density, and typically result in a continuity equation for the density and a time-evolution equation for the current, that is now a functional of the density. The functional that relates the current and the density is generally unknown, but, for a practical implementation of DDFTs it needs to be approximated. A standard approach, for the overdamped dynamics, is to take it as the free-energy functional of a system with the same density at equilibrium. This functional has been well-studied in the statistical mechanics of fluids[@evans_nature_1979; @lutsko_recent_2010]. Also, approximating the functional relating the current and the density with the free energy at equilibrium also ensures that the DDFTs reduce to the corresponding equilibrium DFTs[@goddard_general_2012; @goddard_unification_2013].

The first derivations of a DDFT go back to the early works of @evans_nature_1979 and @dieterich_nonlinear_1990. These initial formulations were mostly phenomenological and subsequent efforts focused on deriving rigorously the same results and also DDFTs that include additional effects such as inertia and hydrodynamic interactions (HIs)[@marconi_dynamic_2000; @archer_dynamical_2004; @archer_dynamical_2009; @espanol_derivation_2009; @goddard_overdamped_2012; @goddard_general_2012; @goddard_unification_2013; @donev_dynamic_2014].

There have also been several efforts to test the derived DDFTs and their predictive capabilities. In fact there have been successful applications to a wide spectrum of physical settings, from hard rod and 