and confining potential. In particular the McKean-Vlasov equation may be recovered in the special case $\bm{Z}_1 = \bm{Z}_2 = V_1 = 0$, see for example [@greg_mckean_vlasov_torus], [@chayes2010mckean]. We will use Corollary \[prop:non\_dimensional\_time\_evolving\_rho\_eqn\], to write the full dynamics including full HI, to obtain our results on weak solutions for $\varrho(\vec{r},t)$ (see Theorem \[thm:eigenfn\_expansion\_of\_flux\], Section \[sec:ex\_uni\_full\_HI\] and Theorem \[thm:existence\_and\_uniqueness\], Section \[sec:existence\_uniqueness\_with\_partial\_HI\]). We continue to the next section by stating the stationary boundary value problem for equilibrium states $\varrho(\vec{r})$.

Stationary Equations.
---------------------

For general $\bm{Z}_2$ we will show in Theorem \[thm:association \_of\_free\_energy\] that the stationary density $\varrho(\vec{r})$ satisfies $$\begin{aligned}
\label{eq:stationary_eqn_for_rho_dimensionless}
\begin{cases}
&\qquad 0=\,\nabla_{\vec{r}}\cdot[\bm{D}\nabla_{\vec{r}}\varrho+\varrho\,\bm{D}\nabla_{\vec{r}}(\kappa _1V_1+\kappa _2\,V_2\star\varrho)],
\\
&\qquad \qquad \qquad \qquad \qquad  \Pi [\varrho]\cdot\vec{n}\big|_{\partial U} = 0,\\
&\qquad\Pi[\varrho]:= \bm{D}\,(\nabla_{\vec{r}}\varrho + 
\varrho\, \nabla_{\vec{r}}(\kappa_1 V_1(\vec{r},t)+ \kappa_2 \,V_2\star \varrho)).
\end{cases}\end{aligned}$$ We now discuss regularity on the potentials and diffusion tensor.

Assumptions & Definitions. {#subsec:assumptions_definitions}
--------------------------

Typically for long range HI the $\boldsymbol{Z}_i$ exhibit singularities at the origin (particle centres) so the correlation function $g$ is a necessary inclusion and provides a way of smoothing $\boldsymbol{D}$ and we assume $g \in L^\infty(U)$. For $\varrho \geq 0$ the diffusion tensor $\boldsymbol{D}$ as a convolution with the density will then be a weakly differentiable function. For the existence and uniqueness 