Variational Principle of Classical Density Functional Theory via Levy's Constrained Search Method

TL;DR

This paper reformulates classical density functional theory using Levy's constrained search method, providing a new explicit expression for the Helmholtz free energy functional and extending the framework to the canonical ensemble.

Contribution

It introduces a variational principle based on Levy's constrained search, offering a novel explicit form of the free energy functional independent of external potentials.

Findings

Derived a variational principle for the grand potential as a functional of trial distributions.

Defined the intrinsic Helmholtz free energy functional explicitly without external potential dependence.

Extended the density functional framework to the canonical ensemble.

Abstract

We show that classical density functional theory can be based on the constrained search method [M. Levy, Proc. Natl. Acad. Sci. 76, 6062 (1979)]. From the Gibbs inequality one first derives a variational principle for the grand potential as a functional of a trial many-body distribution. This functional is minimized in two stages. The first step consists of a constrained search of all many-body distributions that generate a given one-body density. The result can be split into internal and external contributions to the total grand potential. In contrast to the original approach by Mermin and Evans, here the intrinsic Helmholtz free energy functional is defined by an explicit expression that does not refer to an external potential in order to generate the given one-body density. The second step consists of minimizing with respect to the one-body density. We show that this framework can be applied in a straightforward way to the canonical ensemble.