to no-flux boundary conditions. This choice admits the nontrivial effect of the two body forces generated by the potential $V_2$ interacting with density on the boundary of the physical domain. We also seek to understand the asymptotic stability of stationary states. The motivation for this choice of boundary condition is physical; it corresponds to a closed system of particles in which the particle number is conserved over time. It is clear that most applications of such equations will be in confined systems, rather than a periodic domain and, as such, no-flux boundary conditions are natural. We note that the choice of boundary condition is expected to have significant effects on the dynamics, including the form of the bifurcation diagram.

Free Energy Framework. {#subsec:free_energy_framework}
----------------------

Related to the system -, we define the free energy functional $\mathcal{F}:P^+_{\text{ac}}(U)\to \mathbb{R}$ where $P^+_{\text{ac}}$ is the set of strictly positive definite absolutely continuous probability measures on $U$. We define $$\begin{aligned}
\label{eq:def_of_F}
\mathcal{F}[\varrho]&:=\int_U\mathrm{d}\vec{r}\,\varrho(\vec{r},t)\log\varrho(\vec{r},t)+\int_U\mathrm{d}\vec{r}\,\varrho(\vec{r},t)\,\Big[V_1(\vec{r},t)+\tfrac{1}{2}(gV_2)\star\varrho \Big],\end{aligned}$$ where $\star$ denotes convolution in space. Here we assume the probability measure $\varrho$ has density with respect to the Lebesgue measure. Additionally we define the probability measure on $U$ $$\begin{aligned}
\label{eq:def_of_measure}
\mu(\mathrm{d}\vec{r}) = \mathrm{d}\vec{r}\,Z^{-1}e^{-\tfrac{(V_1+(gV_2)\star\varrho)}{k_BT}}\end{aligned}$$ where $Z = \int_U \mathrm{d}\vec{r}\,e^{-\tfrac{(V_1+(gV_2)\star\varrho)}{k_BT}}$ and $\varrho$ (when 