Convergence of Virial Expansions

Abstract

Some bounds are obtained on ℜ(V), the radius of convergence of the density expansion for the logarithm of the grand partition function of a system of interacting particles in a finite volume V, and on ℜ, the radius of convergence of the corresponding infinite-volume expansion (the virial expansion). A common lower bound on ℜ(V) and ℜ is 0.28952/(u+1)B, where u ≡ exp [−Min s−1Σi<j≤s 2φ(xi−xj)]/KT [so that u ≥ 1, with equality for nonnegative φ(r)], B ≡ ∫|e−φ (r)/KT−1| d3 r, and φ(r) is the binary interaction potential; the irreducible Mayer cluster integrals have the related upper bounds βk ≤ [(u+1)B/0.28952]k/k[u = 1, when φ(r) ≥ 0]. For potentials with hard cores the maximum density is an upper bound on ℜ(V), though possibly not on ℜ; an example shows how both ℜ(V) and ℜ can be less than the maximum density, even if there is no phase transition. A theorem is proved, analogous to Yang and Lee's theorem on uniform convergence in the complex z plane, defining a class of domains in the complex ρ plane within which the operations V → ∞ and d/dρ commute. This theorem is used to show that limV→∞ R(V) ≤ R, and that there is no phase transition for 0 ≤ ρ < 0.28952/(u + 1)B.