  In the past decade, classical dynamical density functional theory (DDFT) has
been developed and widely applied to the Brownian dynamics of interacting
colloidal particles. One of the possible derivation routes of DDFT from the
microscopic dynamics is via the Mori-Zwanzig-Forster projection operator
technique with slowly varying variables such as the one-particle density. Here,
we use the projection operator approach to extend DDFT into various directions:
first, we generalize DDFT toward mixtures of $n$ different species of spherical
colloidal particles. We show that there are in general nontrivial
cross-coupling terms between the concentration fields and specify them
explicitly for colloidal mixtures with pairwise hydrodynamic interactions.
Secondly, we treat the energy density as an additional slow variable and derive
formal expressions for an extended DDFT containing also the energy density. The
latter approach can in principle be applied to colloidal dynamics in a nonzero
temperature gradient. For the case without hydrodynamic interactions the
diffusion tensor is diagonal, while thermodiffusion -- the dissipative
cross-coupling term between energy density and concentration -- is nonzero in
this limit. With finite hydrodynamic interactions also cross-diffusion
coefficients assume a finite value. We demonstrate that our results for the
extended DDFT contain the transport coefficients in the hydrodynamic limit
(long wavelengths, low frequencies) as a special case.
