where $d$ is the physical dimension and $\mathrm{A}$ is a characteristic flux scale. The system becomes (after dropping tildes) $$\begin{aligned}
\partial_t\varrho(\vec{r},t) = -\tfrac{1}{Fr}\times \tfrac{\tau^{-1}}{\gamma}\times \mathrm{A}\times \mathrm{L}^{d+1} \nabla_{\vec{r}}\cdot \vec{a}(\vec{r},t),\end{aligned}$$ where we have defined the Froude number $Fr = m\mathrm{U}^2/(k_BT)$. By choosing $Fr = 1$, $\tau = \gamma^{-1}$ and $\mathrm{A} = 1/\mathrm{L}^d$ we simplify the system of equations to the following boundary value problem.\

\[prop:non\_dimensional\_time\_evolving\_flux\_eqn\]

The non-dimensional one-body density $\varrho(\vec{r},t)$ and flux $\vec{a}(\vec{r},t)$ evolve according the the boundary value problem $$\begin{aligned}
\label{eq:evolution_eqn_for_a_dimensionless}
\begin{cases}
&\qquad\qquad\partial_t\varrho = -\nabla_{\vec{r}}\cdot \vec{a}(\vec{r},t), \\
&\vec{a}(\vec{r},t) + \bm{H}[\vec{a},\varrho] +\varrho(\vec{r},t)\bm{D}(\vec{r},[\varrho],t)\nabla_{\vec{r}}\frac{\delta\mathcal{F}}{\delta\varrho}[\varrho]=0,
\\
&\qquad [\bm{H}[\vec{a},\varrho] +\varrho(\vec{r},t)\bm{D}(\vec{r},[\varrho],t)\nabla_{\vec{r}}\frac{\delta\mathcal{F}}{\delta\varrho}[\varrho]]\cdot\vec{n}\big|_{\partial U}=0.
\end{cases}\end{aligned}$$

We note that when the off-diagonal HI tensor $\bm{Z}_{2}=0$, by using the definitions of $\mathcal{F}$ and $\bm{D}$ , the evolution equations in may be written as a nonlinear Smoluchowski equation (such as ) with non-constant diffusion coefficient. However we observe that even when $\bm{Z}_2 \neq 0$ the dynamics may be recast into a Smoluchowski equation for $\varrho$ under an effective drift vector dependent on $\bm{Z}_2$.\

\[prop:non\_dimensional\_time\_evolving\_rho\_eqn\]

The non-dimensional one-body density $\varrho(\vec{r},t)$ evolves according the the boundary value problem $$\begin{aligned}
\label{eq:evolution_eqn_for_rho_dimensionless}
\begin{cases}
&\qquad\partial_t\varrho=\,\nabla_{\vec{r}}\cdot\left[Pe^{-1}\bm{D}\nabla_{\vec{r}}\varrho+\varrho\,\bm{D}\left(\nabla_{\vec{r}} (\kappa _1V_1+\kappa _2\,(gV_2)\star\varrho) + \bm{A}[\bm{a}]\right)\right],
\\
&\qquad \qquad \qquad \qquad \qquad  \Pi [\varrho]\cdot\vec{n}\big|_{\partial U} = 0,\\
&\qquad\Pi[\varrho]:= \bm{D}\,\left(\nabla_{\vec{r}}\varrho + 
\varrho\, \nabla_{\vec{r}}(\kappa_1 V_1(\vec{r},t)+ 