Uniform contractivity in Wasserstein metric for the original 1D Kac's model

We study here a very popular 1D jump model introduced by Kac: it consists of $N$ velocities encountering random binary collisions at which they randomly exchange energy. We show the uniform (in $N$) exponential contractivity of the dynamics in a non-standard Monge-Kantorovich-Wasserstein: precisely the MKW metric of order 2 on the energy. The result is optimal in the sense that for each $N$, the contractivity constant is equal to the $L^2$ spectral gap of the generator associated to Kac's dynamic. As a corollary, we get an uniform but non optimal contractivity in the MKW metric of order $4$. We use a simple coupling that works better that the parallel one. The estimates are simple and new (to the best of our knowledge).