Advertisement

Communications in Mathematical Physics

, Volume 330, Issue 2, pp 723–755 | Cite as

Free Path Lengths in Quasicrystals

  • Jens MarklofEmail author
  • Andreas Strömbergsson
Article

Abstract

Previous studies of kinetic transport in the Lorentz gas have been limited to cases where the scatterers are distributed at random (e.g., at the points of a spatial Poisson process) or at the vertices of a Euclidean lattice. In the present paper we investigate quasicrystalline scatterer configurations, which are non-periodic, yet strongly correlated. A famous example is the vertex set of a Penrose tiling. Our main result proves the existence of a limit distribution for the free path length, which answers a question of Wennberg. The limit distribution is characterised by a certain random variable on the space of higher dimensional lattices, and is distinctly different from the exponential distribution observed for random scatterer configurations. The key ingredients in the proofs are equidistribution theorems on homogeneous spaces, which follow from Ratner’s measure classification.

Keywords

Homogeneous Space Borel Probability Measure Free Path Length Lebesgue Measure Zero Penrose Tiling 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Boca F.P., Gologan R.N., Zaharescu A.: The statistics of the trajectory of a certain billiard in a flat two-torus. Commun. Math. Phys. 240, 53–73 (2003)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    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)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Boca F.P., Gologan R.N.: On the distribution of the free path length of the linear flow in a honeycomb. Ann. Inst. Fourier (Grenoble) 59, 1043–1075 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Boca F.P.: Distribution of the linear flow length in a honeycomb in the small-scatterer limit. New York J. Math. 16, 651–735 (2010)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Boldrighini C., Bunimovich L.A., Sinai Y.G.: On the Boltzmann equation for the Lorentz gas. J. Stat. Phys. 32, 477–501 (1983)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Borel A., Borel A.: Arithmetic subgroups of algebraic groups. Ann. Math. 75, 485–535 (1962)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    de Bruijn, N.G.: Algebraic theory of Penrose’s non-periodic tilings of the plane, Koninklijke Nederlandse Akademie van Wetenschappen. Indagationes Math. 43, 39–52, 53–66 (1981)Google Scholar
  8. 8.
    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)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Caglioti E., Golse F.: On the distribution of free path lengths for the periodic Lorentz gas. III. Commun. Math. Phys. 236, 199–221 (2003)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Caglioti E., Golse F.: On the Boltzmann-Grad limit for the two dimensional periodic Lorentz gas. J. Stat. Phys. 141, 264–317 (2010)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Dahlqvist P.: The Lyapunov exponent in the Sinai billiard in the small scatterer limit. Nonlinearity 10, 159–173 (1997)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Dettmann C.P.: New horizons in multidimensional diffusion: the Lorentz gas and the Riemann hypothesis. J. Stat. Phys. 146, 181–204 (2012)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Gallavotti G.: Divergences and approach to equilibrium in the Lorentz and the Wind-tree-models. Phys. Rev. 185, 308–322 (1969)ADSCrossRefGoogle Scholar
  14. 14.
    Golse F., Wennberg B.: On the distribution of free path lengths for the periodic Lorentz gas. II. M2AN Math. Model. Numer. Anal. 34(6), 1151–1163 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Hof A.: Uniform distribution and the projection method, in Quasicrystals and discrete geometry (Toronto, ON, 1995). Fields Inst. Monogr. 10, 201–206 (1998)MathSciNetGoogle Scholar
  16. 16.
    Kraemer, A.S., Sanders, D.P.: Embedding Quasicrystals in a periodic cell: dynamics in quasiperiodic structures. Phys. Rev. Lett. 111, 125501 (2013)Google Scholar
  17. 17.
    Lang S.: Algebraic Number Theory. Springer, New York (1994)CrossRefzbMATHGoogle Scholar
  18. 18.
    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)CrossRefzbMATHGoogle Scholar
  19. 19.
    Marklof J., Strömbergsson A.: The Boltzmann-Grad limit of the periodic Lorentz gas. Ann. Math. 174, 225–298 (2011)CrossRefzbMATHGoogle Scholar
  20. 20.
    Marklof J., Strömbergsson A.: Kinetic transport in the two-dimensional periodic Lorentz gas. Nonlinearity 21, 1413–1422 (2008)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Marklof J., Strömbergsson A.: The periodic Lorentz gas in the Boltzmann-Grad limit: asymptotic estimates. GAFA 21, 560–647 (2011)zbMATHGoogle Scholar
  22. 22.
    Marklof, J., Strömbergsson, A.: Power-law distributions for the free path length in Lorentz gases. J. Stat. Phys. (2014). doi: 10.1007/s10955-014-0935-9
  23. 23.
    Meyer Y.: Algebraic Numbers and Harmonic Analysis. North-Holland Publishing Co., Amsterdam (1972)zbMATHGoogle Scholar
  24. 24.
    Meyer, Y.: Quasicrystals, Diophantine approximation and algebraic numbers. In: Beyond quasicrystals (Les Houches, 1994), Springer, Berlin, pp. 3–16 (1995)Google Scholar
  25. 25.
    Nandori, P., Szasz, D., Varju, T.: Tail asymptotics of free path lengths for the periodic Lorentz process. On Dettmann’s geometric conjectures, arXiv:1210.2231Google Scholar
  26. 26.
    Pleasants, P.A.B.: Lines and planes in 2- and 3-dimensional quasicrystals, in Coverings of discrete quasiperiodic sets. Tracts Modern Phys., 180, Springer, Berlin, pp. 185–225 (2003)Google Scholar
  27. 27.
    Polya G.: Zahlentheoretisches und Wahrscheinlichkeitstheoretisches über die Sichtweite im Walde. Arch. Math. Phys. 27, 135–142 (1918)zbMATHGoogle Scholar
  28. 28.
    Raghunathan M.S.: Discrete Subgroups of Lie Groups. Springer, New York (1972)CrossRefzbMATHGoogle Scholar
  29. 29.
    Ratner M.: On Raghunathan’s measure conjecture. Ann. Math. 134, 545–607 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    Ratner M.: Raghunathan’s topological conjecture and distributions of unipotent flows. Duke Math. J. 63, 235–280 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    Senechal M.: Quasicrystals and Geometry. Cambridge University Press, Cambridge (1995)zbMATHGoogle Scholar
  32. 32.
    Schlottmann, M.: Cut-and-project sets in locally compact abelian groups. In: Quasicrystals and discrete geometry (Toronto, ON, 1995), Fields Inst. Monogr. 10, 247–264 (1998)Google Scholar
  33. 33.
    Shah N.A.: Limit distributions of expanding translates of certain orbits on homogeneous spaces. Proc. Indian Acad. Sci. Math. Sci. 106(2), 105–125 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  34. 34.
    Siegel C.L.: A mean value theorem in geometry of numbers. Ann. Math. 46, 340–347 (1945)CrossRefzbMATHGoogle Scholar
  35. 35.
    Siegel C.L.: Lectures on the Geometry of Numbers. Springer, Berlin, Heidelberg, New York (1989)CrossRefzbMATHGoogle Scholar
  36. 36.
    Spohn H.: The Lorentz process converges to a random flight process. Commun. Math. Phys. 60, 277–290 (1978)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  37. 37.
    Veech W.A.: Siegel measures. Ann. Math. 148, 895–944 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  38. 38.
    van der Vaart A.W., Wellner J.A.: Weak Convergence and Empirical Processes. Springer, New York (1996)CrossRefzbMATHGoogle Scholar
  39. 39.
    Weil A.: Basic Number Theory, 3rd edn. Springer, New York (1974)CrossRefzbMATHGoogle Scholar
  40. 40.
    Wennberg B.: Free path lengths in quasi crystals. J. Stat. Phys. 147, 981–990 (2012)ADSCrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.School of MathematicsUniversity of BristolBristolUK
  2. 2.Department of MathematicsUppsala UniversityUppsalaSweden

Personalised recommendations