The convergence rates for the superdiffusion in the Boltzmann-Grad limit of the periodic Lorentz gas

TL;DR

This paper derives convergence rates for superdiffusion in the Boltzmann-Grad limit of the periodic Lorentz gas, using Stein's method to quantify the speed of convergence in both discrete and continuous time settings.

Contribution

It provides the first explicit convergence rates for superdiffusion in the Boltzmann-Grad limit of the periodic Lorentz gas, building on prior limit theorems.

Findings

Wasserstein distance convergence rate for discrete time displacement

Berry-Essen type bound for continuous time displacement

Quantitative analysis of superdiffusion convergence speed

Abstract

In this article, we obtain the rates of convergence for superdiffusion in the Boltzmann-Grad limit of the periodic Lorentz gas, which is one of the fundamental models to study diffusions in deterministic systems. In their seminal work, Marklof and Str\"ombergsson proved the Boltzmann-Grad limit of the periodic Lorentz gas~\cite{M-Sannals2}, and then Marklof and T\'oth established a superdiffusive central limit theorem in large time for the Boltzmann-Grad limit~\cite{M-T16}. Based on their work, we apply Stein's method to derive the convergence rates for the superdiffusion in the Boltzmann-Grad limit of the periodic Lorentz gas. For the discrete time displacement the rate of convergence in Wasserstein distance is obtained, while in the context of the continuous time displacement the result is presented for the Berry-Essen type bound.