  The focus of the present work is the application of the random phase
approximation (RPA), derived for inhomogeneous fluids [Frydel and Ma, Phys.
Rev. E 93, 062112 (2016)], to penetrable-spheres. As penetrable-spheres
transform into hard-spheres with increasing interactions, they provide an
interesting case for exploring the RPA, its shortcomings, and limitations, the
weak- versus the strong-coupling limit. Two scenarios taken up by the present
study are a one-component and a two-component fluid with symmetric
interactions. In the latter case, the mean-field contributions cancel out and
any contributions from particle interactions are accounted for by correlations.
The accuracy of the RPA for this case is the result of a somewhat lucky
cancellation of errors.
