Classical density functional theory in the canonical ensemble.

Area of Science:

• Statistical Mechanics
• Physical Chemistry
• Computational Physics

Background:

• Classical density functional theory (DFT) is typically formulated in the grand-canonical ensemble, allowing variable particle numbers.
• Many physical systems are closed, with fixed temperature and particle number (canonical systems).
• Existing DFT methods are often adapted for canonical systems, but a direct formulation was lacking.

Purpose of the Study:

• To revisit and compare fundamental theorems of classical DFT in both canonical and grand-canonical ensembles.
• To develop and demonstrate a practical DFT formulation for canonical systems.
• To derive exact Helmholtz free energy functionals for specific canonical systems.

Main Methods:

• Revisiting and comparing fundamental theorems of classical DFT across ensembles.
• Deriving exact Helmholtz free energy functionals.
• Applying the canonical DFT formulation to model systems: ideal gas, restricted geometries, and hard rods in a cavity.

Main Results:

• Demonstrated that classical DFT in the canonical ensemble has only trivial formal differences compared to the grand-canonical ensemble.
• Derived exact Helmholtz functionals for the ideal gas, systems in restricted geometries, and one-dimensional hard rods.
• Observed strong similarities between canonical and grand-canonical ensembles, even in small systems.

Conclusions:

• Classical density functional theory can be effectively formulated and applied directly within the canonical ensemble.
• This formulation provides a more rigorous framework for studying closed systems using DFT.
• The findings bridge a gap in DFT methodology, offering new possibilities for analyzing systems with fixed particle numbers.