Convergence of cluster and virial expantions. Theory and applications

The central theme of this thesis is the development of strong and efficient convergence criteria for cluster and virial expansions for a number of systems in classical and quantum statistical mechanics. The thesis can be roughly divided in three parts. The first one is the study of cluster expansions for systems of particles subject to positive two-body interactions. We state a rather general convergence theorem that holds in a very general framework that does not require translation invariance and is applicable to models in general measure spaces. The theorem is based on expressing the cluster coefficients is based on expressing the truncated weights in terms of trees and partition schemes, and generalizes to soft repulsions previous approaches for models with hard exclusions. Our results strengthen and generalize existing lower bounds on the radii of convergence and on the value of the pressure. This is illustrated through several applications including, in particular, the high- and low-temperature expansion for Ising model. The second part of the thesis deals with the virial expansion for homogeneous single-space systems involving positive two-body interactions. Our results are based on a refinement of an approach due to Ramawadth and Tate that leads to a diagrammatic expression of the virial coefficients in terms of trees, rather than the doubly connected diagrams traditionally used. We obtain a new virial convergence criterion that strengthens, for repulsive interactions, the best criterion previously available (proposed by Groenveld and proven by Ramawadth and Tate) and compares rather favorably with convergence radii estimated numerically for several models with repulsive interactions proposed in physics. The third part of the thesis deals with the cluster expansion of finite-spin quantum (and classical) models involving suitably summable multi-body interactions. Our approach is based on work by Park that studied these systems through integral equations of Kikwood-Salsburg type. Following Park we use “decoupling parameters" to relate partition functions with successive additional interaction terms, but instead of equations we obtain a novel cluster expansion that permits the explicit evaluation of free energy, reduced correlations and expectations. Furthermore, as shown in the case of two-body interactions, the radius of convergence of our expansion exceeds the region of applicability of Park’s equations. The thesis is concluded with a chapter on work in progress, towards alternative expressions of cluster expansions for classical systems multi-body interactions, including models of colloids.