Geometrically Based Free Energy Functional for Confined Hard Spheres and the Density Functional Theory for Simple Classical Systems

Abstract

In the last two decades there has been a continuous progress in the theory of nonuniform classical fluids [1], as described by Bob Evans in his first lecture. Important approximations and models were developed for the Helmholtz free energy functional, F[ρ(r)], of a inhomogeneous density distribution, ρ(r). The geometrical character of the hard-sphere interactions, which has been a major reason for their long standing central role in the microscopic theory of classical fluids [2], also simplifies the construction of model functional [3]. Pedro Tarazona, in his first lecture, described the geometrically-based so called fundamental-measure functionals (FMFs) [4]–[8], emphasizing several very recent analyses [9]–[13] that revealed the important role played by the dimensional cross-over properties of the functional. This supplementary lecture is divided into two main parts. In the first part (Section II) I list some of the basic physical properties expected from the exact (but unknown!) free-energy functional for the hard-sphere system (especially when applied to densely packed hard-spheres), and then explain how these are achieved by the FMFs. These properties are also important in applications for general interactions.