Limit Laws and Recurrence for the Planar Lorentz Process with Infinite Horizon

Abstract

As Bleher, [B 92] observed the free flight vector of the planar, infinite horizon, periodic Lorentz process {Sn|n = 0, 1, 2, . . . } belongs to the nonstandard domain of attraction of the Gaussian law -actually with the √ n log n scaling. Our first aim is to establish his conjecture that, indeed, Sn √ n log n converges in distribution to the Gaussian law (a Global Limit Theorem). Here the recent method of Bálint and Gouëzel, [BG 06] helped us to essentially simplify the ideas of our earlier sketchy proof [SzV 04b]. Moreover, we can also derive a.) the local version of the Global Limit Theorem, b.) the recurrence of the planar, infinite horizon, periodic Lorentz process, and finally c.) the ergodicity of its infinite invariant measure.- ------------------------

Nonequilibrium Density Profiles in Lorentz Tubes with Thermostated Boundaries

“…We assume that the domain has finite horizon (that is, the particle can not move indefinitely without hitting a scatterer) since an anomalous transport takes place in the infinite horizon case [Bl92,SzV07,ChD09A,MS10]. Moving particles are injected from the left end of the tube according to a Poisson process with constant intensity.…”

Convergence of moments for Axiom A and non-uniformly hyperbolic flows

“…Remark 1.1 Our results are restricted to systems modelled by Young towers with summable decay of correlations (β > 1) where the CLT holds with the standard n 1 2 normalisation (or t 1 2 in the case of flows). For the infinite horizon planar periodic Lorentz gas, Szász & Varjú [40] proved a nonstandard CLT with normalisation (t log t) 1 2 . In this situation convergence of moments seems to be more subtle, see for example [2,16].…”

Universality in Statistical Measures of Trajectories in Classical Billiard Systems

“…In the case of infinite horizon, the CLT has not been proven. Moreover, in this case the diffusion coefficient is infinite and there is no convergence to Brownian motion [51]. Actually, for the periodic Lorentz gas with finite horizon it can be shown rigorously that the distribution of length of trajectories becomes a Gaussian distribution in the limit of many bounces.…”