Mayer Expansion for monomer-dimer system

Statistical mechanics is a branch of physics that applies probability theory, which contains mathematical tools for dealing with large populations, to the study of the thermodynamic behavior of systems composed of a large number of particles. A deep connection between probability theory and physics has been discovered in the research of statistical mechanics. The simplest model in statistical mechanics is the ideal gas, in which molecules have no interaction with each other. The ideal gas is the form used in physics for the Poisson firld in mathematics. Besides the concepts normally usde in this context in mathematics such as N(X) which is the random numbers of molecules in a set X, there is also a formule for the pressure p. Explicit calculation of Poisson Field derives the law for the ideal gas, for example, the Ideal Gas Law p|Λ| = T < N(X) >, where p, |Λ|, T and < N(X) > are pressure, volume, temperature and the expected value of number of molecules respectively. See [1] for more details. What if the molecules have an interaction between each other? Under this assumption, the ideal gas model is no longer correct. The Mayer expansion give a method to calculate the pressure of such system. In particular, the monomer-dimer system is a canonical ensemble that models problems involving a diatomic gas, like hydrogen or oxygen, whose molecules are composed only of two atoms. A monomer-dimer system is defined by a graph G = (E, V ) and a set of dimers (covering an edge with its two vertices) and monomers (covering one vertex) on G. The dimers and monomers have exclusion interaction and thus satisfy the condition of compatibility (which will be defined in section 2). The Mayer expansion give a expression about the pressure of the monomer-dimer system (in section 3). However, the expansion need the conditions of low density to be convergent. Our aim is to find method to extend the Mayer expansion to a larger