The density functional theory of classical fluids revisited

J. -M. Caillol

Published 2001-04-20Version 1

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 $\log \Xi [\phi]$ is a convex functional of the external potential $\phi$ it is shown that the Kohn-Sham free energy ${\cal A}[\rho]$ is a convex functional of the density $\rho$. $\log \Xi [\phi]$ and ${\cal A}[\rho]$ 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 unicity 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 ${\cal A}[\rho]$ as the maximum of a functional of $\phi$ seems to be a new result.