Vagaries of density functional theory of the liquid-vapor interface

The (direct) potential functional and (inverse) density functional formulations of classical equilibrium statistical mechanics are reviewed, and their local thermodynamic limits presented. Both are extended, in particular for interfacial structure, in Van der Waals mean field theory. The direct generator, the grand potential, is transformed by the Kac-Siegert approach and the profile equation approximated. The indirect free energy functional is expanded in interaction strength. Both approaches fail to combine consistency with known profile softening under weak external fields. The Kac-Siegert form is then rewritten in terms of a distribution of interfacial surfaces alone for a special model, and transformed to inverse format for a two-dimensional system, yielding a free energy density functional which involves a deaveraged density field, and has the expected interfacial properties.