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The distribution of the free path lengths in the periodic two-dimensional Lorentz gas in the small-scatterer limit. (English) Zbl 1143.37002

Summary: We study the free path length and the geometric free path length in the model of the periodic two-dimensional Lorentz gas (Sinai billiard). We give a complete and rigorous proof for the existence of their distributions in the small-scatterer limit and explicitly compute them. As a corollary one gets a complete proof for the existence of the constant term \(c=2-3\ln 2+\frac{27\zeta(3)}{2\pi^2}\) in the asymptotic formula \(h(T)=-2 \ln \epsilon +c+o(1)\) of the KS entropy of the billiard map in this model, as conjectured by P. Dahlqvist.

MSC:

37A25 Ergodicity, mixing, rates of mixing
11B57 Farey sequences; the sequences \(1^k, 2^k, \dots\)
37A60 Dynamical aspects of statistical mechanics
37D50 Hyperbolic systems with singularities (billiards, etc.) (MSC2010)
82C05 Classical dynamic and nonequilibrium statistical mechanics (general)
82C40 Kinetic theory of gases in time-dependent statistical mechanics
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[1] Augustin V., Boca F.P., Cobeli C., Zaharescu A. (2001) The h-spacing distribution between Farey points. Math. Proc. Cambridge Phil. Soc. 131, 23–38 · Zbl 1161.11312
[2] Blank S., Krikorian N. (1993) Thom’s problem on irrational flows. Internat. J. Math. 4, 721–726 · Zbl 0813.58048
[3] Bleher P. (1992) Statistical properties of two-dimensional periodic Lorentz with infinite horizon. J. Stat. Phys. 66, 315–373 · Zbl 0925.82147
[4] Boca F.P., Cobeli C., Zaharescu A. (2000) Distribution of lattice points visible from the origin. Commun. Math. Phys. 213, 433–470 · Zbl 0989.11049
[5] Boca F.P., Cobeli C., Zaharescu A. (2001) A conjecture of R.R. Hall on Farey points. J. Reine Angew. Math. 535, 207–236 · Zbl 1006.11053
[6] Boca F.P., Gologan R.N., Zaharescu A. (2003) The average length of a trajectory in a certain billiard in a flat two-torus. New York J. Math. 9, 303–330 · Zbl 1066.37021
[7] Boca F.P., Gologan R.N., Zaharescu A. (2003) The statistics of the trajectory of a billiard in a flat two-torus. Commun. Math. Phys. 240, 53–73 · Zbl 1078.37006
[8] Bouchaud J.-P., Le Doussal P. (1985) Numerical study of a D-dimensional periodic Lorentz gas with universal properties. J. Stat. Phys. 41, 225–248
[9] Bourgain J., Golse F., Wennberg B. (1998) On the distribution of free path lengths for the periodic Lorentz gas. Commun. Math. Phys. 190, 491–508 · Zbl 0910.60082
[10] Bunimovich L.: Billiards and other hyperbolic systems. In: Dynamical systems, ergodic theory and applications, edited by Ya.G. Sinai, Encyclopaedia Math. Sci. Vol. 100, Berlin: Springer-Verlag, 2000, pp. 192–233 · Zbl 1417.37009
[11] Caglioti E., Golse F. (2003) On the distribution of free path lengths for the periodic Lorentz gas III. Commun. Math. Phys. 236, 199–221 · Zbl 1041.82016
[12] Chernov N. (1991) New proof of Sinai’s formula for the entropy of hyperbolic billiard systems. Application to Lorentz gases and Bunimovich stadium. Funct. Anal. and Appl. 25(3): 204–219 · Zbl 0748.58015
[13] Chernov N.: Entropy values and entropy bounds. In: Hard ball systems and the Lorentz gas, edited by D. Szász, Encyclopaedia Math. Sci., Vol. 101, Berlin: Springer-Verlag, 2000, pp. 121–143 · Zbl 0995.37006
[14] Dahlqvist P. (1997) The Lyapunov exponent in the Sinai billiard in the small scatterer limit. Nonlinearity 10, 159–173 · Zbl 0907.58038
[15] Deshouillers J.-M., Iwaniec H. (1982/1983) Kloosterman sums and Fourier coefficients of cusp forms. Invent. Math. 70, 219–288 · Zbl 0502.10021
[16] Dumas H.S., Dumas L., Golse F. (1997) Remarks on the notion of mean free path for a periodic array of spherical obstacles. J. Stat. Phys. 87(3/4): 943–950 · Zbl 0952.82512
[17] Erdös P. (1959) Some results on diophantine approximation. Acta Arith. 5, 359–369 · Zbl 0097.03502
[18] Erdös P., Szüsz P., Turán P. (1958) Remarks on the theory of diophantine approximation. Colloq. Math. 6, 119–126 · Zbl 0087.04305
[19] Estermann T. (1961) On Kloosterman’s sum. Mathematika 8, 83–86 · Zbl 0114.26302
[20] Friedman B. Niven I. (1959) The average first recurrence time. Trans. Amer. Math. Soc. 92, 25–34 · Zbl 0087.42006
[21] Friedman B., Oono Y., Kubo I. (1984) Universal behaviour of Sinai billiard systems in the small-scatterer limit. Phys. Rev. Lett. 52, 709–712
[22] Gallavotti G.: Lectures on the billiard. In: Dynamical systems, theory and applications (Rencontres, Battelle Res. Inst., Seattle, Wash., 1974), edited by J. Moser, Lecture Notes in Phys. Vol. 38, Berlin-Heidelberg-Newyork: Springer-Verlag, 1975, pp. 236–295
[23] Goldfeld D., Sarnak P. (1983) Sums of Kloosterman sums. Invent. Math. 71, 243–250 · Zbl 0507.10029
[24] Golse F. On the statistics of free-path lengths for the periodic Lorentz gas. In: XIV International Congress on Mathematical Physics (Lisbon, 2003), edited by J.-C. Zambrini, River Edge, NJ: World Sci. Publ., 2006, pp. 439–446 · Zbl 1221.82081
[25] Golse F., Wennberg B. (2000) On the distribution of free path lengths for the periodic Lorentz gas I. M2AN Math. Model. Numer. Anal. 34, 1151–1163 · Zbl 1006.82025
[26] Gutzwiller M.: Physics and arithmetic chaos in the Fourier transform. In: The mathematical beauty of physics (Saclay, 1996). edited by J.M. Drouffe J.B. Zuber, Adv. Series in Math. Phys. Vol. 24, River Edge, NJ: World Sci. Publ., 1997, pp. 258–280 · Zbl 1058.81543
[27] Hooley C. (1957) An asymptotic formula in the theory of numbers. Proc. London Math. Soc. 7, 396–413 · Zbl 0079.27301
[28] Kesten H. (1962) Some probabilistic theorems on diophantine approximations. Trans. Amer. Math. Soc. 103, 189–217 · Zbl 0105.03805
[29] Kuznetsov N.V. The Petterson conjecture for forms of weight zero and Linnik’s conjecture. Math. Sb. (N.S.) 111(153): 334–383, 479 (1980) · Zbl 0427.10016
[30] Lewin L., (1958) Dilogarithms and associated functions. London, Macdonald & Co. London · Zbl 0083.35904
[31] Lorentz H.A.: Le mouvement des électrons dans les métaux. Arch. Néerl. 10, 336 (1905). Reprinted in Collected papers. Vol. 3. The Hague: Martinus Nijhoff, 1936
[32] Pólya G. (1918) Zahlentheoretisches und wahrscheinlichkeitstheoretisches über die sichtweite im walde. Arch. Math. Phys. 27, 135–142 · JFM 46.0284.01
[33] Santaló L.A. (1943) Sobre la distribucion probable de corpusculos en un cuerpo. Deducida de la distribucion en sus secciones y problemas analogos. Rev. Un. Mat. Argentina 9, 145–164
[34] Sinai Y.G. (1970) Dynamical systems with elastic reflections. Ergodic properties of dispersing billiards. Russ. Math. Surveys 25, 137–189 · Zbl 0252.58005
[35] Weil A. (1948) On some exponential sums. Proc. Nat. Acad. Sci. USA 34, 204–207 · Zbl 0032.26102
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