Let $n\in\mathbb N$, let $\lambda\in\mathbb R$, and let $V\in C^2(\mathbb R^n;\mathbb R)$ satisfy

\begin{align*} \xi^\top J(\nabla V)_x\xi\ge \lambda |\xi|^2 \end{align*}

for every $x\in\mathbb R^n$ and every $\xi\in\mathbb R^n$. For a Borel probability measure $\mu\in\mathcal P_2(\mathbb R^n)$, define

\begin{align*} \operatorname{Ent}_{\mathcal L^n}(\mu):=\int_{\mathbb R^n}\rho(x)\log \rho(x)\,d\mathcal L^n(x) \end{align*}

if $\mu=\rho\,\mathcal L^n$ for a Borel density $\rho:\mathbb R^n\to[0,\infty)$ and this integral is a finite real number, and set $\operatorname{Ent}_{\mathcal L^n}(\mu):=+\infty$ otherwise. Write $V^+:=\max\{V,0\}$ and $V^-:=\max\{-V,0\}$. Define the potential energy by

\begin{align*} \mathcal V[\mu]:=\int_{\mathbb R^n}V(x)\,d\mu(x) \end{align*}

when both $V^+$ and $V^-$ are $\mu$-integrable, and set $\mathcal V[\mu]:=+\infty$ otherwise. Define the free energy functional $\mathcal F:\mathcal P_2(\mathbb R^n)\to(-\infty,+\infty]$ by

\begin{align*} \mathcal F[\mu]:=\operatorname{Ent}_{\mathcal L^n}(\mu)+\mathcal V[\mu] \end{align*}

when both terms are finite real numbers, and set $\mathcal F[\mu]:=+\infty$ otherwise.

If $\mu_0,\mu_1\in\mathcal P_2(\mathbb R^n)$ are absolutely continuous with respect to $\mathcal L^n$ and satisfy $\mathcal F[\mu_0]\in\mathbb R$ and $\mathcal F[\mu_1]\in\mathbb R$, then for every constant-speed geodesic $\mu:[0,1]\to\mathcal P_2(\mathbb R^n)$, $t\mapsto\mu_t$, from $\mu_0$ to $\mu_1$ with respect to the quadratic Wasserstein distance $W_2$, one has, for every $t\in[0,1]$,

\begin{align*} \mathcal F[\mu_t]\le (1-t)\mathcal F[\mu_0]+t\mathcal F[\mu_1]-\frac{\lambda}{2}t(1-t)W_2(\mu_0,\mu_1)^2. \end{align*}