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Zeitschrift für Physik B Condensed Matter

, Volume 43, Issue 2, pp 119–140 | Cite as

Finite size scaling analysis of ising model block distribution functions

  • K. Binder
Article

Abstract

The distribution functionP L (s) of the local order parameters in finite blocks of linear dimensionL is studied for Ising lattices of dimensionalityd=2, 3 and 4. Apart from the case where the block is a subsystem of an infinite lattice, also the distribution in finite systems with free [P L (f) (s)] and periodic [P L (p)(s) ] boundary conditions is treated. Above the critical pointT c , these distributions tend for largeL towards the same gaussian distribution centered around zero block magnetization, while belowT c these distributions tend towards two gaussians centered at ±M, whereM is the spontaneous magnetization appearing in the infinite systems. However, belowT c the wings of the distribution at small |s| are distinctly nongaussian, reflecting two-phase coexistence. Hence the distribution functions can be used to obtain the interface tension between ordered phases.

At criticality, the distribution functions tend for largeL towards scaled universal forms, though dependent on the boundary conditions. These scaling functions are estimated from Monte Carlo simulations. For subsystem-blocks, good agreement with previous renormalization group work of Bruce is obtained.

As an application, it is shown that Monte Carlo studies of critical phenomena can be improved in several ways using these distribution functions:(i) standard estimates of order parameter, susceptibility, interface tension are improved(ii) T c can be estimated independent of critical exponent estimates(iii) A Monte Carlo “renormalization group” similar to Nightingale's phenomenological renormalization is proposed, which yields fairly accurate exponent estimates with rather moderate effort(iv) Information on coarse-grained hamiltonians can be gained, which is particularly interesting if the method is extended to more general Hamiltonians.

Keywords

Interface Tension Monte Carlo Simulation Renormalization Group Critical Phenomenon Model Block 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Kampen, N.G. van: Phys. Rev.135, A362 (1964).Google Scholar
  2. 2.
    Kadanoff, L.P.: Physics2, 263 (1966)Google Scholar
  3. 3.
    Langer, J.S.: Ann. Phys.65, 53 (1971); Physica73, 61 (1974)Google Scholar
  4. 4a.
    Langer, J.S., Baron, M., Miller, H.D.: Phys. Rev. A11, 1417 (1975)Google Scholar
  5. 4b.
    Billotet, C., Binder, K.: Z. Physik. B—Condensed Matter32, 195 (1979)Google Scholar
  6. 5.
    Kawasaki, K., Imaeda, T., Gunton, J.D.: In: Studies in Statistical Mechanics. Raveche, H. (ed.). Amsterdam: North-Holland 1980Google Scholar
  7. 6.
    Wilson, K.G.: Phys. Rev. B4, 3174 (1971); B4, 3184 (1971)Google Scholar
  8. 7a.
    Fisher, M.E.: Rev. Mod. Phys.46, 587 (1974)Google Scholar
  9. 7b.
    Domb, C., Green, M.S. (eds.): Phase Transitions and Critical Phenomena. Vol. 6. New York: Academic Press 1976Google Scholar
  10. 8.
    Le Guillou, J.C., Zinn-Justin, J.: Phys. Rev. B21, 3976 (1980)Google Scholar
  11. 9.
    Binder, K. (ed.): Monte Carlo Methods in Statistical Physics Berlin, Heidelberg, New York: Springer 1979Google Scholar
  12. 10.
    Bruce, A.D.: Preprint; see also Bruce, A.D., Schneider, T., Stoll, E.: Phys. Rev. Lett. 43, 1284 (1979)Google Scholar
  13. 11.
    Nightingale, M.P.: Physica83A, 561 (1976); Proc. K. Ned. Acad. v. Wet. B82 (3), 235 (1979)Google Scholar
  14. 12.
    Sneddon, L.: J. Phys. C11, 2823 (1978); C12, 3051 (1979) dos Santos, R.R., Sneddon, L.: PreprintGoogle Scholar
  15. 13a.
    Racz, Z.: Phys. Rev. B21, 4012 (1980)Google Scholar
  16. 13b.
    Derrida, B., Vannimenus, J.: J. Phys. Lett.41, (1980)Google Scholar
  17. 13c.
    Derrida, B.: PreprintGoogle Scholar
  18. 13d.
    Nightingale, M.P., Blöte, H.W.J.: Physica104A, 352 (1980)Google Scholar
  19. 13e.
    Schick, M., Kinzel, W.: (unpublished)Google Scholar
  20. 13f.
    Blöte, H.W.J., Nightingale, M.P., Derrida, B.: PreprintGoogle Scholar
  21. 14.
    Ma, S.-K.: Phys. Rev. Lett.37, 461 (1976)Google Scholar
  22. 15.
    Friedman, Z., Felsteiner, J.: Phys. Rev. B15, 5317 (1977)Google Scholar
  23. 16a.
    Reynolds, P.J., Stanley, H.E., Klein, W.: Phys. Rev. B21, 1223 (1980)Google Scholar
  24. 16b.
    Herrmann, H.J., Stauffer, D., Eschbach, P.D.: Phys. Rev. B23, 422 (1981)Google Scholar
  25. 17.
    Swendsen, R.H.: Phys. Rev. Lett.42, 859 (1979); Phys. Rev. B20, 2080 (1979)Google Scholar
  26. 18.
    Blöte, H.W.J., Swendsen, R.H.: Phys. Rev. B20, 2077 (1979)Google Scholar
  27. 19.
    Blöte, H.W.J., Swendsen, R.H.: Phys. Rev. B22, 4481 (1980)Google Scholar
  28. 20a.
    Blöte, H.W.J., Swendsen, R.H.: Phys. Rev. Lett.43, 737 (1979)Google Scholar
  29. 20b.
    Swendsen, R.H., Krinsky, S.: Phys. Rev. Lett.43, 177 (1979)Google Scholar
  30. 20c.
    Rebbi, C., Swendsen, R.H.: preprint; Novotny, M.A., Landau, D.P., Swendsen, R.H.: PreprintGoogle Scholar
  31. 20d.
    Landau, D.P., Swendsen, R.H.: PreprintGoogle Scholar
  32. 21a.
    Baumgärtner, A.: J. Phys. A13, L38 (1980)Google Scholar
  33. 21b.
    Kremer, K., Baumgärtner, A., Binder, K.: Z. Phys. B—Condensed Matter40, 331 (1981)Google Scholar
  34. 22.
    Reoner, S., Reynolds, P.J.: PreprintGoogle Scholar
  35. 23.
    Shenker, S., Tobochnik, J.: Phys. Rev. B22, 4462 (1980)Google Scholar
  36. 24.
    Landau, D.P.: Phys. Rev. B13, 2297 (1976); B14, 255 (1976)Google Scholar
  37. 25.
    For connecting probability theory and renormalization group ideas, see Jona-Lasinio, G.: I. Nuovo Cimento26B, 99 (1975)Google Scholar
  38. 26.
    Binder, K., Rauch, H.: Z. Phys.219, 201 (1969)Google Scholar
  39. 27.
    Schulman, L.S.: J. Phys. A13, 237 (1980)Google Scholar
  40. 28.
    Binder, K., Kalos, M.H.: J. Stat. Phys.22, 363 (1980)Google Scholar
  41. 29a.
    Fisher, M.E.: In: Critical Phenomena Green, M.S. (ed.), p. 1. New York: Academic Press 1971Google Scholar
  42. 29b.
    Suzuki, M.: Prog. Theor. Phys.58, 1142 (1977)Google Scholar
  43. 30.
    The correlation length ζ. which enters the second argument of\(\tilde P\) could differ from the standard definition of ζ by a constant of order unity. For simplicity this constant is here absorbed in the definition of ζGoogle Scholar
  44. 31.
    Patashinskii, A.Z.: Sov. Phys. JETP26, 1126 (1968)Google Scholar
  45. 32.
    Baker, G.A., Jr.: Phys. Rev. B15, 1552 (1977)Google Scholar
  46. 33.
    Baker, G.A., Jr., Kincaid, J.M.: J. Stat. Phys.24, 469 (1981)Google Scholar
  47. 34a.
    Stell, G.: In: Critical Phenomena. Green, M.S. (ed.), p. 188. New York: Academic Press 1971Google Scholar
  48. 34b.
    Fisher, M.E.: In: Proceedings of the Twenty-Fourth Nobel Symposium on Collective Properties of Physical Systems. Lundquist, B., Lundquist, S. (eds.). p. 16. New York: Academic Press 1973Google Scholar
  49. 35.
    Binder, K.: Thin Solid Films20, 367 (1974)Google Scholar
  50. 36.
    Baumgärtner, A., Binder, K.: J. Chem. Phys. (in press)Google Scholar
  51. 37.
    Onsager, L.: Phys. Rev.65, 117 (1944)Google Scholar
  52. 38.
    Because of the symmetryP L (s)=P L(−s) only data fors>0 are shown. The data in the figures arbitrarily are normalized to\(\int\limits_0^\infty {P_L (s)ds = 1} \) rather than 1/2Google Scholar
  53. 39.
    Domb, C.: In: Phase Transitions and Critical Phenomena. Vol. 3. Domb, C., Green, M.S. (eds.), Vol. 3. New York: Academic Press 1974Google Scholar
  54. 40.
    Gaunt, D.S., Sykes, M.F., McKenzie, S.: J. Phys. A12, 871 (1979)Google Scholar
  55. 41.
    Mouritsen, O.G., Knak Jensen, S.J.: Phys. Rev. B19, 3663 (1979)Google Scholar
  56. 42.
    Brezin, E., Le Guillou, J.C., Zinn-Justin, J.: Phys. Rev. D8, 2418 (1973)Google Scholar
  57. 43.
    Binder, K.: (to be published)Google Scholar
  58. 44.
    Kinzel, W.: Phys. Rev. B19, 4584 (1979)Google Scholar
  59. 45a.
    Racz, Z., Rujan, P.: Z. Phys. B—Condensed Matter28, 287 (1977)Google Scholar
  60. 45b.
    Muto, S., Oguchi, T., Ono, I.: J. Phys. A13, 1799 (1980)Google Scholar
  61. 46.
    Leeuwen, J.M.J. van: In: Phase Transitions and Critical Phenomena. Vol. 6. Domb, C., Green, M.S. (eds.), Vol. 6. New York: Academic Press 1976Google Scholar
  62. 47.
    Müller-Krumbhaar, H.: Z. Phys. B—Condensed Matter (1980)Google Scholar
  63. 48.
    Nickel, B.G., Sharpe, B.: J. Phys. A12, 1819 (1979)Google Scholar
  64. 49a.
    Gaunt, D.S., Sykes, M.F.: J. Phys. A12, L25 (1979)Google Scholar
  65. 49b.
    Rehr, J.J.: J. Phys. A12, L179 (1979)Google Scholar
  66. 49c.
    McKenzie, S.: J. Phys. A12, L185 (1979)Google Scholar
  67. 49d.
    Zinn-Justin, J.: J. Phys. (Paris)40, 969 (1979)Google Scholar

Copyright information

© Springer-Verlag 1981

Authors and Affiliations

  • K. Binder
    • 1
  1. 1.Institut für FestkörperforschungKernforschungsanlage JülichJülich 1Germany

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