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

Abstract

We use the Fourier based Gabetta–Toscani–Wennberg 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 $\min \left\{ B e^{-\frac{4 \lambda _1}{n+3}t}, d_2(\mu ,R_\mu )\right\} ,$ 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 Tossounian and Vaidyanathan (J Math Phys 56(8):083301, 2015).