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 $lambda_1 = frac{n+2}{2(n-1)}$ is the gap of the Kac model in $L^2$ and $B$ depends only on the second moment of $mu$. We also construct a family of Schwartz probability densities ${f_0^{(n)}: mathbb{R}^nrightarrow mathbb{R}}$ with finite second moments that shows practically no decrease in $d_2(f_0(t), R_{f_0})$ for time at least $frac{1}{2lambda}$ with $lambda$ the rate of the Kac operator. We also present a propagation of chaos result for the partially thermostated Kac model in [14].