Abstract
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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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
Blank S., Krikorian N. (1993) Thom’s problem on irrational flows. Internat. J. Math. 4, 721–726
Bleher P. (1992) Statistical properties of two-dimensional periodic Lorentz with infinite horizon. J. Stat. Phys. 66, 315–373
Boca F.P., Cobeli C., Zaharescu A. (2000) Distribution of lattice points visible from the origin. Commun. Math. Phys. 213, 433–470
Boca F.P., Cobeli C., Zaharescu A. (2001) A conjecture of R.R. Hall on Farey points. J. Reine Angew. Math. 535, 207–236
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
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
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
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
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
Caglioti E., Golse F. (2003) On the distribution of free path lengths for the periodic Lorentz gas III. Commun. Math. Phys. 236, 199–221
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
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
Dahlqvist P. (1997) The Lyapunov exponent in the Sinai billiard in the small scatterer limit. Nonlinearity 10, 159–173
Deshouillers J.-M., Iwaniec H. (1982/1983) Kloosterman sums and Fourier coefficients of cusp forms. Invent. Math. 70, 219–288
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
Erdös P. (1959) Some results on diophantine approximation. Acta Arith. 5, 359–369
Erdös P., Szüsz P., Turán P. (1958) Remarks on the theory of diophantine approximation. Colloq. Math. 6, 119–126
Estermann T. (1961) On Kloosterman’s sum. Mathematika 8, 83–86
Friedman B. Niven I. (1959) The average first recurrence time. Trans. Amer. Math. Soc. 92, 25–34
Friedman B., Oono Y., Kubo I. (1984) Universal behaviour of Sinai billiard systems in the small-scatterer limit. Phys. Rev. Lett. 52, 709–712
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
Goldfeld D., Sarnak P. (1983) Sums of Kloosterman sums. Invent. Math. 71, 243–250
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
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
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
Hooley C. (1957) An asymptotic formula in the theory of numbers. Proc. London Math. Soc. 7, 396–413
Kesten H. (1962) Some probabilistic theorems on diophantine approximations. Trans. Amer. Math. Soc. 103, 189–217
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)
Lewin L., (1958) Dilogarithms and associated functions. London, Macdonald & Co. London
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
Pólya G. (1918) Zahlentheoretisches und wahrscheinlichkeitstheoretisches über die sichtweite im walde. Arch. Math. Phys. 27, 135–142
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
Sinai Y.G. (1970) Dynamical systems with elastic reflections. Ergodic properties of dispersing billiards. Russ. Math. Surveys 25, 137–189
Weil A. (1948) On some exponential sums. Proc. Nat. Acad. Sci. USA 34, 204–207
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by P. Sarnak
In memory of Walter Philipp
Rights and permissions
About this article
Cite this article
Boca, F.P., Zaharescu, A. The Distribution of the Free Path Lengths in the Periodic Two-Dimensional Lorentz Gas in the Small-Scatterer Limit. Commun. Math. Phys. 269, 425–471 (2007). https://doi.org/10.1007/s00220-006-0137-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-006-0137-7