Abstract
We prove a superdiffusive central limit theorem for the displacement of a test particle in the periodic Lorentz gas in the limit of large times t and low scatterer densities (Boltzmann–Grad limit). The normalization factor is $${\sqrt{t {\rm log} t}}$$ , where t is measured in units of the mean collision time. This result holds in any dimension and for a general class of finite-range scattering potentials. We also establish the corresponding invariance principle, i.e., the weak convergence of the particle dynamics to Brownian motion.
Highlights
The periodic Lorentz gas is one of the iconic models of “chaotic” diffusion in deterministic systems
We prove a superdiffusive central limit theorem for the displacement of a test particle in the periodic Lorentz gas in the limit of large t√imes t and low scatterer densities (Boltzmann–Grad limit)
The normalization factor is t log t, where t is measured in units of the mean collision time
Summary
The periodic Lorentz gas is one of the iconic models of “chaotic” diffusion in deterministic systems. The first seminal result on this subject was the proof of a central limit theorem for the displacement of the test particle at large times t for the finite-horizon Lorentz gas by Bunimovich and Sinai [9]. For notational reasons it is convenient to extend the dynamics to T1(Rd ) := Rd × S1d−1 by setting xt = x0 for all initial conditions x0 ∈/ Kr. We consider the time evolution of a test particle with random initial data (x0, v0) ∈ T1(Rd ), distributed according to a given Borel probability measure on T1(Rd ). The following superdiffusive central limit theorem, valid for small scattering radii and large tim√es, asserts that the normalized particle displacement at time t, and measured in units of t log t, converges weakly to a Gaussian distribution.
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