On the Convergence of the Virial Expansion

Abstract

The virial expansion appears in statistical mechanics, an area where physics and mathematics intersect. Throughout this thesis we will mostly ignore the physics and mainly focus on the mathematical aspects. This is a deliberate choice, made for two reasons. Firstly, there are several books that explain statistical mechanics in a more rigorous manner than would be possible in this thesis, see for example [Rue69]. Secondly, physics is not that important for our eventual goal, which is finding the region of convergence of the virial expansion. The two main objects of this thesis are (1): deriving the cluster expansion; and (2) firstly explaining how the virial expansion came to be, and then deriving the virial expansion.