Classical density functional theory: The local density approximation

We prove that the lowest free energy of a classical interacting system at temperature T with a prescribed density profile ρ(x) can be approximated by the local free energy ∫fT(ρ(x))dx, provided that ρ varies slowly over sufficiently large length scales. A quantitative error on the difference is provided in terms of the gradient of the density. Here fT 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.