Convergence of Fugacity Expansions for Fluids and Lattice Gases

Upper and lower bounds are obtained for R(V), the radius of convergence of the Mayer expansion VΣl bl(V)zl expressing the logarithm of the classical grand partition function for a finite volume V as a power series in the fugacity z. The particles in V interact only through two-body forces whose potential φ(r) satisfies s−1 Σij≤s φ(xi − xj) ≥ const ≡ −Φ for all s, x1 ⋯ xs. The bounds are e1+2Φ/kT ∫ |e−φ(r)/kT−1| d3r−1≤R(V)≤|eΦ/kTl/(l−1) bl(V)|1/(l−1)for any l ≥ 2. For lattice gases the integral becomes a sum. The upper bounds, obtained from the theory of entire functions, include a subsequence converging to R(V) as l → ∞. The lower bound is obtained by using the Kirkwood-Salsburg integral equation to calculate upper bounds on the bl(V)'s and the coefficients in the fugacity expansions of the s-particle distribution functions. For hard-core potentials some of these bounds can be strengthened. For nonnegative potentials, 1/2|b2(V)| is an extra upper bound on R(V). The radius of convergence of the infinite-volume series Σ blzl is shown to be at least limV→∞ R(V), with equality for nonnegative potentials.