Statics and dynamics of inhomogeneous liquids via the internal-energy functional

We give a variational formulation of classical statistical mechanics where the one-body density and the local entropy distribution constitute the trial fields. Using Levy's constrained search method, it is shown that the grand potential is a functional of both distributions, that it is minimal in equilibrium, and that the minimizing fields are those at equilibrium. The functional splits into a sum of entropic, external energetic, and internal energetic contributions. Several common approximate Helmholtz free-energy density functionals, such as the Rosenfeld fundamental measure theory for hard-sphere mixtures, are transformed to internal-energy functionals. The variational derivatives of the internal-energy functional are used to generalize dynamical density-functional theory to include the dynamics of the microscopic entropy distribution, as is relevant for studying heat transport and thermal diffusion.