Refined central limit theorem and infinite density tail of the Lorentz gas from Lévy walk. (10th September 2020) - Fouxon, Itzhak; Ditlevsen, Peter
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 due to infinitely long corridors between the scatterers (the infinite-horizon Lorentz gas, LG). The Bleher central limit theorem (CLT) states that the distribution of the particle displacement divided by t ln t is Gaussian in the limit of infinite time t . However this result, describing the bulk of the distribution, is not as powerful as in normal diffusion cases. The Gaussian peak fails to describe the displacement's moments of order higher than two and gives only half of the true value of the dispersion. These demand the tail of the distribution which is formed by rare long flights along the corridors where the particle propagates much further than in typical diffusive displacements due to collisions with close scatterers. We derive the tail using a Lévy walk (LW) model of the LG, demonstrating extreme sensitivity to single flight event. The tail depends on whether between the steps the particle moves ballistically or jumps instantaneously and it obeys the single-big-jump principle i.e. large deviations are formed fully in only one step of the walk. The validity of the LW model in the LG's description was proved recently in studying another deficiency of the CLT—the slow (logarithmic) convergence to the Gaussian peak hindering its observability. Using the LW model, it was proposed that rescaling the LG's displacement by a Lambert function factor instead of t ln t provides a fast convergent observable CLT, which was confirmed by the LG simulations. Here we demonstrate that this result can be simplified giving 'mixed CLT' where the scaling factor combines normal and anomalous diffusions. This fits the previous observation that if infinite corridors are narrow then for long time the diffusion is normal. We conjecture that for observables determined by single flight events the LW results can be transferred to the LG.