Classical density functional theory: The local density approximation

Abstract

We prove that the lowest free energy of a classical interacting system at temperature [Formula: see text] with a prescribed density profile [Formula: see text] can be approximated by the local free energy [Formula: see text], provided that [Formula: see text] varies slowly over sufficiently large length scales. A quantitative error on the difference is provided in terms of the gradient of the density. Here [Formula: see text] 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.