Rates of convergence for the superdiffusion in the Boltzmann–Grad limit of the periodic Lorentz gas

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 for studying diffusions in deterministic systems. In their seminal work, Marklof and Strömbergsson (2011) proved the Boltzmann–Grad limit of the periodic Lorentz gas, following which Marklof and Tóth (2016) established a superdiffusive central limit theorem in large time for the Boltzmann–Grad limit. 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. The convergence rate in Wasserstein distance is obtained for the discrete-time displacement, while the result for the Berry–Essen type bound is presented for the continuous-time displacement.