  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.
