Infinite horizon billiards: Transport at the border between Gauss and Lévy universality classes
Journal article, Peer reviewed
MetadataShow full item record
Original versionZarfaty L, Peletskyi A, Barkai E, Denysov S. Infinite horizon billiards: Transport at the border between Gauss and Lévy universality classes. Physical review. E. 2019;100 https://dx.doi.org/10.1103/PhysRevE.100.042140
We consider transport in two billiard models, the inﬁnite horizon Lorentz gas and the stadium channel, presentinganalyticalresultsforthespreadingpacketofparticles.Weﬁrstobtainthecumulativedistributionfunction of traveling times between collisions, which exhibits nonanalytical behavior. Using a renewal assumption and the Lévy walk model, we obtain the particles’ probability density. For the Lorentz gas, it shows a distinguished difference when compared with the known Gaussian propagator, as the latter is valid only for extremely long times. In particular, we show plumes of particles spreading along the inﬁnite corridors, creating power-law tails of the density. We demonstrate the slow convergence rate via summation of independent and identically distributed random variables on the border between Lévy and Gauss laws. The renewal assumption works well for the Lorentz gas with intermediate-size scattering centers, but fails for the stadium channel due to strong temporal correlations. Our analytical results are supported with numerical samplings.