Let $n\in\mathbb N$, let $\mathcal P_2(\mathbb R^n)$ denote the space of Borel probability measures on $\mathbb R^n$ with finite second moment, and let $W_2$ denote the quadratic Wasserstein distance on $\mathcal P_2(\mathbb R^n)$. Let $\lambda>0$, and let $V\in C^2(\mathbb R^n)$ satisfy
as quadratic forms for every $x\in\mathbb R^n$, where $D^2V(x)=J(\nabla V)_x$ is the Hessian matrix of $V$ at $x$ and $I_n$ is the identity matrix on $\mathbb R^n$. Define the relative Boltzmann entropy $\operatorname{Ent}_{\mathcal L^n}:\mathcal P_2(\mathbb R^n)\to(-\infty,+\infty]$ by
when $\mu=\rho\mathcal L^n$ for a Borel density $\rho:\mathbb R^n\to[0,\infty)$ and $\rho\log\rho\in L^1(\mathbb R^n)$, and by $+\infty$ otherwise. Define the free energy $\mathcal F:\mathcal P_2(\mathbb R^n)\to(-\infty,+\infty]$ by
whenever the right-hand side is well-defined as a finite real number or $+\infty$, with value $+\infty$ otherwise. Assume that $\mathcal F$ attains a finite minimum on $\mathcal P_2(\mathbb R^n)$; that is, there exists $\mu_*\in\mathcal P_2(\mathbb R^n)$ such that