Abstract
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.
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MRNUMBER = {98b:22024},
MRREVIEWER = {Dave Witte Morris},
DOI = {10.1007/BF02837164},
ZBLNUMBER = {0864.22004},
} -
[Shimura] G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Publications of the Mathematical Society of Japan, No. 11, Iwanami Shoten, Publishers, Tokyo, 1971.
@book {Shimura, MRKEY = {0314766},
AUTHOR = {Shimura, Goro},
TITLE = {Introduction to the Arithmetic Theory of Automorphic Functions},
PUBLISHER = {Publications of the Mathematical Society of Japan, No. 11, Iwanami Shoten, Publishers, Tokyo},
YEAR = {1971},
PAGES = {xiv+267},
MRCLASS = {10D10 (12-02 14K22)},
MRNUMBER = {47 \#3318},
MRREVIEWER = {A. N. Andrianov},
ZBLNUMBER = {0221.10029},
} -
[Shiryaev] A. N. Shiryaev, Probability, Second ed., New York: Springer-Verlag, 1996, vol. 95.
@book {Shiryaev, MRKEY = {1368405},
AUTHOR = {Shiryaev, A. N.},
TITLE = {Probability},
SERIES = {Grad. Texts in Math.},
VOLUME = {95},
EDITION = {Second},
NOTE = {Translated from the first (1980) Russian edition by R. P. Boas},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1996},
PAGES = {xvi+623},
ISBN = {0-387-94549-0},
MRCLASS = {60-01},
MRNUMBER = {97c:60003},
ZBLNUMBER = {0870.00038},
} -
[siegel] C. L. Siegel, Lectures on the Geometry of Numbers, New York: Springer-Verlag, 1989.
@book {siegel, MRKEY = {1020761},
AUTHOR = {Siegel, Carl Ludwig},
TITLE = {Lectures on the Geometry of Numbers},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1989},
PAGES = {x+160},
ISBN = {3-540-50629-2},
MRCLASS = {11Hxx (01A75)},
MRNUMBER = {91d:11070},
MRREVIEWER = {Rudolf Scharlau},
ZBLNUMBER = {0691.10021},
} -
[Spohn78]
H. Spohn, "The Lorentz process converges to a random flight process," Comm. Math. Phys., vol. 60, iss. 3, pp. 277-290, 1978.
@article {Spohn78, MRKEY = {0496299},
AUTHOR = {Spohn, Herbert},
TITLE = {The {L}orentz process converges to a random flight process},
JOURNAL = {Comm. Math. Phys.},
FJOURNAL = {Communications in Mathematical Physics},
VOLUME = {60},
YEAR = {1978},
NUMBER = {3},
PAGES = {277--290},
ISSN = {0010-3616},
MRCLASS = {82.60},
MRNUMBER = {58 \#14861},
MRREVIEWER = {Burton H. Voorhees},
URL = {http://projecteuclid.org/getRecord?id=euclid.cmp/1103904131},
ZBLNUMBER = {0381.60099},
} -
[SV]
A. Strömbergsson and A. Venkatesh, "Small solutions to linear congruences and Hecke equidistribution," Acta Arith., vol. 118, iss. 1, pp. 41-78, 2005.
@article {SV, MRKEY = {2136220},
AUTHOR = {Str{ö}mbergsson, Andreas and Venkatesh, Akshay},
TITLE = {Small solutions to linear congruences and {H}ecke equidistribution},
JOURNAL = {Acta Arith.},
FJOURNAL = {Acta Arithmetica},
VOLUME = {118},
YEAR = {2005},
NUMBER = {1},
PAGES = {41--78},
ISSN = {0065-1036},
CODEN = {AARIA9},
MRCLASS = {11P21 (11F72)},
MRNUMBER = {2006g:11204},
MRREVIEWER = {A. Raghuram},
DOI = {10.4064/aa118-1-4},
ZBLNUMBER = {1082.11061},
}