  We present the exact adiabatic theory for the dynamics of the inhomogeneous
density distribution of a classical fluid. Erroneous particle number
fluctuations of dynamical density functional theory are absent, both for
canonical and grand canonical initial conditions. We obtain the canonical free
energy functional, which yields the adiabatic interparticle forces of
overdamped Brownian motion. Using an exact and one of the most advanced
approximate hard core free energy functionals, we obtain excellent agreement
with simulations. The theory applies to finite systems in and out of
equilibrium.
