Dimensional Cross-Over, Close-Packed Configurations, Symmetry Breaking, and Freezing in Density Functional Theory

Abstract

The theory of nonuniform classical fluids has made continuous progress in the last two decades, along with the development of important approximations and model Helmholtz free energy functionals, for a inhomogeneous density distribution, The density functional theory should eventually provide a unified description of classical systems, including the bulk liquid of uniform density, the bulk solid of narrow density peaks at lattice sites, and the crystallization of a fluid as a strong self-sustained inhomogeneity. The geometrical character of the hard-sphere interactions, which is one of the main reasons for their long standing central role in the microscopic theory of classical fluids, also simplifies the construction of model functionals. Good results for the equation of state of the fcc solid and the freezing transition have been obtained by many approaches for the hard spheres with nonlocal dependence on the density through weight functions. The direct extension of these functionals to continuous (“soft”) potentials brought mixed success and results of sometimes questionable quality. In turn, standard perturbation expansions around the hard-sphere (HS) reference density functional proved successful for both bcc and fcc classical solids with simple soft interactions. In either case, however, elementary properties like the analytic connection of the density functional theory to basic standard models of simple classical solids (notably the free-volume cell model, or the harmonic approximation!) have not been demonstrated, mainly due to the intrinsic limitations of the functionals that were employed. On the other hand, several very recent analyses6–10 of the geometrically-based so called fundamental measure functionals (FMFs) revealed that they have many of the basic physical properties expected from the exact (but unknown!) free-energy functional when applied to densely packed hard-spheres. Moreover, these properties are important also for applications to continuous (“soft”) potentials in general, and to charged-particle systems (including plasmas) in particular. These are distinguishing features of the FMF’s, shared by none of the other functionals that were proposed in the literature. In particular, in order to describe correctly densely packed configurations for soft interactions, the hard-sphere functional must feature a true divergence of the equation of state at configurations of close packing. The singularity which they possess, and their unique geometrically-based structure, enable the FMFs to achieve this as well as other important properties. Configurations of densely packed hard spheres, confined in different effective dimensions D, provide the ultimate test for model free energy functionals. The exact functional exhibits