Renormalized density-functional theory for inhomogeneous liquids

A density-functional model for fluids, containing a one-point and a two-point coupling function, is studied. The field variables of the model are the local fluid density ρ(r) and a coarse-grained density ${\rho }_0$(r). Using the method developed by Meister and Kroll [Phys. Rev. A 31, 4055 (1985)], the coarse-grained density is eliminated as a free variable, which defines a hypersurface in the function space ρ⊗${\rho }_0$. From the variations of the free-energy functional along this hypersurface, we derive a renormalization equation for the two-point coupling, using the direct correlation function as a normalization condition. We solve this equation numerically for the hard-sphere fluid, but the given algorithm can be applied to any other fluid, of which the direct correlation function is known.