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Equilibrium Configurations for Generalized Frenkel–Kontorova Models on Quasicrystals

Abstract

I study classes of generalized Frenkel–Kontorova models whose potentials are given by almost-periodic functions which are closely related to aperiodic Delone sets of finite local complexity. Since such Delone sets serve as models for quasicrystals, this setup presents Frenkel–Kontorova models for the type of aperiodic crystals which have been discovered since Shechtman’s discovery of quasicrystals. Here I consider models with configurations \(u:{\mathbb {Z}}^r \rightarrow {\mathbb {R}}^d\), where d is the dimension of the quasicrystal, for any r and d. The almost-periodic functions used for potentials are called pattern-equivariant and I show that if the interactions of the model satisfies a mild \(C^2\) requirement, and if the potential satisfies a mild non-degeneracy assumption, then there exist equilibrium configurations of any prescribed rotation rotation number/vector/plane. The assumptions are general enough to satisfy the classical Frenkel–Kontorova models and its multidimensional analoges. The proof uses the idea of the anti-integrable limit.

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References

  1. Aubry, S., Abramovici, G.: Chaotic trajectories in the standard map. The concept of anti-integrability. Physica D 43(2–3), 199–219 (1990)

    ADS  MathSciNet  Article  Google Scholar 

  2. Aliste-Prieto, J., Coronel, D., Cortez, M.I., Durand, F., Petite, S.: Linearly Repetitive Delone Sets, Mathematics of Aperiodic Order. Progress in Mathematics, vol. 309, pp. 195–222. Birkhäuser, Basel (2015)

    Book  Google Scholar 

  3. Baake, M., Grimm, U.: Aperiodic Order. Encyclopedia of Mathematics and Its Applications. A Mathematical Invitation, with a Foreword by Roger Penrose, vol. 149, vol. 1. Cambridge University Press, Cambridge (2013)

    Book  Google Scholar 

  4. Braun, O.M., Kivshar, Y.S.: The Frenkel–Kontorova Model. Concepts, Methods, and Applications. Texts and Monographs in Physics. Springer, Berlin (2004)

    Book  Google Scholar 

  5. Bolotin, S.V., Treshchëv, D.V.: Anti-integrable limit. Uspekhi Mat. Nauk 70(6), 3–62 (2015)

    MathSciNet  Article  Google Scholar 

  6. Candel, A., de la Llave, R.: On the Aubry–Mather theory in statistical mechanics. Commun. Math. Phys. 192(3), 649–669 (1998)

    ADS  MathSciNet  Article  Google Scholar 

  7. de la Llave, R.: A tutorial on KAM theory, Smooth Ergodic theory and its applications (Seattle, WA, 1999). In: Proceedings of Symposia in Pure Mathematics, vol. 69, pp. 175–292. American Mathematical Society, Providence, RI (2001)

  8. de la Llave, R., Valdinoci, E.: Ground states and critical points for Aubry–Mather theory in statistical mechanics. J. Nonlinear Sci. 20(2), 153–218 (2010)

    ADS  MathSciNet  Article  Google Scholar 

  9. Dworkin, S.: Spectral theory and X-ray diffraction. J. Math. Phys. 34(7), 2965–2967 (1993)

    ADS  MathSciNet  Article  Google Scholar 

  10. Frenkel, J., Kontorova, T.: On the theory of plastic deformation and twinning. Acad. Sci. U.S.S.R. J. Phys. 1, 137–149 (1939)

    MathSciNet  MATH  Google Scholar 

  11. Gambaudo, J.-M., Guiraud, P., Petite, S.: Minimal configurations for the Frenkel–Kontorova model on a quasicrystal. Commu. Math. Phys. 265(1), 165–188 (2006)

    ADS  MathSciNet  Article  Google Scholar 

  12. Golé, C.: Symplectic Twist Maps: Global Variational Techniques. Advanced Series in Nonlinear Dynamics, vol. 18. World Scientific Publishing Co., Inc., River Edge, NJ (2001)

    Book  Google Scholar 

  13. Garibaldi, E., Petite, S., Thieullen, P.: Calibrated configurations for Frenkel–Kontorova type models in almost periodic environments. Ann. Henri Poincaré 18(9), 2905–2943 (2017)

    ADS  MathSciNet  Article  Google Scholar 

  14. Kellendonk, J., Putnam, I.F.: The Ruelle–Sullivan map for actions of \({\mathbb{R}}^n\). Math. Ann. 334(3), 693–711 (2006)

    Google Scholar 

  15. Lagarias, J.C.: Geometric models for quasicrystals I. Delone sets of finite type. Discrete Comput. Geom. 21(2), 161–191 (1999)

    MathSciNet  Article  Google Scholar 

  16. Lifshitz, R.: What is a crystal? Z. Kristallogr 222(6), 313–317 (2007)

    Article  Google Scholar 

  17. Mather, J.N., Forni, G.: Action Minimizing Orbits in Hamiltonian Systems. Transition to Chaos in Classical and Quantum Mechanics (Montecatini Terme, 1991), Lecture Notes in Mathematics, vol. 1589, pp. 92–186. Springer, Berlin (1994)

    MATH  Google Scholar 

  18. Shechtman, D., Blech, I., Gratias, D., Cahn, J.W.: Metallic phase with long-range orientational order and no translational symmetry. Phys. Rev. Lett. 53, 1951–1953 (1984)

    ADS  Article  Google Scholar 

  19. Schmieding, S., Treviño, R.: Self affine delone sets and deviation phenomena. Commun. Math. Phys. 357(3), 1071–1112 (2018)

    ADS  MathSciNet  Article  Google Scholar 

  20. Schmieding, S., Treviño, R.: Random Substitution Tilings and Deviation Phenomena. arXiv e-prints (2019). arXiv:1902.08996

  21. Treviño, R.: Tilings, Traces and Triangles. arXiv e-prints (2019). arXiv:1906.00466

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Acknowledgements

I first learned about the Frenkel–Kontorova model from Rafael de la Llave. He also suggested that an approach using the anti-integrable limit may be fruitful. I am grateful to him for many conversations about the Frenkel–Kontorova, math, and life in general over many years. I am also grateful to two anonymous referees who have helped make the paper and its exposition much better.

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Correspondence to Rodrigo Treviño.

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Dedicated to Rafael de la Llave on the occasion of his 60th birthday

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Treviño, R. Equilibrium Configurations for Generalized Frenkel–Kontorova Models on Quasicrystals. Commun. Math. Phys. 371, 1–17 (2019). https://doi.org/10.1007/s00220-019-03557-7

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  • DOI: https://doi.org/10.1007/s00220-019-03557-7