Refined central limit theorem and infinite density tail of the Lorentz gas from Levy walk

Abstract

We consider point particle that collides with a periodic array of hard-core elastic scatterers where the length of the free flights is unbounded (the infinite-horizon Lorentz gas, LG). The Bleher central limit theorem (CLT) states that the distribution of the particle displacement divided by $\sqrt{t\ln t}$ is Gaussian in the limit of infinite time $t$. However it was stressed recently that the slow convergence makes this result unobservable. Using a L\'{e}vy walk model (LW) of the LG, it was proposed that the use of a rescaled Lambert function instead of $\sqrt{t\ln t}$ provides a fast convergent, observable CLT, which was confirmed by the LG simulations. We demonstrate here that this result can simplified to a mixed CLT where the scaling factor combines normal and anomalous diffusions. For narrow infinite corridors the particle for long time obeys the usual normal diffusion, which explains the previous numerical observations. In the opposite limit of small scatterers the Bleher CLT gives a good guiding. In the intermediate cases the mixed CLT applies. The Gaussian peak determines moments of order smaller than two. In contrast, the CLT gives only half the coordinate dispersion. The missing half of the dispersion and also moments of order higher than two are described by the distribution's tail (the infinite density) which we derive here. The tail is supported along the infinite corridors and formed by anomalously long flights whose duration is comparable with the whole time of observation. The moments' calculation from the tail is confirmed by direct calculation of the fourth moment from the statistics of the backward recurrence time defined as time that elapsed since the last collision. This completes the solution of the LW model allowing full comparison with the LG.