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Aperiodicity and Disorder — Do They Play a Role?

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Computational Statistical Physics

Summary

The effects of an aperiodic order or a random disorder on phase transitions in statistical mechanics are discussed. A heuristic relevance criterion based on scaling arguments as well as specific results for Ising models with random disorder or certain kinds of aperiodic order are reviewed. In particular, this includes an exact real-space renormalization treatment of the Ising quantum chain with coupling constants modulated according to substitution sequences, related to a two-dimensional classical Ising model with layered disorder.

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References

  1. R: Balian: From Microphysics to Macrophysics, Vol. I (Springer, Berlin 1991 )

    Google Scholar 

  2. D. Frenkel: Physica A 263, 26 (1999)

    Article  ADS  Google Scholar 

  3. R.J. Baxter: Exactly Solved Models in Statistical Mechanics ( Academic Press, London 1982 )

    MATH  Google Scholar 

  4. W. Lenz: Physikalische Zeitschrift 21, 613 (1920)

    Google Scholar 

  5. E. Ising: Zeitschrift für Physik 31, 253 (1925)

    Article  ADS  Google Scholar 

  6. L. Onsager: Phys. Rev. 65, 117 (1944)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  7. J.-B. Suck, M. Schreiber, P. Häussler (eds.): Quasicrystals (Springer, Berlin 2002) to appear

    Google Scholar 

  8. L.P. Kadanoff: ‘Scaling, universality and operator algebras’. In: Phase Transitions and Critical Phenomena Vol. 5a, ed. by C. Domb, J.L. Lebowitz ( Academic Press, London 1976 ) pp. 1–34

    Google Scholar 

  9. J.L. Cardy: Scaling and Renormalization in Statistical Physics ( Cambridge University Press, Cambridge 1996 )

    Google Scholar 

  10. P. DiFrancesco, P. Mathieu, D. Sénéchal: Conformal Field Theory ( Springer, New York 1997 )

    Book  Google Scholar 

  11. B.M. McCoy, T.T. Wu: The Two-Dimensional Ising Model (Harvard University Press, Cambridge, Massachusetts 1973 )

    Google Scholar 

  12. H.-O. Georgii: Gibbs Measures and Phase Transitions (de Gruyter, Berlin 1988 )

    Google Scholar 

  13. S. Kobe: J. Stat. Phys. 88, 991 (1997);

    Article  MathSciNet  ADS  MATH  Google Scholar 

  14. S. Kobe: Phys. Bl. 54, 917 (1998)

    Google Scholar 

  15. U. Grimm, M. Baake: ‘Aperiodic Ising models’. In: The Mathematics of Long-Range Aperiodic Order, ed. by R.V. Moody ( Kluwer, Dordrecht 1997 ) pp. 199–237

    Google Scholar 

  16. R. Peierls: Proc. Cambridge Philos. Soc. 32, 477 (1936)

    MATH  Google Scholar 

  17. R.S. Ellis: Entropy, Large Deviations, and Statistical Mechanics ( Springer, New York 1985 )

    Google Scholar 

  18. H.A. Kramers, G.H. Wannier: Phys. Rev. 60, 252 (1941)

    Article  MathSciNet  ADS  Google Scholar 

  19. C.N. Yang: Phys. Rev. 85, 808 (1952)

    Article  ADS  MATH  Google Scholar 

  20. R.B. Stinchcombe: J. Phys. C 6, 2459 (1973)

    Article  ADS  Google Scholar 

  21. E. Lieb, T. Schultz, D. Mattis: Ann. Phys. (NY) 16, 407 (1961)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  22. J.B. Kogut: Rev. Mod. Phys. 51, 659 (1979)

    Article  MathSciNet  ADS  Google Scholar 

  23. C. Chatelain, P.E. Berche, B. Berche: Eur. Phys. J. B 7, 439 (1999)

    Article  ADS  Google Scholar 

  24. J. Hermisson: Aperiodische Ordnung und Magnetische Phasenübergänge ( Shaker, Aachen 1999 )

    Google Scholar 

  25. A.B. Harris: J. Phys. C: Solid State Phys. 7, 1671 (1974)

    Article  ADS  Google Scholar 

  26. J.M. Luck: J. Stat. Phys. 72, 417 (1993);

    Article  MathSciNet  ADS  MATH  Google Scholar 

  27. J.M. Luck: Europhys. Lett. 24, 359 (1993)

    Article  ADS  Google Scholar 

  28. R.B. Stinchcombe: ‘Dilute magnetism’. In: Phase Transitions and Critical Phenomena Vol. 7, ed. by C. Domb, J.L. Lebowitz ( Academic Press, London 1983 ) pp. 151–280

    Google Scholar 

  29. W. Selke: ‘Monte Carlo simulations of dilute Ising models’. In: Computer Simulation Studies in Condensed Matter Physics IV, ed. by D.P. Landau, K.K. Mon, H.-B. Schüttler ( Springer, Berlin 1993 ) pp. 18–27

    Chapter  Google Scholar 

  30. W. Selke, L.N. Shchur, A.L. Talapov: ‘Monte Carlo simulations of dilute Ising models’. In: Annual Reviews of Computational Physics I, ed. by D. Stauffer ( World Scientific, Singapore 1994 ) pp. 17–54

    Google Scholar 

  31. Vik. S. Dotsenko, Vl. S. Dotsenko: Adv. Phys. 32, 129 (1983)

    Article  MathSciNet  ADS  Google Scholar 

  32. B.N. Shalaev: Sov. Phys. Solid State 26, 1811 (1984);

    Google Scholar 

  33. B.N. Shalaev: Phys. Rep. 237, 129 (194)

    Google Scholar 

  34. R. Shankar: Phys. Rev. Lett. 58, 2466 (1987);

    Article  MathSciNet  ADS  Google Scholar 

  35. R. Shankar: Phys. Rev. Lett. 61, 2390 (1988)

    Article  ADS  Google Scholar 

  36. A.W.W. Ludwig: Phys. Rev. Lett. 61, 2388 (1988);

    Article  ADS  Google Scholar 

  37. A.W.W. Ludwig: Nucl. Phys. B 330, 639 (1990)

    Article  MathSciNet  ADS  Google Scholar 

  38. A. Roder, J. Adler, W. Janke: Phys. Rev. Lett. 80, 4697 (1998)

    Article  ADS  Google Scholar 

  39. W. Janke, R. Villanova, ‘Ising spins on 3D random lattices’. In: Computer Simulation Studies in Condensed Matter Physics XI, ed. by D.P. Landau, H.-B. Schüttler ( Springer, Berlin 1999 ) pp. 22–26

    Chapter  Google Scholar 

  40. C.M. Newman: Topics in Disordered Systems ( Birkhäuser, Basel 1997 )

    Book  MATH  Google Scholar 

  41. A. Bovier, P. Picco (eds.): Mathematical Aspects of Spin Glasses and Neural Networks, Progress in Probability 41 (Birkhäuser, Boston 1998 )

    Google Scholar 

  42. H.-O. Georgii, O. Häggström, C. Maes: ‘The random geometry of equilibrium phases’. In: Phase Transitions and Critical Phenomena, Vol. 18, ed. by C. Domb, J.L. Lebowitz ( Academic Press, London 2001 ) pp. 1–142

    Chapter  Google Scholar 

  43. H.-O. Georgii, Y. Higuchi: J. Math. Phys. 41, 1153 (2000)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  44. M. Baake: ‘A guide to mathematical quasicrystals’, math-ph/9901014, to appear in [7]

    Google Scholar 

  45. S. Wolfram: Mathematica: A System for Doing Mathematics by Computer, 2nd ed. ( Addison-Wesley, Reading, Massachusetts 1991 )

    Google Scholar 

  46. U. Grimm, M. Schreiber: ‘Aperiodic tilings on the computer’, cond-mat/9903010, to appear in [7]

    Google Scholar 

  47. R. Ammann, B. Grünbaum, G. Shephard: Discrete Comput. Geom. 8, 1 (1992)

    MATH  Google Scholar 

  48. M. Baake, U. Grimm, R.J. Baxter: Int. J. Mod. Phys. B 8, 3579 (1994)

    Article  MathSciNet  ADS  Google Scholar 

  49. C. Domb: ‘Ising model’ in: Phase Transitions and Critical Phenomena Vol. 3, ed. by C. Domb, M.S. Green ( Academic Press, London 1974 ) pp. 357–458

    Google Scholar 

  50. P. Repetowicz, U. Grimm, M. Schreiber: J. Phys. A: Math. Gen. 32, 4397 (1999)

    Article  ADS  MATH  Google Scholar 

  51. P. Repetowicz, U. Grimm, M. Schreiber: Mat. Sci. Eng. A 294–296, 638 (2000)

    Google Scholar 

  52. O. Redner, M. Baake: J. Phys. A: Math. Gen. 33, 3097 (2000)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  53. F. Iglói, L. Turban: Phys. Rev. Lett. 77, 1206 (1996)

    Article  ADS  Google Scholar 

  54. F. Iglói, L. Turban, D. Karevski, F. Szalma: Phys. Rev. B 56, 11031 (1997)

    Article  ADS  Google Scholar 

  55. J. Hermisson, U. Grimm, M. Baake: J. Phys. A: Math. Gen. 30, 7315 (1997)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  56. J. Hermisson, U. Grimm: Phys. Rev. B 57, R673 (1998)

    Article  ADS  Google Scholar 

  57. J. Hermisson: J. Phys. A: Math. Gen. 33, 57 (2000)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  58. See http://www.tu-chemnitz.de/physik/HERAUS/2000/Springer.html

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Authors
  1. Uwe Grimm
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Grimm, U. (2002). Aperiodicity and Disorder — Do They Play a Role?. In: Computational Statistical Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-04804-7_12

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  • DOI: https://doi.org/10.1007/978-3-662-04804-7_12

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-07571-1

  • Online ISBN: 978-3-662-04804-7

  • eBook Packages: Springer Book Archive

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