Equilibration in the Kac Model using the GTW Metric $d_2$

We use the Fourier based Gabetta-Toscani-Wennberg (GTW) metric $d_2$ to study the rate of convergence to equilibrium for the Kac model in $1$ dimension. We take the initial velocity distribution of the particles to be a Borel probability measure $\mu$ on ${\mathbb{R}}^n$ that is symmetric in all its variables, has mean $\vec{0}$ and finite second moment. Let ${\mu }_t(dv)$ denote the Kac-evolved distribution at time $t$, and let $R_{\mu }$ be the angular average of $\mu$. We give an upper bound to $d_2({\mu }_t,R_{\mu })$ of the form $\min \{B{e}^{-\frac{4{\lambda }_1}{n+3}t},d_2(\mu ,R_{\mu })\}$, where $\lambda_1 = \frac{n+2}{2(n-1)}$isthegapoftheKacmodelin$L^2$and$B$ depends only on the second moment of $\mu$. We also construct a family of Schwartz probability densities $\{f_0^{(n)}:{\mathbb{R}}^n\to \mathbb{R}\}$ with finite second moments that shows practically no decrease in $d_2(f_0(t), R_{f_0})$fortimeatleast$\frac{1}{2\lambda}$with$\lambda$ the rate of the Kac operator. We also present a propagation of chaos result for the partially thermostated Kac model in [14].