Let $n\in\mathbb N$, and equip $\mathcal P_2(\mathbb R^n)$ with the $2$-Wasserstein distance $W_2$. Define the Boltzmann entropy functional $\operatorname{Ent}:\mathcal P_2(\mathbb R^n)\to(-\infty,+\infty]$ as follows. If $\mu\in\mathcal P_2(\mathbb R^n)$ is absolutely continuous with respect to $\mathcal L^n$, write $\mu=\rho\,\mathcal L^n$ for its density. Then

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

whenever $\rho\log\rho\in L^1(\mathbb R^n,\mathcal L^n)$, with the convention $0\log 0=0$. In all other cases set $\operatorname{Ent}[\mu]=+\infty$. Let $\operatorname{Dom}(\operatorname{Ent})=\{\mu\in\mathcal P_2(\mathbb R^n):\operatorname{Ent}[\mu]<+\infty\}$ denote the effective domain of the entropy.

For every $\mu_0\in\mathcal P_2(\mathbb R^n)$, the unique metric Wasserstein gradient flow of $\operatorname{Ent}$ starting from $\mu_0$, understood as the unique $\operatorname{EVI}_0$ gradient flow in $(\mathcal P_2(\mathbb R^n),W_2)$, is the heat semigroup. More precisely, for every $t>0$,

\begin{align*} \mu_t=G_t*\mu_0 \end{align*}

where $G_t:\mathbb R^n\to(0,\infty)$ is given by

\begin{align*} G_t(x)=(4\pi t)^{-n/2}\exp\left(-\frac{|x|^2}{4t}\right). \end{align*}

Equivalently, for every $t>0$ one has $\mu_t=\rho_t\,\mathcal L^n$, where $\rho_t:\mathbb R^n\to[0,\infty)$ is given by

\begin{align*} \rho_t(x)=\int_{\mathbb R^n}G_t(x-y)\,d\mu_0(y). \end{align*}

The map $\rho:(0,\infty)\times\mathbb R^n\to[0,\infty)$, $\rho(t,x)=\rho_t(x)$, is smooth, satisfies the heat equation

\begin{align*} \partial_t\rho(t,x)=\Delta\rho(t,x) \end{align*}

for every $(t,x)\in(0,\infty)\times\mathbb R^n$, and $\mu_t\to\mu_0$ in $(\mathcal P_2(\mathbb R^n),W_2)$ as $t\downarrow0$.