines in the complementary diagram contributing a term , with corresponding results for the `rigid-square' and `rigid-cube' models. These results enable the consequences of the `Gaussian' model to be worked out almost completely, and provide some justification for approximate treatments that neglect all but the more `open' diagrams. The `rigid-square' model is more difficult analytically, and only a few preliminary deductions have been made.
