  The dynamical density functional theory of Marconi and Tarazona [J. Chem.
Phys., 110, 8032 (1999)], a theory for the non-equilibrium dynamics of the
one-body density profile of a colloidal fluid, is applied to a binary fluid
mixture of repulsive Gaussian particles confined in a spherical cavity of
variable size. For this model fluid there exists an extremely simple Helmholtz
free energy functional that provides a remarkably accurate description of the
equilibrium fluid properties. We therefore use this functional to test the
assumptions implicit in the dynamical density functional theory, rather than
any approximations involved in constructing the free energy functional. We find
very good agreement between the theory and Brownian dynamics simulations,
focusing on cases where the confined fluid exhibits phase separation in the
cavity. We also present an instructive derivation of the Smoluchowski equation
(from which one is able to derive the dynamical density functional theory)
starting from the Liouville equation -- a fully microscopic treatment of the
colloid and solvent particles. This `coarse-graining' is, of course, not exact
and thus the derivation demonstrates the physical assumptions implicit in the
Smoluchowski equation and therefore also in the dynamical density functional
theory.
