an analogue of a local pressure gradient for the particle density. We will show that $\mathcal{F}[\varrho]$ is associated to the PDE even when $\bm{Z}_2 \neq 0$, that is $\partial_t \varrho = 0$ implies $\varrho$ is a critical point of $\mathcal{F}$.

[0.48]{} ![(a). The bifurcation diagram for (a). $V_2(x,y) = x\cdot y$ and (b). $V_2(x,y) = -\cos\left(\frac{2\pi(x-y)}{\mathrm{L}}\right)$ in Section \[subsec:numericalexperiments\]: the solid [blue]{} line denotes the stable branch of solutions while the dotted [red]{} line denotes the unstable branch of solutions. In (a) the stationary density $e^{-x^2}/Z$ changes stability at the critical interaction energy $\kappa_2 = \kappa_{2\sharp}= -2.4$ and the new stable density is asymmetric adhering to one wall (Figure \[fig:bif\_fig\_right\]). In (b), in the absence of a confining potential, the uniform density becomes unstable at the critical interaction energy $\kappa_2 = \kappa_{2\sharp} = 0.4$ and the density may become multi-modal (Figure \[fig:bif\_stable\_equilibria\_V2\_cosine\]).[]{data-label="fig:bifurcation_diagram"}](bifurcation_diagram_for_Phi2.pdf "fig:"){width="\textwidth"}

 

[0.48]{} ![(a). The bifurcation diagram for (a). $V_2(x,y) = x\cdot y$ and (b). $V_2(x,y) = -\cos\left(\frac{2\pi(x-y)}{\mathrm{L}}\right)$ in Section \[subsec:numericalexperiments\]: the solid [blue]{} line denotes the stable branch of solutions while the dotted [red]{} line denotes the unstable branch of solutions. In (a) the stationary density $e^{-x^2}/Z$ changes stability at the critical interaction energy $\kappa_2 = \kappa_{2\sharp}= -2.4$ 