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.