# CSM

## Seminar

### Mayer polytopes and divided differences

Seminar Room 1, Newton Institute

#### Abstract

We start with a short review of Mayer’s theory of cluster integrals : first and second Mayer graph weights W(g) and w(c) and their application to virial expansion for non ideal gases. We then describe some basic general methods for the exact or approximate evaluation of w(c) for individual connected graphs c. Emphasis is put on the use of covering rooted trees and Fourier transforms. In the context of hard-core continuous gas in one dimension, the weight w(c) is a signed volume of a multidimensional polytope naturally associated with the graph c. Making use of the extra tool of divided differences we give a recursive algorithm for the exact computation of this volume. Explicit examples, tables and asymptotic formulas are also given.

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