Advertisement

Journal of Statistical Physics

, Volume 150, Issue 6, pp 1183–1200 | Cite as

The Analyticity Breakdown for Frenkel-Kontorova Models in Quasi-periodic Media: Numerical Explorations

  • Timothy BlassEmail author
  • Rafael de la Llave
Article

Abstract

We study numerically the “analyticity breakdown” transition in 1-dimensional quasi-periodic media. This transition corresponds physically to the transition between pinned down and sliding ground states. Mathematically, it corresponds to the solutions of a functional equation losing their analyticity properties.

We implemented some recent numerical algorithms that are efficient and backed up by rigorous results so that we can compute with confidence even close to the breakdown.

We have uncovered several phenomena that we believe deserve a theoretical explanation: (A) The transition happens in a smooth surface. (B) There are scaling relations near breakdown. (C) The scaling near breakdown is very anisotropic. Derivatives in different directions blow up at different rates.

Similar phenomena seem to happen in other KAM problems.

Keywords

Quasi-periodic solutions Quasi-crystals Hull functions KAM theory 

Notes

Acknowledgements

Part of this work was done while the authors were affiliated with Univ. of Texas. We thank S. Hernandez and X. Su for many discussions about this problem and about numerical issues. We also thank the Center for Nonlinear Analysis (NSF Grants No. DMS-0405343 and DMS-0635983), where part of this research was carried out.

References

  1. 1.
    Aubry, S., André, G.: Analyticity breaking and Anderson localization in incommensurate lattices. In: Group Theoretical Methods in Physics (Proc. Eighth Int. Colloq., Kiryat Anavim, 1979). Annals of the Israel Physical Society, vol. 3, pp. 133–164. Hilger, Bristol (1980) Google Scholar
  2. 2.
    Aubry, S., Le Daeron, P.Y.: The discrete Frenkel-Kontorova model and its extensions. I. Exact results for the ground-states. Physica D 8(3), 381–422 (1983) MathSciNetADSzbMATHCrossRefGoogle Scholar
  3. 3.
    Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Universitext. Springer, New York (2011) zbMATHGoogle Scholar
  4. 4.
    Calleja, R., Celletti, A.: Breakdown of invariant attractors for the dissipative standard map. Chaos 20(1), 013121 (2010) MathSciNetADSCrossRefGoogle Scholar
  5. 5.
    Calleja, R., de la Llave, R.: Fast numerical computation of quasi-periodic equilibrium states in 1D statistical mechanics, including twist maps. Nonlinearity 22(6), 1311–1336 (2009) MathSciNetADSzbMATHCrossRefGoogle Scholar
  6. 6.
    Calleja, R., de la Llave, R.: Computation of the breakdown of analyticity in statistical mechanics models: numerical results and a renormalization group explanation. J. Stat. Phys. 141(6), 940–951 (2010) MathSciNetADSzbMATHCrossRefGoogle Scholar
  7. 7.
    Calleja, R., de la Llave, R.: A numerically accessible criterion for the breakdown of quasi-periodic solutions and its rigorous justification. Nonlinearity 23(9), 2029–2058 (2010) MathSciNetADSzbMATHCrossRefGoogle Scholar
  8. 8.
    Calleja, R., Figueras, J.-L.: Collision of invariant bundles of quasi-periodic attractors in the dissipative standard map. Chaos 23, 02123 (2012) Google Scholar
  9. 9.
    Celletti, A., Falcolini, C., Locatelli, U.: On the break-down threshold of invariant tori in four dimensional maps. Regul. Chaotic Dyn. 9(3), 227–253 (2004) MathSciNetADSzbMATHCrossRefGoogle Scholar
  10. 10.
    de la Llave, R.: KAM theory for equilibrium states in 1-D statistical mechanics models. Ann. Henri Poincaré 9(5), 835–880 (2008) ADSzbMATHCrossRefGoogle Scholar
  11. 11.
    Fox, A., Meiss, J.D.: Critical asymmetric tori in the multiharmonic standard map (2012) Google Scholar
  12. 12.
    Frenkel, J., Kontorova, T.: On the theory of plastic deformation and twinning. Acad. Sci. URSS, J. Phys. 1, 137–149 (1939) MathSciNetzbMATHGoogle Scholar
  13. 13.
    Gambaudo, J.-M., Guiraud, P., Petite, S.: Minimal configurations for the Frenkel-Kontorova model on a quasicrystal. Commun. Math. Phys. 265(1), 165–188 (2006) MathSciNetADSzbMATHCrossRefGoogle Scholar
  14. 14.
    Gentile, G., van Erp, T.S.: Breakdown of Lindstedt expansion for chaotic maps. J. Math. Phys. 46(10), 102702 (2005) MathSciNetADSCrossRefGoogle Scholar
  15. 15.
    Greene, J.M.: A method for determining a stochastic transition. J. Math. Phys. 20, 1183–1201 (1979) ADSCrossRefGoogle Scholar
  16. 16.
    Haro, A., Simó, C.: A numerical study of the breakdown of invariant tori in 4D symplectic maps. In: XIV CEDYA/IV Congress of Applied Mathematics (Vic, 1995), 9 pp. (electronic, in Spanish). Univ. Barcelona, Barcelona (1996) Google Scholar
  17. 17.
    Herman, M.-R.: Sur les courbes invariantes par les difféomorphismes de l’anneau, vol. 1. Astérisque, vol. 103. Société Mathématique de France, Paris (1983) (with an appendix by Albert Fathi, with an English summary) zbMATHGoogle Scholar
  18. 18.
    Israel, R.B.: Convexity in the Theory of Lattice Gases. Princeton Series in Physics. Princeton University Press, Princeton (1979) (with an introduction by Arthur S. Wightman) zbMATHGoogle Scholar
  19. 19.
    Katok, A., Hasselblatt, B.: Introduction to the Modern Theory of Dynamical Systems. Encyclopedia of Mathematics and Its Applications, vol. 54. Cambridge University Press, Cambridge (1995) (with a supplementary chapter by Katok and Leonardo Mendoza) zbMATHCrossRefGoogle Scholar
  20. 20.
    Koch, H.: A renormalization group fixed point associated with the breakup of golden invariant tori. Discrete Contin. Dyn. Syst. 11(4), 881–909 (2004) MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Lions, P.-L., Souganidis, P.E.: Correctors for the homogenization of Hamilton-Jacobi equations in the stationary ergodic setting. Commun. Pure Appl. Math. 56(10), 1501–1524 (2003) MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    MacKay, R.S.: Renormalisation in Area-Preserving Maps. Advanced Series in Nonlinear Dynamics, vol. 6. World Scientific, River Edge (1993) zbMATHGoogle Scholar
  23. 23.
    Mather, J.N.: Existence of quasiperiodic orbits for twist homeomorphisms of the annulus. Topology 21(4), 457–467 (1982) MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Mattis, D.C. (ed.): The Many-Body Problem. An Encyclopedia of Exactly Solved Models in One Dimension, 3rd edn. (with revisions and corrections). World Scientific, Hackensack (2009) zbMATHGoogle Scholar
  25. 25.
    Mestel, B.D., Osbaldestin, A.H.: Periodic orbits of renormalisation for the correlations of strange nonchaotic attractors. Math. Phys. Electron. J. 6, 5 (2000) MathSciNetGoogle Scholar
  26. 26.
    Moser, J.: A rapidly convergent iteration method and non-linear partial differential equations. I. Ann. Sc. Norm. Super. Pisa (3) 20, 265–315 (1966) zbMATHGoogle Scholar
  27. 27.
    Moser, J.: Monotone twist mappings and the calculus of variations. Ergod. Theory Dyn. Syst. 6(3), 401–413 (1986) zbMATHCrossRefGoogle Scholar
  28. 28.
    Percival, I.C.: A variational principle for invariant tori of fixed frequency. J. Phys. A 12(3), L57–L60 (1979) MathSciNetADSzbMATHCrossRefGoogle Scholar
  29. 29.
    Ruelle, D.: Statistical Mechanics: Rigorous Results. Benjamin, New York (1969) zbMATHGoogle Scholar
  30. 30.
    Rüssmann, H.: On optimal estimates for the solutions of linear partial differential equations of first order with constant coefficients on the torus. In: Dynamical Systems, Theory and Applications (Rencontres, Battelle Res. Inst., Seattle, 1974). Lecture Notes in Physics, vol. 38, pp. 598–624. Springer, Berlin (1975) CrossRefGoogle Scholar
  31. 31.
    Rüssmann, H.: Note on sums containing small divisors. Commun. Pure Appl. Math. 29(6), 755–758 (1976) CrossRefGoogle Scholar
  32. 32.
    Su, X., de la Llave, R.: KAM theory for quasi-periodic equilibria 1-D quasiperiodic media. SIAM J. Math. Anal. 14, 3901–3927 (2012) CrossRefGoogle Scholar
  33. 33.
    Su, X., de la Llave, R.: KAM theory for quasi-periodic equilibria in one dimensional quasiperiodic media II: extended range interactions. J. Phys. A 45, 45203 (2012) CrossRefGoogle Scholar
  34. 34.
    Tompaidis, S.: Numerical study of invariant sets of a quasiperiodic perturbation of a symplectic map. Exp. Math. 5(3), 211–230 (1996) MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    van Erp, T.S.: Frenkel-Kontorova model on quasiperiodic substrate potentials. Thesis (1999) Google Scholar
  36. 36.
    van Erp, T.S., Fasolino, A.: Aubry transition studied by direct evaluation of the modulation functions of infinite incommensurate systems. Europhys. Lett. 59(3), 330–336 (2002) ADSCrossRefGoogle Scholar
  37. 37.
    van Erp, T.S., Fasolino, A., Radulescu, O., Janssen, T.: Pinning and phonon localization in Frenkel-Kontorova models on quasiperiodic substrates. Phys. Rev. B 60, 6522–6528 (1999) ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Center for Nonlinear Analysis, Department of Mathematical SciencesCarnegie Mellon UniversityPittsburghUSA
  2. 2.School of MathematicsGeorgia Institute of TechnologyAtlantaUSA

Personalised recommendations