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

被引：82

作者：

Marklof, Jens ^{[1
]}

Strombergsson, Andreas ^{[2
]}

机构：

[1] Univ Bristol, Sch Math, Bristol BS8 1TW, Avon, England

[2] Royal Inst Technol, Dept Math, SE-10044 Stockholm, Sweden

来源：

ANNALS OF MATHEMATICS | 2010年 / 172卷 / 03期

基金：

英国工程与自然科学研究理事会;

关键词：

SURE INVARIANCE-PRINCIPLE; STATISTICAL PROPERTIES; INFINITE-HORIZON; LIMIT; EQUIDISTRIBUTION; RECURRENCE; SYSTEMS;

D O I：

10.4007/annals.2010.172.1949

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

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.

引用

页码：1949 / 2033

页数：85

共 37 条

[1]

[Anonymous], 1971, PUBLICATIONS MATH SO

[2] Exponential Decay of Correlations in Multi-Dimensional Dispersing Billiards [J].

Balint, Peter ;

Toth, Imre Peter .

ANNALES HENRI POINCARE, 2008, 9 (07) :1309-1369

[3] STATISTICAL PROPERTIES OF 2-DIMENSIONAL PERIODIC LORENTZ GAS WITH INFINITE HORIZON [J].

BLEHER, PM .

JOURNAL OF STATISTICAL PHYSICS, 1992, 66 (1-2) :315-373

[4] The distribution of the free path lengths in the periodic two-dimensional Lorentz gas in the small-scatterer limit [J].

Boca, Florin P. ;

Zaharescu, Alexandru .

COMMUNICATIONS IN MATHEMATICAL PHYSICS, 2007, 269 (02) :425-471

[5] On the correlations of directions in the Euclidean plane [J].

Boca, FP ;

Zaharescu, A .

TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, 2006, 358 (04) :1797-1825

[6] The correlations of Farey fractions [J].

Boca, FP ;

Zaharescu, A .

JOURNAL OF THE LONDON MATHEMATICAL SOCIETY-SECOND SERIES, 2005, 72 :25-39

[7] Distribution of lattice points visible from the origin [J].

Boca, FP ;

Cobeli, C ;

Zaharescu, A .

COMMUNICATIONS IN MATHEMATICAL PHYSICS, 2000, 213 (02) :433-470

[8] The statistics of the trajectory of a certain billiard in a flat two-torus [J].

Boca, FP ;

Gologan, RN ;

Zaharescu, A .

COMMUNICATIONS IN MATHEMATICAL PHYSICS, 2003, 240 (1-2) :53-73

[9] ON THE BOLTZMANN-EQUATION FOR THE LORENTZ GAS [J].

BOLDRIGHINI, C ;

BUNIMOVICH, LA ;

SINAI, YG .

JOURNAL OF STATISTICAL PHYSICS, 1983, 32 (03) :477-501

[10] On the distribution of free path lengths for the periodic Lorentz gas [J].

Bourgain, J ;

Golse, F ;

Wennberg, B .

COMMUNICATIONS IN MATHEMATICAL PHYSICS, 1998, 190 (03) :491-508

← 1 2 3 4 →