\kappa_2 (gV_2)\star \varrho)+\bm{A}[\bm{a}]\right),
\end{cases}\end{aligned}$$ where $\bm{A}[\bm{a}]$ is an effective background flow induced by the hydrodynamic interactions defined by $$\begin{aligned}
\label{eq:def_of_V1_eff}
\bm{A}[\vec{a}]:=\int_U\mathrm{d}\vec{r}'\, g(\vec{r},\vec{r}') \bm{Z}_2(\vec{r},\vec{r}')\vec{a}(\vec{r'},t),\end{aligned}$$ $\kappa_1$, $\kappa_2$ are non-dimensional constants measuring the strength of confining and interaction potentials respectively, $Pe=\mathrm{L}\text{U}/\alpha$ is the P[é]{}clet number measuring the ratio of advection rates to diffusive rates and $\alpha = k_B\mathrm{T}/(m\gamma)$.

Corollary \[prop:non\_dimensional\_time\_evolving\_rho\_eqn\] is the general formulation of the nondimensional equations - when $\bm{Z}_2\neq0$, including a non-constant diffusion coefficient and an effective drift. Throughout this paper, to study the intermediate regime of equally strong advection and diffusion, we set $Pe = 1$. Additionally, we redefine the two-body potential to absorb the correlation function $g$ to ease notation, $V_2(\vec{r},\vec{r}'):= g(\vec{r},\vec{r}') V_2(\vec{r},\vec{r}')$. In practice, there are many choices for $g$, for example the hard sphere approximation takes $g(|\vec{r}- \vec{r}'|) = 0$ for $|\vec{r}-\vec{r}'| < 1$ and unity otherwise. Alternatively $g$ may be obtained numerically from microscopic dynamics. We consolidate the choices for equations , in Section \[subsec:assumptions\_definitions\].

The effective drift $\bm{A}[\vec{a}]$, dependent on $\bm{Z}_2$ and $\vec{a}$ may be determined once $\vec{a}(\vec{r},t)$ is solved from the second equation in . Note that the evolution equation in may be viewed as a generalised McKean-Vlasov equation with a non-constant diffusion tensor 