We present a rigorous derivation of the free energy functional for inhomogeneous systems, i.e. with a density that depends on the position, orientation or other internal degrees of freedom. It can be viewed as an extension of the virial inversion (developed for homogeneous systems) to uncountably many species. The key technical tool is a combinatorial identity for a special type of trees which allows us to implement the inversion step as well as to prove its convergence. 
Applications include classical density functional theory, Onsager's functional for liquid crystals, hard spheres of different sizes and shapes. Furthermore, the method can be generalized in order to provide convergence for other expansions commonly used in the liquid state theory. The validity is always in the gas regime, but with the new method we improve the original radius of convergence for the hard spheres as proved by Lebowitz and Penrose and subsequent works. This is joint work with Sabine Jansen and Tobias Kuna.
org: GIULIANI Alessandro