Convergence of Virial Expansions

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 $\mathrm{u\hspace{0.17em}\equiv \hspace{0.17em}}\exp \mathrm{\hspace{0.17em}[-}\mathrm{Min}{\mathrm{\hspace{0.17em}s}}^{\text{−1}}{\Sigma }_{\mathrm{i<j\le s}}\text{\hspace{0.17em}2φ(}{\mathrm{x}}_i-{\mathrm{x}}_j{\mathrm{)]/}}_{K}T$ [so that u ≥ 1, with equality for nonnegative φ(r)], ${\mathrm{B\hspace{0.17em}\equiv \hspace{0.17em}\int |e}}^{\mathrm{-\phi \; (}\mathrm{r})}{/}_K{\text{T−1|\hspace{0.17em}d}}^3\hspace{0.17em}\mathrm{r}$ , and φ(r) is the binary interaction potential; the irreducible Mayer cluster integrals have the related upper bounds ${\beta }_k{\text{\hspace{0.17em}≤\hspace{0.17em}[(u+1)B/0.28952]}}^k\text{/k[u\hspace{0.17em}=\hspace{0.17em}1}$ , 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 ${\lim }_{\mathrm{V\to \infty }}\hspace{0.17em}\mathfrak{R}\mathrm{(V)\hspace{0.17em}\le \hspace{0.17em}}\mathfrak{R}$ , and that there is no phase transition for 0 ≤ ρ < 0.28952/(u + 1)B.