Abstract
A generalized version of Groeneveld’s convergence criterion for the virial expansion and generating functionals for weighted two-connected graphs is proven. This criterion works for inhomogeneous systems and yields bounds for the density expansions of the correlation functions ρs (a.k.a. distribution functions or factorial moment measures) of grand-canonical Gibbs measures with pairwise interactions. The proof is based on recurrence relations for graph weights related to the Kirkwood–Salsburg integral equation for correlation functions. The proof does not use an inversion of the density-activity expansion; however, a Möbius inversion on the lattice of set partitions enters the derivation of the recurrence relations.
Highlights
Graphical expansions of thermodynamic functionals and correlation functions play an important role in statistical mechanics and liquid state theory
Relevant quantities are expanded in powers of the activity z or the density ρ, leading, for example, to the Mayer expansion and virial expansion for the pressure
The expansions are used to establish the absence of phase transitions and exponential decay of correlations,1 in this regard disagreement percolation2–5 and Dobrushin uniqueness,6 approaches based on Glauber birth and death dynamics or other algorithms7–9 or recursive approaches and complex analysis10,11 often yield better results; in addition to expansions, there are bounds
Summary
Graphical expansions of thermodynamic functionals and correlation functions play an important role in statistical mechanics and liquid state theory. The principal difference is an additional expression with a Möbius inversion on the lattice of set partitions, in part, inspired by recent work by Dorlas, Rebenko, and Savoie.30 This corresponds to a version of the Kirkwood–Salsburg equation from which the activity is eliminated; a similar variant of the Kirkwood–Salsburg equation has, been derived from the canonical ensemble and applied to prove convergence of density expansions by Bogoljubov et al.. The results and proofs in the main body of this article are phrased in terms of generating functions of weighted labeled graphs; in principle, they do not require any knowledge in statistical mechanics This choice of presentation is motivated by interest in the virial expansions from a combinatorial point of view.. This choice of presentation is motivated by interest in the virial expansions from a combinatorial point of view. The relation of these generating functions with correlation functions of grand-canonical Gibbs measures is recalled in the Appendix
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