Zeitschrift für Physik B Condensed Matter

, Volume 63, Issue 4, pp 521–535 | Cite as

Fluctuations and lack of self-averaging in the kinetics of domain growth

  • A. Milchev
  • K. Binder
  • D. W. Heermann
Article

Abstract

The fluctuations occurring when an initially disordered system is quenched at timet=0 to a state, where in equilibrium it is ordered, are studied with a scaling theory. Both the mean-sizel(t)d of thed-dimensional ordered domains and their fluctuations in size are found to increase with the same power of the time; their relative size fluctuations are independent of the total volumeLd of the system. This lack of self-averaging is tested for both the Ising model and the φ4 model on the square lattice. Both models exhibit the same lawl(t)=(Rt)x withx=1/2, although the φ4 model has “soft walls”. However, spurious results withx≷1/2 are obtained if “bad” pseudorandom numbers are used, and if the numbern of independent runs is too small (n itself should be of the order of 103). We also predict a critical singularity of the rateR∝(1−T/Tc)v(z−1/x),v being the correlation length exponent,z the dynamic exponent.

Also quenches to the critical temperatureTc itself are considered, and a related lack of self-averaging in equilibrium computer simulations is pointed out for quantities sampled from thermodynamic fluctuation relations.

Keywords

Neural Network Computer Simulation Nonlinear Dynamics Relative Size Correlation Length 

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References

  1. 1.
    Extensive lists of references can be found in the recent review articles Refs. 2, 3.Google Scholar
  2. 2.
    Binder, K., Heermann, D.W.: In: Scaling phenomena in disordered systems. Pynn, R., Skjeltorp, T. (eds.). New York: Plenum Press 1985Google Scholar
  3. 3.
    Binder, K.: In: Proceedings of the Discussion Meeting on “Phase Transitions on Solid Surfaces”, Erlangen 1985 (in press)Google Scholar
  4. 4.
    Sadiq, A., Binder, K.: J. Statist. Phys.35, 617 (1984)Google Scholar
  5. 5.
    Gawlinski, E.T., Grant, M., Gunton, J.D., Kaski, K.: Phys. Rev. B3, 281 (1985)Google Scholar
  6. 6.
    Mouritsen, O.G.: Phys. Rev. B28, 3150 (1983); B31, 2613 (1985); B32, 1632 (1985)Google Scholar
  7. 7.
    Grest, G.S., Srolovitz, D.J., Anderson, M.P.: Phys. Rev. Lett.52, 1321 (1984); Grest, G.S., Srolovitz, D.J.: Phys. Rev. B32, 3014 (1985); Sahni, P.S., Grest, G.S., Safran, S.A.: Phys. Rev. Lett.50, 60 (1983); Grest, G.S., Safran, S.A., Sahni, P.S.: J. Appl. Phys.55, 2432 (1984)Google Scholar
  8. 8.
    Mouritsen, O.G.: Phys. Rev. Lett.56, 850 (1986)Google Scholar
  9. 9.
    Lifshitz, I.M.: Sov. Phys. JETP15, 939 (1962)Google Scholar
  10. 10.
    Lifshitz, I.M., Slyozov, V.V.: J. Phys. Chem. Solids19, 35 (1961)Google Scholar
  11. 11.
    Allen, S.W., Cahn, J.W.: Acta Metall27, 1085 (1979)Google Scholar
  12. 12.
    Ohta, T., Jasnow, D., Kawasaki, K.: Phys. Rev. Lett.49, 1223 (1982)Google Scholar
  13. 13.
    Mazenko, G.F., Zannetti, M.: Phys. Rev. Lett.53, 2106 (1984)Google Scholar
  14. 14.
    Binder, K. (ed.): Monte Carlo methods in statistical physics. Berlin, Heidelberg, New York: Springer 1979; Applications of the Monte Carlo method in statistical physics. Berlin, Heidelberg, New York: Springer 1984Google Scholar
  15. 15.
    Vinals, J., Gunton, J.D.: Surf. Sci.157, 473 (1985)Google Scholar
  16. 16.
    Grest, G.S., Sahni, P.S.: Phys. Rev. B30, 2261 (1984)Google Scholar
  17. 17.
    This type of scaling behaviour and universality was first emphasized in the context of the later stages of spinodal decomposition, see Binder, K., Stauffer, D.: Phys. Rev. Lett.33, 1006 (1974); Binder, K.: Phys. Rev. B15, 4425 (1977); Binder, K., Billotet, C., Mirold, P.: Z. Phys. B-Condensed Matter30, 183 (1978); Billotet, C., Binder, K.: Z. Phys. B-Condensed Matter32, 195 (1979)Google Scholar
  18. 18.
    Furukawa, H.: Phys. Rev. A29, 2160 (1984); A30, 1052 (1984); and preprintsGoogle Scholar
  19. 19.
    Weeks, J.D.: J. Chem. Phys.67, 3106 (1977)Google Scholar
  20. 20.
    Milchev, A., Heermann, D.W., Binder, K.: J. Statist. Phys. (in press)Google Scholar
  21. 21.
    Filk, Th., Marcu, M., Fredenhagen, K.: DESY preprint 85-098Google Scholar
  22. 22.
    Feller, W.: Introduction to probability theory and its application. New York 1966Google Scholar
  23. 23.
    Landau, L.D., Lifshitz, E.M.: Statistical physics. London: Pergamon Press 1959Google Scholar
  24. 24.
    Binder, K.: Z. Phys. B-Condensed Matter43, 119 (1981)Google Scholar
  25. 25.
    For recent reviews, see Barber, M.N.: In: Phase transitions and critical phenomena. Domb, C., Lebowitz, J.L. (eds.), Vol. 8, Chap. 2. New York, London: Academic Press 1983; Binder, K.: Ferroelectrics (in press)Google Scholar
  26. 26.
    Hohenberg, P.C., Halperin, B.I.: Rev. Mod. Phys.49, 435 (1977)Google Scholar
  27. 27.
    In this section we assume the validity of hyperscaling. If hyperscaling is not valid, but one considers a situation with fully periodic boundary conditions, already (19) must be modified and the correlation length ζ is replaced by a thermodynamic lengthl∝|1−T/T c|−(y+2β)/d: then\(P_L (\psi ) = L^{{{\beta d} \mathord{\left/ {\vphantom {{\beta d} {(y + 2\beta )}}} \right. \kern-\nulldelimiterspace} {(y + 2\beta )}}} \tilde P(\psi L^{{{\beta d} \mathord{\left/ {\vphantom {{\beta d} {(y + 2\beta )}}} \right. \kern-\nulldelimiterspace} {(y + 2\beta )}}} \), |1−T/T c|−(y+2β)/d/L). For background on this point, see Binder, K., Nauenberg, M., Privman, V., Young, A.P.: Phys. Rev. B31, 1498 (1985); Brézin, E., Zinn-Justin, J.: Nucl. Phys. B257 [FS 14], 867 (1985)Google Scholar
  28. 28.
    If the scaling considerations of this section are translated into a description in terms of the renormalization group, 3″ would be a “dangerous irrelevant variable”: see Fisher, M.E.: Critical phenomena. In: Lecture Notes in Physics. Hahne, F.J.W. (ed.), Vol. 186. Berlin, Heidelberg, New York: Springer 1983Google Scholar
  29. 29.
    Mazenko, G.F., Valls, O.T.: Phys. Rev. B30, 6732 (1984)Google Scholar
  30. 30.
    Morf, R., Schneider, T., Stoll, E.: Phys. Rev. B16, 462 (1977)Google Scholar
  31. 31.
    Bruce, A.D.: J. Phys. A18, L-873 (1985)Google Scholar
  32. 32.
    See e.g. a brief remark in Ref. 5Google Scholar
  33. 33.
    Kirkpatrick, S., Stoll, E.: J. Comput. Phys.40, 517 (1981)Google Scholar
  34. 34.
    Wang, G.-C., Lu, T.-M.: Phys. Rev. Lett.50, 2014 (1983); Wu, P.K., Perepezko, J.H., Mc Kinney, J.T., Lagally, M.G.: Phys. Rev. Lett.51, 1577 (1983)Google Scholar
  35. 35.
    Lehmer, D.H.: Proceedings of the 2nd Symposium on Large Scale Digital Computing Machines. Vol. 142, Cambridge: Harvard University Press 1951Google Scholar
  36. 36.
    Tausworth, R.C.: Math. Compt.19, 201 (1968)Google Scholar
  37. 37.
    In MULTICS “Subroutines andI/0 Modules”, Honeywell Information Systems Inc. 1983, Louveciennes (France)Google Scholar

Copyright information

© Springer-Verlag 1986

Authors and Affiliations

  • A. Milchev
    • 1
  • K. Binder
    • 1
  • D. W. Heermann
    • 1
  1. 1.Institut für PhysikJohannes Gutenberg-Universität MainzMainzGermany

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