  Equations of motion for the microscopic number density $\hat{\rho}({\bf
x},t)$ and the momentum density $\hat{\bf g}({\bf x},t)$ of a fluid have been
obtained in the past from the corresponding Langevin equations representing the
dynamics of the fluid particles. In the present work we average these exact
equations of microscopic dynamics over the local equilibrium distribution to
obtain stochastic partial differential equations for the coarse grained
densities with smooth spatial and temporal dependence. In particular, we
consider Dean's exact balance equation for the microscopic density of a system
of interacting Brownian particles to obtain the basic equation of the dynamic
density functional theory. In the thermally averaged equation for the coarse
grained density ${\rho}({\bf x},t)$, the related dependence on the bare
interaction potential in Dean's equation is converted to that on the
corresponding direct correlation functions of the density functional theory.
