systems in the overdamped limit, the HI are taken to be the long range forces given by the Rotne-Prager-Yamakawa tensor [@rotne1969variational]. However, we do not make any such assumptions on the form of the tensors here.

We have described a general set of coupled Langevin equations with spatially-dependent friction tensor $\boldsymbol{\Gamma}(\vec{r}^N)$. As we will see, the dynamics – tend towards an equilibrium given by the Gibbs probability measure, which we will show to be independent of the friction tensor. Instead of computing the trajectories of individual particles we consider the evolution of the density of particles $\varrho(\vec{r},t)$ given by the Smoluchowski equation in the high friction limit $\gamma\to \infty$, $$\begin{aligned}
\label{eq:mk-eq}
\qquad \partial_{t}\varrho(\vec{r},t) =-\tfrac{k_B\mathrm{T}}{m\gamma} \nabla_{\vec{r}}\cdot\vec{a}(\vec{r},[\varrho], t)  \qquad \text{ for } \vec{r} \in  U,\,t\in [0,T]\end{aligned}$$ where $\vec{a}(\vec{r},[\varrho], t)$ is the flux, $[\varrho]$ denotes functional dependence, $U\subseteq\mathbb{R}^d$ and $T<\infty$. Equation was derived rigorously as a solvability condition of the corresponding Vlasov-Fokker-Planck equation for the one-body density in position and momentum space $f(\vec{r},\vec{p},t)$ by writing $f$ as a Hilbert expansion in a small nondimensional parameter $\epsilon\propto\gamma^{-1}$ [@goddard2012overdamped]. Therein, $\epsilon$ has units length, and therefore a problem specific length scale must be introduced to make it truly nondimensional.

We are interested in global existence, 