Note that $\mu_{c}$ is a potential, so by raising it one may force more particles into the system. We will assume that $\mu_c$ is constant to fix the number of particles. To ensure $\varrho$ (and $\varrho_0$) is a probability density one may rescale $\varrho/N = \tilde{\varrho}$, $N g = \tilde{g}$ and $\vec{a}/N^2 = \tilde{\vec{a}}$, where the tilde denotes the new variable, so that $\int_U\mathrm{d}\vec{r}\, \varrho_0(\vec{r})=1$ and equations - become independent of $N$.

This provides a method of converting back to the number density which is typically used in numerical modelling of finite colloidal systems [@goddard2016dynamical], [@goddard2012unification], [@goddard2012general]. Throughout however, since we will frequently use the integral of the density, we will assume $\varrho$ and $\varrho_0$ are probability densities to ease notation. With this, one has three equations for three unknowns $\mu_{c}$, $\varrho_0$, $\vec{a}_0$ and the initial density $\varrho_0$ can be computed. For the rest of paper it is convenient to work in dimensionless units. We now nondimensionalise the governing equations.

Evolution Equations.
--------------------

We now nondimensionalise our equations. Let $\mathrm{L}$, $\tau$, $\text{U}$ be characteristic length, time and velocity scales respectively, then by nondimensionalising $$\begin{aligned}
\vec{r}\sim \mathrm{L} \tilde{\vec{r}},\quad t\sim\tau\tilde{t}, \quad  \mathrm{U} = \tfrac{\mathrm{L}}{\tau}, \quad \varrho\sim\tfrac{1}{\mathrm{L}^d}\tilde{\varrho},\quad \mathcal{F}\sim k_BT \tilde{\mathcal{F}},\quad \vec{a}\sim \mathrm{A}\tilde{ \vec{a}}.\end{aligned}$$ 