Cluster Expansions

Abstract

The method of cluster expansions allows to write the grand-canonical thermo-dynamic potential as a convergent perturbation series, where the small parameter is related to the temperature (usually high), the chemical potential (usually low — small densities), and the interactions (small). It was pioneered by Mayer in the 1930's, and made rigorous both by Penrose and Ruelle in 1963. Subsequent works, especially by Koteck and Preiss, have simplified the method, allowing many gen-eralizations. We introduce the method in the context of the classical gas in Section 1. We explain the combinatorics, and give the result. We ignore the problems of conver-gence of various series, until Section 2, where a theorem is provided that rigorizes the computations of Section 1. This theorem applies to a very broad class of physi-cal systems, including lattice spin systems where it helps proving the occurrence of phase transitions, and quantum systems. Assumptions involve the " Koteck-Preiss criterion " , a condition that has proved convenient and rather optimal in many situations. 1. Weakly interacting classical gas The method is very general and is actually an intriguing piece of combinatorics. Roughly summarized, a sum over arbitrary graphs can be written as the exponential of a sum over connected graphs. The interactions in the partition functions can be expressed using graphs; the logarithm of the partition function involves a sum over connected graphs. Recall that the Hamiltonian of the classical gas is H({p i , q i }) = N i=1