Phase space dynamical density functional theory for colloids with hydrodynamic interactions

We study the dynamics of a colloidal fluid in the full position- momentum phase space. These dynamics are modelled by stochastic equations of motion for a large number of identical spherical particles. We include the full hydrodynamic interactions, which strongly influence the non-equilibrium properties of the system. For large systems, the number of degrees of freedom prohibits a direct solution of the equations and a reduced model is necessary. Under certain assumptions, we derive a dynamical density functional theory (DDFT), i.e. a reduction to the dynamics of the reduced one-body distribution. Our formulation includes the case where the momentum distribution is not a local Maxwellian. Near equilibrium, it reduces to a Navier-Stokes-like equation with additional non-local terms. In the high friction limit, we show rigorously that it reduces to a previously-derived DDFT, describing only the position distribution, but with a novel definition of the diffusion tensor.