starting from the full system of bath and colloidal particles). A consequence of this coarse graining down to the dynamics of only the colloidal-particle degrees of freedom is that we no longer deal with deterministic but with random variables. The advantage is that it removes the enormous number of variables related to the dynamical variables of the bath, traditionally via projection-operator techniques which effectively allow to average out the bath[@kirkwood_statistical_1946; @murphy_brownian_1972; @deutch_molecular_1971; @wilemski_derivation_1976; @ermak_brownian_1978; @grabert_microdynamics_1980; @darve_computing_2009]. Despite such a simplification, the resultant equations are still computationally prohibitive because of the large number of particles that need to be taken into consideration to describe the behavior of colloidal fluids. A theoretical way out is to obtain mean field approximations which are independent of the number of particles. For an $N$-particle system, a standard procedure in the statistical mechanics of classical fluids is to average over $N-n$ (with $n < N$) colloidal particles, resulting in a time-evolution equation for the $n$-body reduced distribution function. However, to obtain a closed set of equations for a given $n$ requires knowledge of the relationships between the full and the reduced distributions. Dynamical density functional theories (DDFTs) perform this procedure for $n=1$, i.e. for the one-body 