Relationship between free energy density functional, Born-Green-Yvon, and potential distribution approaches for inhomogeneous fluids

For hard spheres in an external field it is shown that free energy density functionals of the type suggested by Percus [J. Chem. Phys. 75, 1316 (1981)] lead to certain approximations to the Born–Green–Yvon equation. For two functionals these approximations are worked out explicitly as well as are the expressions for the chemical potential and the direct correlation function of the homogeneous fluid. This formal relationship explains why results obtained from a Born–Green–Yvon approach and from coarse grained free energy density functionals are mostly rather similar. Moreover, a functional for the chemical potential derived by Robledo [J. Chem. Phys. 72, 1701 (1980)] from potential distribution theory is compared with the expressions from the Percus approach.