The distribution of free path lengths in the periodic Lorentz gas and related lattice point problems

• 成果类型: Article
• 署名作者: Marklof, Jens; Strombergsson, Andreas
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2010.172.1949
• 发表日期: 2010
• 页码: 1949-2033
• 关键词: sure invariance-principle; STATISTICAL PROPERTIES; infinite-horizon; limit; equidistribution; recurrence; systems

摘要

The periodic Lorentz gas describes the dynamics of a point particle in a periodic array of spherical scatterers, and is one of the fundamental models for chaotic diffusion. In the present paper we investigate the Boltzmann-Grad limit, where the radius of each scatterer tends to zero, and prove the existence of a limiting distribution for the free path length. We also discuss related problems, such as the statistical distribution of directions of lattice points that are visible from a fixed position.