Journal of Statistical Physics

, Volume 66, Issue 3–4, pp 1147–1153 | Cite as

How should one define a (weak) crystal?

  • A. C. D. van Enter
  • Jacek Miekisz
Short Communications


We compare two proposals for the study of positional long-range order: one in terms of the spectrum of the translation operator, the other in terms of the Fourier spectrum. We point out that only the first one allows for the consideration of molecular, as opposed to atomic, (weakly) periodic structures. We illustrate this point on the Thue-Morse system.

Key words

Positional long-range order Fourier spectrum spectrum of the shift 


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  1. 1.
    C. Radin, Low temperature and the origin of crystalline symmetry,Int. J. Mod. Phys. B 1:1157 (1987).Google Scholar
  2. 2.
    C. Radin, Global order from local sources,Bull. Am. Math. Soc., to appear.Google Scholar
  3. 3.
    D. Shechtman, I. Blech, D. Gratias, and J. W. Cahn, Metallic phase with long-range orientational order and no translation symmetry,Phys. Rev. Lett. 53:1951 (1984).Google Scholar
  4. 4.
    D. Levine and P. J. Steinhardt, Quasicrystals: A new class of ordered structures,Phys. Rev. Lett. 53:2477 (1984).Google Scholar
  5. 5.
    D. Ruelle, Do turbulent crystals exist?,Physica 113A:619 (1982).Google Scholar
  6. 6.
    S. Aubry, Devil's staircase and order without periodicity in classical condensed matter,J. Phys. (Paris)44:147 (1983).Google Scholar
  7. 7.
    D. Ruelle,Statistical Mechanics; Rigorous Results (Benjamin, Reading, Massachusetts, 1969), esp. Chapter 6.Google Scholar
  8. 8.
    D. Ruelle, States of physical systems,Commun. Math. Phys. 3:133 (1966).Google Scholar
  9. 9.
    D. Ruelle, Integral representation of states on aC * algebra,J. Funct. Anal. 6:116 (1970).Google Scholar
  10. 10.
    D. Kastler and D. W. Robinson, Invariant states in statistical mechanics,Commun. Math. Phys. 3:151 (1966).Google Scholar
  11. 11.
    G. G. Emch, TheC *-algebra to phase transitions, inPhase Transitions and Critical Phenomena, Vol. 1, C. Domb and M. L. Green, eds. (Academic Press, New York, 1972).Google Scholar
  12. 12.
    G. G. Emch, H. J. F. Knops, and E. J. Verboven, Breaking of Euclidean symmetry with an application to the theory of crystallization,J. Math. Phys. 11:1165 (1970).Google Scholar
  13. 13.
    O. Bratteli and D. W. Robinson,Operator Algebras and Quantum Statistical Mechanics, Vols. 1 and 2 (Springer, Berlin, 1979/1981).Google Scholar
  14. 14.
    C. Radin, Correlations in classical ground states,J. Stat. Phys. 43:707 (1986).Google Scholar
  15. 15.
    E. Bombieri and J. E. Taylor, Quasicrystals, tilings, and algebraic number theory,Contemp. Math. 64 (1987).Google Scholar
  16. 16.
    C. Radin, Disordered ground states of classical lattice models,Rev. Math. Phys. 3:125 (1991).Google Scholar
  17. 17.
    R. I. Jewett, The prevalence of uniquely ergodic systems,J. Math. Mech. 19:717 (1970).Google Scholar
  18. 18.
    W. Krieger, On unique ergodicity, inProceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (1970), pp. 327–346.Google Scholar
  19. 19.
    M. Denker, C. Grillenberger, and K. Sigmund,Ergodic Theory on Compact Spaces (Springer, Berlin, 1976).Google Scholar
  20. 20.
    M. Queffélec,Substitution Dynamical Systems—Spectral Analysis (Springer, Berlin, 1987).Google Scholar
  21. 21.
    S. Aubry, Weakly periodic structures and example,J. Phys. (Paris)Coll. C3-50:97 (1984).Google Scholar
  22. 22.
    S. Aubry, Weakly periodic structures with a singular continuous spectrum, inProceedings of the NATO Advanced Research Workshop on Common Problems of Quasi-Crystals,Liquid-Crystals and Incommensurate Insulators, Preveza 1989, J. I. Toledano, ed.Google Scholar
  23. 23.
    S. Aubry, C. Godréche, and J. M. Luck, Scaling properties of a structure intermediate between quasiperiodic and random,J. Stat. Phys. 51:1033 (1988).Google Scholar
  24. 24.
    J. W. Cahn and J. E. Taylor, An introduction to quasicrystals,Contemp. Math. 64 (1987).Google Scholar
  25. 25.
    Z. Cheng, R. Savit, and R. Merlin, Structure and electronic properties of Thue-Morse lattices,Phys. Rev. B 37:4375 (1988).Google Scholar
  26. 26.
    Z. Cheng and R. Savit, Structure factor of substitutional sequences,J. Stat. Phys. 60:383 (1990).Google Scholar
  27. 27.
    M. Kolar, M. K. Ali, and F. Nori, Generalized Thue-Morse chains and their physical properties,Phys. Rev. B 43:1034 (1991).Google Scholar
  28. 28.
    M. Keane, Generalized Morse sequences,Z. Wahr. 10:335 (1968).Google Scholar
  29. 29.
    C. Gardner, J. Micekisz, C. Radin, and A. C. D. van Enter, Fractal symmetry in an Ising model,J. Phys. A 22:L1019 (1989).Google Scholar
  30. 30.
    A. C. D. van Enter and J. Micekisz, Breaking of periodicity at positive temperatures,Commun. Math. Phys. 134:647 (1990).Google Scholar
  31. 31.
    R. Merlin, K. Bajemu, J. Nagle, and K. Ploog, Raman scattering by acoustic phonons and structural properties of Fibonacci, Thue-Morse, and random superlattices,J. Phys. Coll. (Paris)C5:503 (1987).Google Scholar
  32. 32.
    F. Axel and H. Terauchi, High resolutionX-ray diffraction spectra of Thue-Morse GaAs-AlAs heterostructures: Towards a novel description of disorder,Phys. Rev. Lett. 66:2223 (1991).Google Scholar
  33. 33.
    C. Radin, Crystals and quasicrystals: A lattice gas model,Phys. Lett. 114A:385 (1986).Google Scholar
  34. 34.
    K. Mahler, On the translation properties of a simple class of arithmetical functions,J. Math. Phys. 6:150 (1927).Google Scholar
  35. 35.
    S. Kakutani, Ergodic properties of shift transformations, inProceedinigs of the Fifth Berkeley Symposium on Mathematical Statistics and Probability (1967), pp. 404–414.Google Scholar
  36. 36.
    S. Mozes, Tilings, substitutions, and dynamical systems generated by them,J. Anal. Math. 53:139 (1989).Google Scholar
  37. 37.
    J. Slawny, Ergodic properties of equilibrium states,Commun. Math. Phys. 80:477 (1981).Google Scholar

Copyright information

© Plenum Publishing Corporation 1992

Authors and Affiliations

  • A. C. D. van Enter
    • 1
  • Jacek Miekisz
    • 1
  1. 1.Institute for Theoretical PhysicalUniversity of GroningenGroningenThe Netherlands

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