Infinite horizon billiards: Transport at the border between Gauss and Lévy universality classes

Abstract

We consider transport in two billiard models, the infinite horizon Lorentz gas and the stadium channel, presenting analytical results for the spreading packet of particles. We first obtain the cumulative distribution function of traveling times between collisions, which exhibits non-analytical 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 identically distributed random variables on the border between Lévy and Gauss laws. The renewal assumption works well for the Lorentz gas with intermediately sized scattering centers, but fails for the stadium channel due to strong temporal correlations. Our analytical results are supported with numerical samplings.

arXiv comment14 pages, 13 figures, 5 appendixes (25 pages in total)

References in corpus (12)

• Lévy walks
• Non-normalizable densities in strong anomalous diffusion: beyond the central limit theorem
• The Lyapunov exponent in the Sinai billiard in the small scatterer limit
• Superdiffusive dispersals impart the geometry of underlying random walks
• Normal and Anomalous Diffusion in Soft Lorentz Gases
• New horizons in multidimensional diffusion: The Lorentz gas and the Riemann Hypothesis
• Measuring logarithmic corrections to normal diffusion in infinite-horizon billiards
• Dispersion of particles in an infinite-horizon Lorentz gas
• Machta-Zwanzig regime of anomalous diffusion in infinite-horizon billiards
• Limit theorems for Lévy walks in $d$ dimensions: rare and bulk fluctuations
• Slow Convergence in Generalized Central Limit Theorems
• Refined central limit theorem and infinite density tail of the Lorentz gas from Levy walk

Cited by in corpus (10)

• Observing emergent hydrodynamics in a long-range quantum magnet
• Non-local emergent hydrodynamics in a long-range quantum spin system
• Lévy Walks and Path Chaos in the Dispersal of Elongated Structures Moving across Cellular Vortical Flows
• The Fluctuation-Dissipation Relations: Growth, Diffusion, and Beyond
• Labyrinth chaos: Revisiting the elegant, chaotic and hyperchaotic walks
• Universality in the Onset of Super-Diffusion in Lévy Walks
• Anomalous Diffusion in the Square Soft Lorentz Gas
• Origin of universality in the onset of superdiffusion in Lévy walks
• Asymptotic densities of planar Lévy walks: a non-isotropic case
• Anomalous fluctuations of renewal-reward processes with heavy-tailed distributions