The density functional theory of classical fluids revisited

We reconsider the density functional theory of nonuniform classical fluids from the point of view of convex analysis. From the observation that the logarithm of the grand-partition function log E[rho] is a convex functional of the external potential phi it is shown that the Kohn-Sham free energy A[rho] is a convex functional of the density rho. log Xi[phi] and A[rho] constitute a pair of Legendre transforms and each of these functionals can therefore be obtained as the solution of a variational principle. The convexity ensures the uniqueness of the solution in both cases. The variational principle which gives log Xi[phi] as the maximum of a functional of rho is precisely that considered in the density functional theory while the dual principle, which gives A[rho] as the maximum of a functional of phi, seems to be a new result.