  A variety of self-consistent (SC) equations have been proposed for
non-uniform states of liquid particles under external fields, including
adsorbed states at solid substrates and confined states in pores. External
fields represent not only confining geometries but also fixed solutes. We
consider SC equations ranging from the modified Poisson-Boltzmann equations for
the Coulomb potential to the hydrostatic linear response equation for the
equilibrium density distribution of Lennard-Jones fluids. Here, we present a
unified equation that explains the apparent diversity of previous forms and
proves the equivalence of various SC equations. This unified description of SC
equations is obtained from a hybrid method combining the conventional density
functional theory and statistical field theory. The Gaussian approximation of
density fluctuations around a mean-field distribution is performed based on the
developed hybrid framework, allowing us to derive a novel form of the
grand-potential density functional that provides the unified SC equation for
equilibrium density.
