Classical Density Functional Theory: The Local Density Approximation

TL;DR

This paper proves that the free energy of a classical interacting system with a slowly varying density can be approximated locally by the free energy of a homogeneous gas, with quantifiable error bounds.

Contribution

It provides a rigorous proof linking the global free energy to a local approximation using the local density, with explicit error estimates.

Findings

Validates the local density approximation for slowly varying densities.

Provides quantitative error bounds based on the density gradient.

Uses Ruelle bounds to support the approximation proof.

Abstract

We prove that the lowest free energy of a classical interacting system at temperature $T$ with a prescribed density profile $\rho(x)$ can be approximated by the local free energy $\int f_T(\rho(x))dx$, provided that $\rho$ varies slowly over sufficiently large length scales. A quantitative error on the difference is provided in terms of the gradient of the density. Here $f_T$ is the free energy per unit volume of an infinite homogeneous gas of the corresponding uniform density. The proof uses quantitative Ruelle bounds (estimates on the local number of particles in a large system), which are derived in an appendix.