Let $n\in\mathbb N$, and let $\Omega\subset\mathbb R^n$ be a connected open set. Let $U\in C^1((0,\infty);\mathbb R)$, let $V\in C(\Omega;\mathbb R)$, and let $W\in C(\mathbb R^n;\mathbb R)$ be even, meaning that $W(z)=W(-z)$ for every $z\in\mathbb R^n$.
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For every smooth positive probability density $\rho\in C^\infty(\Omega;(0,\infty))$ for which all three constituent integrals below are finite real numbers and the interaction double integral is absolutely convergent, define
that $\mathcal E[\rho_\infty]<\infty$ in the preceding sense, and that $\rho_\infty$ minimizes $\mathcal E$ among all smooth positive probability densities on $\Omega$ with finite energy in the same sense. Define $\Phi_\infty:\Omega\to\mathbb R$ by
Assume that $\Phi_\infty(x)$ is finite for every $x\in\Omega$, that $\Phi_\infty\in C(\Omega;\mathbb R)$, and that for every compact set $K\subset\Omega$,