Macroscopic Variables in Commensurate and Incommensurate Condensed Phases, Quasicrystals and Phasmids

  • Harald Pleiner
Part of the NATO ASI Series book series (NSSB, volume 166)


Macroscopic variables, necessary for a useful dynamical description of many condensed phases, are discussed. The nature of these macroscopic variables, their origin and their implications for the dynamics are considered. The somewhat special case of phasmids is treated separately.


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  1. 1.
    L. D. Landau and E. M. Lifshitz, “Fluid Mechanics” (Pergamon, London), 1959.MATHGoogle Scholar
  2. 2.
    L. P. Kadanoff and P. C. Martin, Ann. Phys. 24, 419 (1963).ADSCrossRefGoogle Scholar
  3. 3.
    P. Hohenberg and P. C. Martin, Ann. Phys. 34, 291 (1965).ADSCrossRefGoogle Scholar
  4. 4.
    I. M. Khalatnikov, “Introduction to the Theory of Superfluidity” (Benjamin, New York, 1965).Google Scholar
  5. 5.
    B. I. Halperin and P. Hohenberg, Phys. Rev. 188, 898 (1968).ADSCrossRefGoogle Scholar
  6. 6.
    R. Graham, Phys. Rev. Lett. 23, 1431 (1974).ADSCrossRefGoogle Scholar
  7. 7.
    R. Graham and H. Pleiner, Phys. Rev. Lett. 24, 792 (1975).ADSCrossRefGoogle Scholar
  8. 8.
    R. Graham and H. Pleiner, J. Phys. C9, 279 (1976).ADSGoogle Scholar
  9. 9.
    M. Liu,. Phys. Rev. Lett. 43, 1740 (1979).ADSCrossRefGoogle Scholar
  10. 10.
    P. C. Martin, O. Parodi and P. S. Pershan, Phys. Rev. A6, 2401 (1972).ADSCrossRefGoogle Scholar
  11. 11.
    D. Forster, Ann. Phys. 84, 505 (1974).ADSCrossRefGoogle Scholar
  12. 12.
    T. C. Lubensky, Phys. Rev. A6, 452 (1972).ADSCrossRefGoogle Scholar
  13. 13.
    H. Brand and H. Pleiner, J. Physique 41, 553 (1980).MathSciNetCrossRefGoogle Scholar
  14. 14.
    H. Pleiner and H. Brand, J. Physique Lett. 41, L491 (1980).CrossRefGoogle Scholar
  15. 15.
    H. Brand and H. Pleiner, Phys. Rev. A24, 2777 (1981).ADSCrossRefGoogle Scholar
  16. 16.
    H. Pleiner and H. Brand, Phys. Rev. A25, 995 (1982).ADSCrossRefGoogle Scholar
  17. 17.
    H. Brand and H. Pleiner, J. Physique 45, 563 (1984).CrossRefGoogle Scholar
  18. 18.
    H. Pleiner and H. R. Brand, Phys. Rev. A29, 911 (1984).ADSCrossRefGoogle Scholar
  19. 19.
    Except the angular momentum density, which is dynamically not an independent variable (cf. Ref. 10).Google Scholar
  20. 20.
    Except when long-ranged forces are present.Google Scholar
  21. 21.
    D. Forster, “Hydrodynamic Fluctuations, Broken Symmetry, and Correlation Functions” (Benjamin, Reading, 1975).Google Scholar
  22. 22.
    F. Jähnig and H. Schmidt, Ann. Phys. 71, 129 (1972).ADSCrossRefGoogle Scholar
  23. 23.
    W. L. McMillan, Phys. Rev. A9, 1720 (1974).ADSCrossRefGoogle Scholar
  24. 24.
    J. Swift, Phys. Rev. A13, 2274 (1976).ADSCrossRefGoogle Scholar
  25. 25.
    M. Liu, Phys. Rev. A19, 2090 (1979).ADSCrossRefGoogle Scholar
  26. 26.
    B. Julia, and G. Toulouse, J. Physique Lett. 40, L395 (1979).CrossRefGoogle Scholar
  27. 27.
    I. E. Dzyaloshinskii and G. E. Volovik, Ann Phys. 125, 67 (1980).ADSCrossRefGoogle Scholar
  28. 28.
    K. Kawasaki, Ann. Phys. 154, 319 (1984).ADSCrossRefGoogle Scholar
  29. 29.
    H. R. Brand and K. Kawasaki, J. Phys. A17, L905 (1984).ADSGoogle Scholar
  30. 30.
    Usually rotational symmetry; but if a field is modulated in space, it also breaks translational symmetry.Google Scholar
  31. 31.
    P. G. de Gennes, “The Physics of Liquid Crystals” (Clarendon, Oxford, 1974).MATHGoogle Scholar
  32. 32.
    H. Pleiner, J. Phys. C10, 4241 (1977).ADSGoogle Scholar
  33. 33.
    D. R. Nelson and B. I. Halperin, Phys. Rev. B21, 5312 (1980).ADSCrossRefGoogle Scholar
  34. 34.
    R. Pindak, D. E. Moncton, S. C. Davey and J. W. Goodby, Phys. Rev. Lett. 46, 1135 (1981).ADSCrossRefGoogle Scholar
  35. 35.
    J. J. Benattar, F. Moussa and M. Lambert, J. Chim. Phys. 80, 99 (1983).CrossRefGoogle Scholar
  36. 36.
    J. D. Brock, A. Aharony, R. J. Birgeneau, K. W. Evans-Lutterodt, J. D. Litster, P. M. Horn, G. B. Stephenson and A. R. Tajbakhsh, Phys. Rev. Lett. 57, 98 (1986).ADSCrossRefGoogle Scholar
  37. 37.
    H. Pleiner, Mol. Cryst. Liq. Cryst. 114, 103 (1984).CrossRefGoogle Scholar
  38. 38.
    H. Pleiner and H. R. Brand, Phys. Rev. A29, 911 (1984).ADSCrossRefGoogle Scholar
  39. 39.
    This definition is not restricted to translational broken symmetries.Google Scholar
  40. 40.
    Except near the smectic A-nematic phase transition (ref. 25).Google Scholar
  41. 41.
    J. Prost in “Liquid Crystals of One- and Two-Dimensional Order,” W. Helfrich and G. Heppke, eds. (Springer, Berlin, 1980), p. 125.CrossRefGoogle Scholar
  42. 42.
    This definition avoids the physically somewhat unsatisfactory distinction between rational and irrational numbers; on the other hand, it may be difficult to distinguish between no interaction and extremely small interaction.Google Scholar
  43. 43.
    J. Prost and P. Barois, J. Chim Phys. 80, 65 (1983).CrossRefGoogle Scholar
  44. 44.
    H. R. Brand and H. Pleiner (to be published).Google Scholar
  45. 45.
    H. Brand and P. Bak, Phys. Rev. A27, 1062 (1983).ADSCrossRefGoogle Scholar
  46. 46.
    In that case, I would call these systems commensurate.Google Scholar
  47. 47.
    The real and imaginary parts of the dispersion relation vanish with equal power of the wave vector.Google Scholar
  48. 48.
    The masses of the different atomic species are different.Google Scholar
  49. 49.
    J. D. Axe and P. Bak, Phys. Rev. B26, 4963 (1982).ADSCrossRefGoogle Scholar
  50. 50.
    D. Schechtman, I. Blech, D. Gratias and J. W. Cahn, Phys. Rev. Lett. 53, 1951 (1984).ADSCrossRefGoogle Scholar
  51. 51.
    P. Bak, Phys. Rev. Lett. 54, 1517 (1985).ADSCrossRefGoogle Scholar
  52. 52.
    T. C. Lubensky, S. Ostlund, S. Ramaswamy, P. J. Steinhardt, and J. Toner, Phys. Rev. Lett. 54, 1520 (1985).ADSCrossRefGoogle Scholar
  53. 53.
    N. D. Mermin and S. M. Troian, Phys. Rev. Lett. 54, 1524 (1985).ADSCrossRefGoogle Scholar
  54. 54.
    M. Kleman, Y. Gefen and Y. Pavlovitch, Europhys. Lett. 1, 61 (1986).ADSCrossRefGoogle Scholar
  55. 55.
    The latter is proposed in T. Janssen, these proceedings.Google Scholar
  56. 56.
    J. Malthete, A. M. Levelut and Nguyen Hu Tinh, J. Physique Lett. 46, L876 (1985).CrossRefGoogle Scholar

Copyright information

© Plenum Press, New York 1987

Authors and Affiliations

  • Harald Pleiner
    • 1
  1. 1.Department of PhysicsUniversity of ColoradoBoulderUSA

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