compared to a pure bath [@Einstein06], the blurring of laning that arises in driven flow [@WysockiLowen11], the migration of molecules away from a wall [@HodaKumar07], and are particularly complex in confined systems [@happel2012low; @LaugaSquires05], and for active particles and microswimmers, which result in additional HI [@HuberKoehlerYang11].

Mathematically, the inter-particle forces and HI can be described through the hydrodynamic fields $\varrho$ and $\vec{v}$, the one-body density and one-body velocity fields, respectively. These fields, inherent to a continuum description of a collection of particles, are derived by considering successive moments (density, velocity, heat flux, …) of the underlying kinetic system [@gorban2014hilbert]. In particular, for systems of interacting Newtonian particles, when the momenta are non-negligible, the evolution of the phase space density $f(\vec{r}^N,\vec{p}^N, t)$ for a system of $N$ colloids determining the probability of finding the system in the state $(\vec{r}^N,\vec{p}^N)$ at time $t$ is described by the $N$-body Fokker-Planck equation and the dynamics of the hydrodynamic fields are defined by obtaining closed equations for $\{\varrho, \varrho\times \vec{v}\} := \int \mathrm{d}\vec{r}^{N-1}\,\mathrm{d}\vec{p}^N\, \{1, \vec{p}/m\}f(\vec{r}^N,\vec{p}^N, t)$, where $m$ is the particle mass. Here, $\vec{r}^N$ and $\vec{p}^N$ denote the $3N$-dimensional position and momentum vectors of all $N$ particles.

The inclusion of HI leads to a much 