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

Abstract: 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 ${f_0^{(n)}: \mathbb{R}^n\rightarrow \mathbb{R}}$ with finite second moments that shows practically no decrease in $d_2(f_0(t), R_{f_0})$ for time at least $\frac{1}{2\lambda}$ with $\lambda$ the rate of the Kac operator. We also present a propagation of chaos result for the partially thermostated Kac model in [14].