Interacting hard rods on a lattice: distribution of microstates and density functionals

Area of Science:

• Statistical Mechanics
• Condensed Matter Physics
• Theoretical Chemistry

Background:

• Understanding the behavior of interacting particles in confined spaces is crucial.
• Density functional theory (DFT) is a powerful tool for studying many-body systems.
• Previous models often simplified interactions or lattice structures.

Purpose of the Study:

• To derive exact density functionals for one-dimensional hard rod systems.
• To model systems with first-neighbor interactions of arbitrary shape and limited range.
• To provide a theoretical framework for calculating thermodynamic properties.

Main Methods:

• Utilized conditional probabilities within a Markov chain approach.
• Derived the exact joint probability distribution as a functional of the density profile.
• Developed a lattice fundamental measure theory for contact interactions.

Main Results:

• Obtained exact density functionals for hard rods on a lattice.
• Provided explicit results for contact correlations, entropy, free energy, and chemical potential.
• The framework accommodates inhomogeneous couplings and external potentials.

Conclusions:

• The derived density functionals offer an exact description of 1D hard rod systems.
• This work advances the application of DFT to lattice-based statistical mechanics models.
• The findings are applicable to systems with short-range, first-neighbor interactions.