Variational principle of classical density functional theory via Levy's constrained search method

Area of Science:

• Quantum mechanics
• Statistical mechanics
• Condensed matter physics

Background:

• Classical density functional theory (DFT) is a powerful tool for studying many-body systems.
• Existing DFT formulations often rely on implicit definitions of energy functionals.
• A rigorous foundation for DFT is essential for its accurate application.

Purpose of the Study:

• To reformulate classical density functional theory using the constrained search method.
• To derive a variational principle for the grand potential based on Gibbs inequality.
• To establish an explicit expression for the intrinsic Helmholtz free-energy functional.

Main Methods:

• Application of the constrained search method to derive DFT.
• Utilizing Gibbs inequality to establish a variational principle for the grand potential.
• Two-stage minimization involving a constrained search for many-body distributions and minimization with respect to one-body density.

Main Results:

• Demonstration that classical DFT can be founded on the constrained search method.
• Derivation of a variational principle for the grand potential as a functional of trial many-body distributions.
• Definition of an intrinsic Helmholtz free-energy functional independent of external potentials.

Conclusions:

• The constrained search method provides a rigorous foundation for classical density functional theory.
• The derived framework offers an explicit and versatile approach to DFT calculations.
• The method is readily extendable to the canonical ensemble, enhancing its applicability.