I am interested in the $2$-Wasserstein distance for probabilities over ${\mathbb R}^n$, $$W_2(\mu,\nu)=\left(\inf\int_{{\mathbb R}^n\times{\mathbb R}^n}|w-v|^2\pi(v,w)\right)^{1/2}$$ where the infimum is taken over the probabilities $\pi$ having marginals $\mu$ and $\nu$.

Let $GP$ denote the subset of probabilities that have a Gaussian density. Is there any close formula, or approximate formula, for the distance (in terms of $W_2$) from a given $\mu$ to $GP$ ? How can it be related to the entropy dissipation rate encountered in the Boltzman equation, where one integrate the non-negative quantity $$\left(f(v^*)f(v'^*)-f(v)f(v')\right)\left(\log\big(f(v^*)f(v'^*)\big)-\log\big(f(v)f(v')\big)\right)$$ where $f$ is the density of the velocity distribution and the arguments are constrained by (conservation of momentum and energy) $$v+v'=v*+v'^*,\qquad\|v\|^2+\|v'\|^2=\|v^*\|^2+\|v'^*\|^2.$$ Notice that, for consistency, the Wasserstein distance should be applied to measures of the form $$\frac1\rho\,f(v)\,dv,\qquad\rho:=\int f(v)\,dv.$$