richer hierachy of fluid equations compared to systems without HI; compare e.g. [@goddard2012unification] and [@archer2009dynamical]. In particular, see e.g. [@goddard2012unification], by integration over all but one particle position, the one-body Fokker-Planck equation may be obtained. If, in addition, two-body HI and interparticle interactions are assumed and the inertia of the colloids is considered small, a high friction limit $\gamma\to \infty$ may be taken [@goddard2012overdamped]. The result is that the velocity distribution converges to a Maxwellian, and one can eliminate the momentum variable through an adiabatic elimination process that is based on multiscale analysis [@pavliotis2008multiscale]. The final one-body Smoluchowski equation for $\varrho$ is a novel, nonlinear, nonlocal PDE shown to be independent of the unknown kinetic pressure term $\int \mathrm{d}\vec{r}\,\mathrm{d}\vec{p}\, m^{-2} \vec{p}\otimes\vec{p} f(\vec{r},\vec{p}, t)$, which normally persists at $\gamma = O(1)$ (see[@goddard2012overdamped], Theorem 4.1).

Existence, uniqueness and global asymptotic stability of the novel Smoluchowski equation in this overdamped limit has, until this work, remained unproven. It is the inclusion of HI that provides richness through additional nonlinearities in both the dissipation and convection terms. The inclusion of HI is interesting from both physical and mathematical standpoints. Physically, as above, the HI give rise to a much more complex evolution in the 