flux through the boundary $\partial U$ is zero for all time $t\in [0,T]$. If $\varrho$ is a number density then $\int \mathrm{d}\vec{r}\, \varrho = N$ for all time, however for the analysis in Section \[sec:ex\_uni\_full\_HI\] and onwards we will assume $\varrho$ is a probability density so that $\int \mathrm{d}\vec{r}\, \varrho = 1$. The rescaling between number and probability densities is discussed in the following section.

Initial Conditions.
-------------------

We will assume that the initial data has finite free energy and is consistent with the imposed boundary conditions. For example, one could prescribe initial data $(\varrho_0,\vec{a}_0)^\top$ such that $$\begin{aligned}
\label{eq:initial_data_for_a_and_rho}
\frac{\delta\mathcal{F}}{\delta\varrho}[\varrho_0](\vec{r}) = \mu_{c},\qquad
\vec{a}_0 = \vec{0}.\end{aligned}$$ where $\mu_{c}$ is the chemical potential, constant at equilibrium. It is straightforward to check that $(\varrho_0,\vec{a}_0)^\top$ is an equilibrium point of the system . Commonly, one then drives the system out of equilibrium via a time-dependent external potential. In principle $\mu_{c}$ may be given and the equations , are well defined. In practice, for complicated particle configurations, $\mu_{c}$ is not known but can be computed by minimising the free energy along with the additional constraint $\int_U\mathrm{d}\vec{r}\, \varrho_0(\vec{r})=N$, where $N$ is the (expected) number of particles for a finite system and $\varrho_0$ is a number density. 