Communications in Mathematical Physics

, Volume 331, Issue 1, pp 111–137 | Cite as

Tail Asymptotics of Free Path Lengths for the Periodic Lorentz Process: On Dettmann’s Geometric Conjectures

Article

Abstract

In the simplest case, consider a \({\mathbb{Z}^d}\)-periodic (d ≥ 3) arrangement of balls of radii < 1/2, and select a random direction and point (outside the balls). According to Dettmann’s first conjecture, the probability that the so determined free flight (until the first hitting of a ball) is larger than t >  > 1 is \({\sim\frac{C}{t}}\), where C is explicitly given by the geometry of the model. In its simplest form, Dettmann’s second conjecture is related to the previous case with tangent balls (of radii 1/2). The conjectures are established in a more general setup: for \({\mathcal{L}}\)-periodic configuration of—possibly intersecting—convex bodies with \({\mathcal{L}}\) being a non-degenerate lattice. These questions are related to Pólya’s visibility problem (Arch Math Phys Ser 2:135–142, 1918), to theories of Bourgain et al. (Commun Math Phys 190:491–508,1998), and of Marklof–Strömbergsson (Ann Math 172:1949–2033,2010). The results also provide the asymptotic covariance of the periodic Lorentz process assuming it has a limit in the super-diffusive scaling, a fact if d = 2 and the horizon is infinite.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. B92.
    Bleher P.M.: Statistical properties of two-dimensional periodic Lorentz gas with infinite horizon. J. Stat. Phys. 66(1), 315–373 (1992)ADSCrossRefMATHMathSciNetGoogle Scholar
  2. BS81.
    Bunimovich L.A., Sinai Ya.G.: Statistical properties of Lorentz gas with periodic configuration of scatterers. Commun. Math. Phys. 78, 479–497 (1981)ADSCrossRefMATHMathSciNetGoogle Scholar
  3. BSCh91.
    Bunimovich L.A., Sinai Ya.G., Chernov N.I.: Statistical properties of two dimensional dispersing billiards. Russian Math. Surv. 46, 47–106 (1991)ADSCrossRefMathSciNetGoogle Scholar
  4. BGW98.
    Bourgain J., Golse F., Wennberg B.: On the distribution of free path lengths for the periodic Lorentz gas. Commun. Math. Phys. 190, 491–508 (1998)ADSCrossRefMATHMathSciNetGoogle Scholar
  5. BT08.
    Bálint P., Tóth I.P.: Exponential decay of correlations in multi-dimensional dispersing billiards. Annales Henri Poincaré 9, 1309–1369 (2008)ADSCrossRefMATHGoogle Scholar
  6. ChD09.
    Chernov N., Dolgopyat D.: Anomalous current in periodic Lorentz gases with infinite horizon. Russian Math. Surv. 64, 73–124 (2009)MathSciNetGoogle Scholar
  7. D12.
    Dettmann C.P.: New horizons in multidimensional diffusion: The Lorentz gas and the Riemann Hypothesis. J. Stat. Phys. 146, 181–204 (2012)ADSCrossRefMATHMathSciNetGoogle Scholar
  8. GW00.
    Golse F., Wennberg B.: On the distribution of free path lengths for the periodic Lorentz gas II. ESAIM M2AN 34, 1151–1163 (2000)CrossRefMATHMathSciNetGoogle Scholar
  9. KS12.
    Kraemer, A.S., Sanders, D.P.: Periodizing quasi-crystals: Anomalous diffusion in quasi-periodic systems. http://arxiv.org/abs/1206.1103
  10. KSSz89.
    Krámli A., Simányi N., Szász D.: Ergodic properties of semi-dispersing billiards. I. Two cylindric scatterers in the 3-D torus. Nonlinearity 2, 311–326 (1989)ADSCrossRefMATHMathSciNetGoogle Scholar
  11. K08.
    Kruskal C.P.: The orchard visibility problem and some variants. J. Comput. Syst. Sci. 74, 587–597 (2008)CrossRefMATHMathSciNetGoogle Scholar
  12. L05.
    Lorentz H.: Le mouvement des électrons dans les métaux. Arch. Néerl. 10, 336–371 (1905)Google Scholar
  13. M10.
    Marklof, J.: Kinetic transport in crystals. In: Proceedings of the XVI International Congress on Mathematical Physics, Prague 2009, World Scientific, pp. 162–179 (2010)Google Scholar
  14. MS10.
    Marklof J., Strömbergsson A.: The distribution of free path lengths in the periodic Lorentz gas and related lattice point problems. Ann. Math. 172, 1949–2033 (2010)CrossRefMATHGoogle Scholar
  15. MS11.
    Marklof J., Strömbergsson A.: The Boltzmann-Grad limit of the periodic Lorentz gas. Ann. Math. 174, 225–298 (2011)CrossRefMATHGoogle Scholar
  16. MS11a.
    Marklof J., Strömbergsson A.: The periodic Lorentz gas in the Boltzmann-Grad limit: asymptotic estimates. GAFA Geom. Funct. Anal. 21, 560–647 (2011)CrossRefMATHGoogle Scholar
  17. P18.
    Pólya G.: Zahlentheoretisches und wahrscheinlichkeitstheoretisches über die Sichtweite im Walde. Arch. Math. Phys. Ser. 2(27), 135–142 (1918)Google Scholar
  18. S05.
    Sanders, D.P.: Deterministic Diffusion in Periodic Billiard Models, Thesis, U. of Warwick, pp. 204. (2005).arXiv:0808.2252[cond-mat.stat-mech]
  19. S08.
    Sanders D.P.: Normal diffusion in crystal structures and higher-dimensional billiard models with gaps. Phys. Rev. E 78, 060101 (2008)ADSCrossRefGoogle Scholar
  20. Sch68.
    Schmidt W.: Asymptotic formulae for point lattices of bounded determinant and subspaces of bounded height. Duke Math. J. 35, 327–339 (1968)CrossRefMATHMathSciNetGoogle Scholar
  21. SSz00.
    Simányi N., Szász D.: Non-integrability of cylindric billiards and transitive Lie-group actions. Ergod. Theory Dynam. Syst. 20, 593–610 (2000)CrossRefMATHGoogle Scholar
  22. Sz94.
    Szász, D.: The K-property of ‘orthogonal’ cylindric billiards. Commun. Math. Phys. 160, 581–597 (1994)Google Scholar
  23. Sz08.
    Szász D.: Some challenges in the theory of (semi)-dispersing billiards. Nonlinearity 21, 187–193 (2008)ADSCrossRefMathSciNetGoogle Scholar
  24. SzV07.
    Szász D., Varjú T.: Limit laws and recurrence for the planar Lorentz process with infinite horizon. J. Stat. Phys. 129, 59–80 (2007)ADSCrossRefMATHMathSciNetGoogle Scholar
  25. W12.
    Wennberg B.: Free path lengths in quasi crystals. J. Stat. Phys. 147(5), 981–990 (2012)ADSCrossRefMATHMathSciNetGoogle Scholar
  26. Y98.
    Young L.S.: Statistical properties of dynamical systems with some hyperbolicity. Ann. Math. 147, 585–650 (1998)CrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Péter Nándori
    • 1
    • 2
  • Domokos Szász
    • 1
  • Tamás Varjú
    • 1
  1. 1.Institute of MathematicsBudapest University of Technology and EconomicsBudapestHungary
  2. 2.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA

Personalised recommendations