  When a fluid is subject to an external field, as is the case near an
interface or under spatial confinement, then the density becomes spatially
inhomogeneous. Although the one-body density provides much useful information,
a higher level of resolution is provided by the two-body correlations. These
give a statistical description of the internal microstructure of the fluid and
enable calculation of the average interparticle force, which plays an essential
role in determining both the equilibrium and dynamic properties of interacting
fluids. We present a theoretical framework for the description of inhomogeneous
(classical) many-body systems, based explicitly on the two-body correlation
functions. By consideration of local Noether-invariance against spatial
distortion of the system we demonstrate the fundamental status of the
Yvon-Born-Green (YBG) equation as a local force-balance within the fluid. Using
the inhomogeneous Ornstein-Zernike equation we show that the two-body
correlations are density functionals and, thus, that the average interparticle
force entering the YBG equation is also a functional of the one-body density.
The force-based theory we develop provides an alternative to standard density
functional theory for the study of inhomogeneous systems both in- and
out-of-equilibrium. We compare force-based density profiles to the results of
the standard potential-based (dynamical) density functional theory. In
equilibrium, we confirm both analytically and numerically that the standard
approach yields profiles that are consistent with the compressibility pressure,
whereas the force-density functional gives profiles consistent with the virial
pressure. The structure of the theory offers deep insights into the nature of
correlation in dense and inhomogeneous systems.
