Estimates of general Mayer graphs. I: Construction of upper bounds for a given graph by means of sets of subgraphs

A large number of physical quantities (thermodynamic and correlation functions, scattering amplitudes, intermolecular potentials, etc. ...) can be expressed as sums of an infinite number of multiple integrals of the following type: $$Gamma left( {x_1 ,. . . , x_n } right) = int {prod {f_L left( {x_{i,} x_j } right)dx_{n + 1} . . . dx_{n + k} } }$$ These are called Mayer graphs in statistical mechanics, Feynman graphs in quantum field theory, and multicenter integrals in quantum chemistry. We call themn-graphs here. In a preceding note [Physics Letters 62A:295 (1977)], we have proposed a new estimation method which provides upper bounds for arbitraryn-graphs. This article is devoted to developing in detail the foundations of this method. As a first application, we prove that all virial coefficients of polar systems are finite. More generally, we show on some examples that our estimation method can givefinite upper bounds forn-graphs occurring in the perturbative developments of thermodynamic functions of neutral, polar, and ionized gases and of Green's functions of Euclidean quantum field theories (thus improving Weinberg's theorem), as also in variational approximations of intermolecular potentials. Our estimation method is based on the Hölder inequality which is an improvement over the mean value estimation method, employed by Riddell, Uhlenbeck, and Groeneveld, except for the hard-sphere gas, where both methods coincide. The method is applied so far only to individual graphs and not to thermodynamic functions.