  For classical many-body systems subject to Brownian dynamics we develop a
superadiabatic dynamical density functional theory (DDFT) for the description
of inhomogeneous fluids out-of-equilibrium. By explicitly incorporating the
dynamics of the inhomogeneous two-body correlation functions we obtain
superadiabatic forces directly from the microscopic interparticle interactions.
We demonstrate the importance of these nonequilibrium forces for an accurate
description of the one-body density by numerical implementation of our theory
for three-dimensional hard-spheres in a time-dependent planar potential. The
relaxation of the one-body density in superadiabatic-DDFT is found to be slower
than that predicted by standard adiabatic DDFT and significantly improves the
agreement with Brownian dynamics simulation data. We attribute this improved
performance to the correct treatment of structural relaxation within the
superadiabatic-DDFT. Our approach provides fundamental insight into the
underlying structure of dynamical density functional theories and makes
possible the study of situations for which standard approaches fail.
