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Izv. RAN. Ser. Mat., 2009, Volume 73, Issue 4, Pages 17–36 (Mi izv2731)  

This article is cited in 13 scientific papers (total in 13 papers)

The statistics of particle trajectories in the inhomogeneous Sinai problem for a two-dimensional lattice

V. A. Bykovskii, A. V. Ustinov*

Institute for Applied Mathematics, Khabarovsk Division, Far-Eastern Branch of the Russian Academy of Sciences

Abstract: In connection with the two-dimensional model known as the ‘periodic Lorentz gas’, we study the asymptotic behaviour of statistical characteristics of a free path interval of a point particle before its first occurrence in an $h$-neighbourhood (a circle of radius $h$) of a non-zero integer point as $h\to 0$ given that the particle starts from the $h$-neighbourhood of the origin. We evaluate the limit distribution function of the free path length and of the input aimed parameter (the distance from the trajectory to the integer point we are interested in) for a given value of the output aimed parameter. This problem was studied earlier for a particle starting from the origin (the homogeneous case).

Keywords: analytic number theory, dynamical systems, continued fractions, Kloosterman sums, billiards, geometry of numbers.
* Author to whom correspondence should be addressed

DOI: https://doi.org/10.4213/im2731

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English version:
Izvestiya: Mathematics, 2009, 73:4, 669–688

Bibliographic databases:

UDC: 511.33+519.21
MSC: Primary 82B20; Secondary 37D50, 37N20
Received: 04.10.2007
Revised: 21.01.2008

Citation: V. A. Bykovskii, A. V. Ustinov, “The statistics of particle trajectories in the inhomogeneous Sinai problem for a two-dimensional lattice”, Izv. RAN. Ser. Mat., 73:4 (2009), 17–36; Izv. Math., 73:4 (2009), 669–688

Citation in format AMSBIB
\by V.~A.~Bykovskii, A.~V.~Ustinov
\paper The statistics of particle trajectories in the inhomogeneous Sinai problem for a~two-dimensional lattice
\jour Izv. RAN. Ser. Mat.
\yr 2009
\vol 73
\issue 4
\pages 17--36
\jour Izv. Math.
\yr 2009
\vol 73
\issue 4
\pages 669--688

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    1. A. V. Ustinov, “O raspredelenii tochek tselochislennoi reshetki”, Dalnevost. matem. zhurn., 9:1-2 (2009), 176–181  mathnet  elib
    2. Caglioti E., Golse F., “On the Boltzmann-Grad limit for the two dimensional periodic Lorentz gas”, J. Statist. Phys., 141:2 (2010), 264–317  crossref  mathscinet  zmath  adsnasa  isi  elib
    3. Marklof J., “Kinetic transport in crystals”, XVIth International Congress on Mathematical Physics, World Sci. Publ., Hackensack, NJ, 2010, 162–179  crossref  mathscinet  zmath  isi
    4. Boca F.P., “Distribution of the linear flow length in a honeycomb in the small-scatterer limit”, New York J. Math., 16 (2010), 651–735  mathscinet  zmath  isi  elib
    5. A. V. Ustinov, “O statistikakh Gaussa — Kuzmina v korotkikh intervalakh”, Dalnevost. matem. zhurn., 11:1 (2011), 93–98  mathnet
    6. Marklof J., Strömbergsson A., “The periodic Lorentz gas in the Boltzmann-Grad limit: asymptotic estimates”, Geom. Funct. Anal., 21:3 (2011), 560–647  crossref  mathscinet  zmath  isi
    7. Shparlinski I.E., “Modular hyperbolas”, Jap. J. Math., 7:2 (2012), 235–294  crossref  mathscinet  zmath  isi  elib
    8. A. V. Ustinov, “Spin chains and Arnold's problem on the Gauss-Kuz'min statistics for quadratic irrationals”, Sb. Math., 204:5 (2013), 762–779  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    9. Dettmann C.P., “Diffusion in the Lorentz Gas”, Commun. Theor. Phys., 62:4 (2014), 521–540  crossref  mathscinet  zmath  isi
    10. Marklof J., Strombergsson A., “Power-Law Distributions For the Free Path Length in Lorentz Gases”, J. Stat. Phys., 155:6 (2014), 1072–1086  crossref  mathscinet  zmath  isi
    11. A. V. Ustinov, “Three-dimensional continued fractions and Kloosterman sums”, Russian Math. Surveys, 70:3 (2015), 483–556  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    12. Marklof J., Toth B., “Superdiffusion in the Periodic Lorentz Gas”, Commun. Math. Phys., 347:3 (2016), 933–981  crossref  mathscinet  zmath  isi  elib  scopus
    13. M. A. Korolev, A. V. Ustinov, “Distribution of rational points on the circle of unit radius”, Izv. Math., 83:5 (2019), 1008–1049  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
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