Revisiting Groeneveld's approach to the virial expansion
Journal of Mathematical Physics 62 (2), 023302 (2021).
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 rho (s) (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 Mobius inversion on the lattice of set partitions enters the derivation of the recurrence relations.
Lagrange Inversion and Combinatorial Species with Uncountable Color Palette
S. Jansen, T. Kuna, D. Tsagkarogiannis
Annales Henri Poincare
We prove a multivariate Lagrange-Good formula for functionals of uncountably many variables and investigate its relation with inversion formulas using trees. We clarify the cancellations that take place between the two aforementioned formulas and draw connections with similar approaches in a range of applications.
Thermodynamics of a hierarchical mixture of cubes
Journal of Statistical Physics (2020).
We investigate a toy model for phase transitions in mixtures of incompressible droplets. The model consists of non-overlapping hypercubes in Zd of sidelengths 2j, j?N0. Cubes belong to an admissible set B such that if two cubes overlap, then one is contained in the other. Cubes of sidelength 2j have activity zj and density ?j. We prove explicit formulas for the pressure and entropy, prove a van-der-Waals type equation of state, and invert the density-activity relations. In addition we explore phase transitions for parameter-dependent activities zj(?)=exp(2dj??Ej). We prove a sufficient criterion for absence of phase transition, show that constant energies Ej?? lead to a continuous phase transition, and prove a necessary and sufficient condition for the existence of a first-order phase transition.
Cluster Expansions with Renormalized Activities and Applications to Colloids
S. Jansen, D. Tsagkarogiannis
Annales Henri Poincare 21 (1), 45-79 (2020).
We consider a binary system of small and large objects in the continuous space interacting via a nonnegative potential. By integrating over the small objects, the effective interaction between the large ones becomes multi-body. We prove convergence of the cluster expansion for the grand canonical ensemble of the effective system of large objects. To perform the combinatorial estimate of hypergraphs (due to the multi-body origin of the interaction), we exploit the underlying structure of the original binary system. Moreover, we obtain a sufficient condition for convergence which involves the surface of the large objects rather than their volume. This amounts to a significant improvement in comparison to a direct application of the known cluster expansion theorems. Our result is valid for the particular case of hard spheres (colloids) for which we rigorously treat the depletion interaction.