Statistical mechanics of fluids confined by polytopes: The hidden geometry of the cluster integrals

This paper, about a fluid-like system of spatially confined particles, reveals the analytic structure for both, the canonical and grand canonical partition functions. The studied system is inhomogeneously distributed in a region whose boundary is made by planar faces without any particular symmetry. This type of geometrical body in the $d$-dimensional space is a polytope. The presented result in the case of $d=3$ gives the conditions under which the partition function is a polynomial in the volume, surface area, and edges length of the confinement vessel. Equivalent results for the cases $d=1,2$ are also obtained. Expressions for the coefficients of each monomial are explicitly given using the cluster integral theory. Furthermore, the consequences of the polynomial shape of the partition function on the thermodynamic properties of the system, away from the so-called thermodynamic limit, is studied. Some results are generalized to the $d$-dimensional case. The theoretical tools utilized to analyze the structure of the partition functions are largely based on integral geometry.