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

TL;DR

This paper investigates the rate of convergence to equilibrium in the 1D Kac model using the Fourier-based GTW metric, providing bounds, constructing near-non-converging examples, and discussing propagation of chaos.

Contribution

It introduces bounds on the convergence rate in the GTW metric, constructs examples with minimal convergence, and extends results to the thermostated Kac model.

Findings

Derived an exponential upper bound for the convergence in $d_2$ metric.

Constructed Schwartz densities with negligible convergence over time.

Presented a propagation of chaos result for the thermostated Kac model.

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].