Let $n\in\mathbb N$, let $T\in(0,\infty)$, and let $V\in C^2(\mathbb R^n;\mathbb R)$. For every strictly positive probability density $r:\mathbb R^n\to(0,\infty)$ for which the following integrals are finite, define the entropy plus potential energy functional by
that $\mathcal E[\rho_t]$ is finite, and that $\rho_t$, $\partial_t\rho_t$, and all spatial derivatives appearing below have sufficient decay at infinity to justify differentiating under the integral sign and all integrations by parts used in the formal Otto calculus.
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Then the formal Wasserstein gradient-flow equation for $\mathcal E$, with the convention