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Dynamics of the formation of ordered domains out of initially disordered configurations

  • I. Spin Glasses
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Heidelberg Colloquium on Glassy Dynamics

Part of the book series: Lecture Notes in Physics ((LNP,volume 275))

Abstract

This talk reviews phenomenological theories and Monte Carlo simulations on the dynamics of ordering processes, and the associated statistical fluctuations, both in pure systems and in systems with quenched disorder. Particular attention is paid to understand size effects and the approach to the thermodynamic limit: we discuss time-scales on which fluctuations around an equilibrium state decay, the time-scale for growth of a domain size comparable to the system volume, and the “ergodic time” on which a system dynamically averages over its various ordered configurations.

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References

  1. K. Binder, in Fundamental Problems in Statistical Mechanics (E. G. D. Cohen, ed.) p. 21 (North-Holland Publ. Comp., Amsterdam 1980)

    Google Scholar 

  2. Y. Imry and S.-K. Ma, Phys. Rev. Lett. 35, 1399 (1975)

    Article  ADS  Google Scholar 

  3. J. Villain, J. Phys. Lett. (Paris) 43, L 551 (1982); G. Grinstein and S.-K. Ma, Phys. Rev. Lett. 49, 685 (1982)

    Google Scholar 

  4. For reviews, see Y. Imry, J. Stat. Phys. 34, 849 (1984); R. A. Cowley, J. Magn. Magn. Mater. 31–34, 1439 (1983); and J. Villain, Ref. [9].

    Article  ADS  Google Scholar 

  5. K. Binder, Z. Physik B 50, 343 (1983)

    ADS  Google Scholar 

  6. I. Morgenstern, K. Binder and R. M. Hornreich, Phys. Rev. B 23, 287 (1981); see also K. Binder, Phys. Rev. B 29, 5184 (1984)

    MathSciNet  ADS  Google Scholar 

  7. A. Aharony, Phys. Rev. B 18, 3318, 3328 (1978)

    ADS  Google Scholar 

  8. We disregard here the recent suggestion (A. P.Young and M. Nauenberg, Phys. Rev. Lett. 54, 2429 (1985); D. Andelman and J. F. Joanny, in Scaling Phenomena in Disordered Systems (ed. by R. Pynn and A. Skjeltorp, p. 163, Plenum Publ.Corp., New York 1985) that for d = 3 and h ≠ Ø the transition is always of first order, which has been disputed (A. T. Ogielski and A. Huse, Phys. Rev.Lett. 56, 1298 (1986))

    Article  ADS  Google Scholar 

  9. J. Villain, in Scaling Phenowena in Disordered System (ed. by R. Pynn and A. Skjeltorp) p. 423 (Plenum Publ. Corp., New York (1985); Phys. Rev. Lett. 52, 1543 (1984)

    Google Scholar 

  10. E. T. Gawlinski, K. Kaski, M. Grant, J. D. Gunton, Phys. Rev. Lett. 53, 2264 (1984); M. Grant and J. G. Gunton, Phys. Rev. B 29, 6266 (1984); see also C. RD, G. S. Grest, C. M. Souvkoulis, and K. Levan, Phys. Rev. B 31, 1682 (1984); Y. Yoshizarra and D. P. Belanger, Phys. Rev. B 30, 5220 (1984); D. Chowdhury and D. Stauffer, preprint; S. R. Anderson and G. F. Mazenko, Phys. Rev. B 33, 2007 (1986)

    Article  ADS  Google Scholar 

  11. G. Grinstein and J. F. Fernandez, Phys. Rev. B 29, 389 (1984)

    Google Scholar 

  12. R. Bruinsma and G. Aeppli, Phys. Rev. Lett. 52, 1543 (1984)

    Article  ADS  Google Scholar 

  13. E. Pytte and J. F. Fernandez, Phys. Rev. B 31, 616 (1985)

    ADS  Google Scholar 

  14. D. Stauffer, C. Hartzstein, K. Binder and A. Aharony, Z. Phys. B 55, 325 (1984)

    Article  ADS  Google Scholar 

  15. For an extensive review, see K. Binder and A. P. Young, Rev. Mod. Phys. (1986, in press)

    Google Scholar 

  16. S. F. Edwards and P. W. Anderson, J. Phys. F 5, 965 (1975)

    Article  ADS  Google Scholar 

  17. G. Toulouse, Commun. Phys. 2, 115 (1977)

    Google Scholar 

  18. D. Sherrington and S. Kirkpatrick, Phys. Rev. Lett. 35, 1972 (1975)

    Article  Google Scholar 

  19. D. C. Mattis, Phys. Lett. 56 A, 421 (1976)

    ADS  Google Scholar 

  20. J. L. van Hemmen, Phys. Rev. Lett. 49, 409 (1982) and in Heidelberg Colloqium on Spin Glasses (ed. by J. L. van Hemmen and I. Morgenstern) p. 203 (Springer, Berlin-Heidelberg-New York 1983)

    Article  MathSciNet  ADS  Google Scholar 

  21. K. Binder, W. Kinzel and D. Stauffer, Z. Physik B 36, 161 (1979)

    ADS  Google Scholar 

  22. W. Kinzel and K. Binder, Phys. Rev. B 24, 2701 (1981)

    ADS  Google Scholar 

  23. I. Morgenstern and K. Binder, Phys. Rev. B. 22, (1980)

    Google Scholar 

  24. M. Schwartz, preprint

    Google Scholar 

  25. A. T. Ogielski and I. Morgenstern, Phys. Rev. Lett. 54, 928 (1985)

    Article  ADS  Google Scholar 

  26. M. Moore and A. J. Bray, J. Phys. C 18, L 699 (1985)

    MathSciNet  ADS  Google Scholar 

  27. R. N. Bhall and A. P. Young, Phys. Rev. Lett. 54, 324 (1985)

    ADS  Google Scholar 

  28. A. T. Ogielski, Phys. Rev. B 32, 7384 (1985)

    ADS  Google Scholar 

  29. N. Sourlas, Europhys. Lett. 1, 189 (1986)

    Article  ADS  Google Scholar 

  30. D. S. Fisher and D. Huse, Phys. Rev. Lett. 56, 1601 (1986)

    Article  ADS  Google Scholar 

  31. K. Binder and W. Kinzel, in Heidelberg Colloquium on Spin Glasses (ed. by J. L. van Hemmen and I. Morgenstern) p. 279 (Springer, Berlin-Heidelberg-New York 1983)

    Chapter  Google Scholar 

  32. R. Binder and A. P. Young, Phys. Rev. B 29, 2864 (1984)

    ADS  Google Scholar 

  33. The model Eq.(6) can be viewed as a generalization of Eq.(5), with 4 instead of 2 ordered states which are easily identified I. Morgenstern and J. L. van Hemmen, Phys. Rev. B 32, 6058 (1985)). Since it does contain frustration, it might be interesting to study by kinetics of ordering of this model. To the author's knowledge, this has not yet been done.

    Google Scholar 

  34. A. Milchev, K. Binder and D. W. Heermann, Z. Phys. B 63, (1986)

    Google Scholar 

  35. A. Sadiq and K. Binder, J. Stat. Phys. 35, 617 (1984)

    Article  MathSciNet  Google Scholar 

  36. A. Sadiq and K. Binder, Phys. Rev. Lett. 51, 674 (1983)

    Article  ADS  Google Scholar 

  37. As far as static critical properties are concerned, this model belongs to the universality class of the XY model with cubic anisotropy: S. Krinsky and D. Mukamel, Phys. Rev. B 16, 2313 (1977)

    Google Scholar 

  38. M. N. Barber, in Phase Transitions and Critical Phenomena, Vol. 8 (ed. by C. Domb and J. L. Lebowitz) p. 146 (Academic Press, New York 1983)

    Google Scholar 

  39. A general review on finite size effects on phase transitions is found in Ref. [38] or in K. Binder, Ferroelectrics (1986)

    Google Scholar 

  40. R. G. Palmer, Adv. Phys. 31, 669 (1982)

    Article  ADS  Google Scholar 

  41. K. Binder and D. W. Heermann, in Scaling Phenomena in Disordered Systems (ed. by R. Pynn and A. Skjeltorp) p. 207 (Plenum Publ. Corp, New York (1985)

    Google Scholar 

  42. For reviews, see Ref. [41] and H. Furukawa, Adv. Phys. 34, 703 (1986)

    Google Scholar 

  43. K. Binder, Z. Phys. B 43, 119 (1981)

    Article  ADS  Google Scholar 

  44. K. Binder, Phys. Rev. A 25, 1699 (1982)

    ADS  Google Scholar 

  45. R. J. Glauber, J. Math. Phys. 4, 294 (1963)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  46. S. Miyashita and H. Takano, Progr. Theor. Phys. 73, 1122 (1985)

    Article  ADS  Google Scholar 

  47. G. F. Mazenko, D. T. Valls, and F. C. Zhang, Phys. Rev. 31 4453 (1985); G. F. Mazenko and O. T. Valls, Phys. Rev. B 33, 1823 (1986); see also S. A. Safran, Phys. Rev. Lett. 46, 1581 (1981)

    Article  ADS  Google Scholar 

  48. M. Suzuki, Progr. Theor. Phys. 58, 1142 (1977)

    Article  ADS  Google Scholar 

  49. P. C. Hohenberg and B. I. Halperin, Revs. Mod. Phys. 49, 435 (1977)

    Article  ADS  Google Scholar 

  50. K. Kawasaki, in Phase Transitions and Critical Phenomena, Vol 2 (ed. by C. Dumb and M. S. Green) p. 443 (Academic Press, New York 1972)

    Google Scholar 

  51. If the Ising model is interpreted as a lattice pas (local density ci = (1-Si)/2 = 0,1), this conservation law means that the density is held constant, which is a rather natural assumption for adsorbed layers on surfaces, for instance.

    Google Scholar 

  52. M. V. Feigel'man and L. B. Ioffe, J. Physique Lett. 46, L 695 (1985); For a discussion of slow relaxation near the random-field Ising critical point, see J. Villain, J. Phys. (Paris) 46, 1643 (1985); D. S. Fisher, Phys. Rev. Lett. 56, 416 (1986)

    Article  Google Scholar 

  53. D. Stauffer and K. Binder, Z. Physik B 30, 313 (1978)

    ADS  Google Scholar 

  54. I. M. Lifshitz, Sov. Phys. JETP 15, 939 (1962)

    Google Scholar 

  55. S. W. Cahn and J. W. Allen, Acta Metall 27, 1085 (1979)

    Article  Google Scholar 

  56. T. Ohta, D. Jasnow and K. Kawasaki, Phys. Rev. Lett. 49, 1223 (1982)

    Article  ADS  Google Scholar 

  57. P. S. Sahni, G. S. Grest, M. P. Anderson, and D. J. Srolovitz, Phys. Rev. Lett. 50, 263 (1983); Phys. Rev. B 28, 2705 (1983); P. S. Sahni, G. S. Grest and S. A. Safran, Phys. Rev. Lett. 50, 60 (1983); S. A. Safran, P. S. Sahni, and G. S. Grest, Phys. Rev. B 28, 2693 (1983)

    Article  ADS  Google Scholar 

  58. K. Binder, Ber. Bunsenges. Phys. Chem 90, (1986); for earlier work see also Ref. [59] and references contained therein.

    Google Scholar 

  59. K. Binder, J. Stat. Phys. 24, 69 (1981)

    Article  ADS  Google Scholar 

  60. Different conclusions, however, have been reached by G. F. Mazenko and O.T. Valls, Phys. Rev. B 30, 6732 (1984), who find <ϕ2>t α t7/8,<5E) t α t−5/8 for the d = 2 Ising model where v = 1, α = 0, Y = 7/4, from a real-space renormalization group approach. If z = 2, Eq.(23) would agree with this result, but Eq.(30) then would rather inply <5E> t α t−1/2.

    Google Scholar 

  61. If the scaling considerations of this section are translated into a description in terms of the renormalization group, would be a “dangerous irrelevant variable”; see M. E. Fisher, in Critical Phenomena (Lecture Notes in Physics, Vol 186), edited by F. J. W. Hahne (Springer, Berlin-Heidelberg-New York 1983)

    Google Scholar 

  62. G. S. Chest, S. A. Safran, and P. S. Sahni, J. Appl. Phys. 55, 2432 (1984)

    Article  ADS  Google Scholar 

  63. G. S. Grest, D. J. Srolovitz, and M. P. Anderson, Phys. Rev. Lett. 52, 1321 (1984)

    Article  ADS  Google Scholar 

  64. E. T. Gawlinski, M. Grant, J. D. Gunton and K. Kaski, Phys. Rev. B 31, 281 (1985)

    ADS  Google Scholar 

  65. M. K. Phani, J. L. Lebowitz, M. H. Kalos and O. Penrose, Phys. Rev. Lett. 45, 410 (1981)

    Google Scholar 

  66. P. S. Sahni and J. D. Gunton, Phys. Rev. Lett. 45, 368 (1980); K. Kaski and J. D. Gunton, Phys. Rev. B 28, 5371 (1983); K. Kaski, M. D. Yalabik, J. D. Gunton, and P. S. Sahni, Phys. Rev. B 26, 5263 (1983); K. Kaski, S. Kumar, J. D. Gunton and P. A. Rikvold, Phys. Rev. B 29, 4420 (1984); H. Kaski, T. Ala-Nissila and J. D. Gunton, Phys. Rev. B 31, 310 (1985)

    Article  ADS  Google Scholar 

  67. A. Milchev, D. W. Heermann and H. Binder, J. Statist. Phys. (1986)

    Google Scholar 

  68. O. G. Mouritsen, Phys. Rev. B 28, 3150 (1983); B 31, 2613 (1985), B 32, 1632 (1985)

    ADS  Google Scholar 

  69. O. G. Mouritsen, Phys. Rev. Lett. 56, 850 (1986)

    Article  ADS  Google Scholar 

  70. G. S. Grest and D. J. Srolovitz, Phys. Rev. B 32, 3014 (1985) D. J. Srolovitz and G. S. Crest, Phys. Rev. B 32, 3021 (1985)

    ADS  Google Scholar 

  71. Experimental evidence on this dependence on direction is seen in spinodal decomposition experiments (R. J. Simon, P. Guyot and A. Ghilarducci de Salva, Phil. Mag. A 49, 151 (1984))

    Google Scholar 

  72. Computer simulation evidence on the dependence of ST(k) on direction is seen in studies of Eq.(48) with conserved magnetization <Ψ>, see D. W. Heermann, A. Milchev and K. Binder, in preparation

    Google Scholar 

  73. This scaling assumption was first written down explicitly in the context of spinodal decomposition, see H. Binder, C. Billotet and P. Mirold, Z. Physik B 30, 183 (1978), and Ref. [75]. It first also was verified in simulations of spinodal decomposition, see J. Marro, J. L. Lebowitz and M. H. Halos, Phys. Rev. Lett. 43, 282 (1979)

    Google Scholar 

  74. B. B. Mandelbrot: Fractals: Form, Chance and Dimension (Freeman, San Francisco 1977)

    MATH  Google Scholar 

  75. K. Binder and D. Stauffer, Phys. Rev. Lett. 33, 1006 (1974), H. Binder, Phys. Rev. B 15, 4425 (1977)

    Article  ADS  Google Scholar 

  76. M. Grant and J. D. Gunton, Phys. Rev. B 28, 5496 (1983)

    Article  ADS  Google Scholar 

  77. G. F. Mazenko and M. Zannetti, Phys. Rev. Lett. 53, 2106 (1984)

    Article  ADS  Google Scholar 

  78. F. Y. Wu, Rev. Mod. Phys. 54, 235 (1982); R. B. Potts, Proc. Camb. Phil. Soc. 46, 106 (1952)

    Article  ADS  Google Scholar 

  79. K. Kaski, J. Nieminem and J. D. Gunton, Phys. Rev. B 31, 2998 (1985)

    ADS  Google Scholar 

  80. H. Furukawa, Phys. Rev. A 29, 2160 (1984); A 30, 1052 (1984)

    ADS  Google Scholar 

  81. G. S. Grest and P. S. Sahni, Phys. Rev. B 30, 226 (1984)

    ADS  Google Scholar 

  82. I. M. Lifshitz and V. V. Slyozov, J. Phys. Chem. Solids 19, 35 (1961)

    Article  ADS  Google Scholar 

  83. J. Vinals and J. D. Gunton, Surf. Sci 157, 473 (1985)

    Article  ADS  Google Scholar 

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J. L. van Hemmen I. Morgenstern

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Binder, K., Heermann, D.W., Milchev, A., Sadiq, A. (1987). Dynamics of the formation of ordered domains out of initially disordered configurations. In: van Hemmen, J.L., Morgenstern, I. (eds) Heidelberg Colloquium on Glassy Dynamics. Lecture Notes in Physics, vol 275. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0057516

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  • DOI: https://doi.org/10.1007/BFb0057516

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