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Finite size effects at thermally-driven first order phase transitions: A phenomenological theory of the order parameter distribution

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Zeitschrift für Physik B Condensed Matter

Abstract

We consider the rounding and shifting of a firstorder transition in a finited-dimensional hypercubicL d geometry,L being the linear dimension of the system, and surface effects are avoided by periodic boundary conditions. We assume that upon lowering the temperature the system discontinuously goes to one ofq ordered states, such as it e.g. happens for the Potts model ind=3 forq≧3, with the correlation length ξ of order parameter fluctuation staying finite at the transition. We then describe each of theseq ordered phases and the disordered phase forL≫ξ by a properly weighted Gaussian. From this phenomenological ansatz for the total distribution of the order parameter, all moments of interest are calculated straight-forwardly. In particular, it is shown that forL exceeding a characteristic minimum sizeL min the forthorder cumulantg L (T) exhibits a minimum atT min>T c, withT min−T c∝L −d and the value of the cumulant and the minimum (g(T min)) behaving asg(T min)∝L −d. All cumulantsg L (T) forL≫ξ approximately intersect at a common crossing pointT cross∝L −2d, with a universal valueg(T cross)=1−n/2q, wheren is the order parameter dimensionality. By searching for such a behavior in numerical simulation data, the first order character of a phase transition can be asserted. The usefulness of this approach is shown using data for theq=3,d=3 Potts ferromagnet.

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References

  1. Binder, K. (ed.): Monte Carlo methods in statistical physics Berlin, Heidelberg, New York: Springer 1979

    Google Scholar 

  2. Binder, K. (ed.): Applications of the Monte Carlo method in statistical physics. Berlin, Heidelberg, New York: Springer 1984.

    Google Scholar 

  3. Heermann, D.W.: Computer simulation methods in theoretical physics. Berlin, Heidelberg, New York: Springer 1986

    Google Scholar 

  4. Hoover, W.G.: Molecular dynamics. Heidelberg, Berlin, New York: Springer 1986

    Google Scholar 

  5. Binder, K., Heermann, D.W.: Monte Carlo simulation in statistical physics: an introduction. Berlin, Heidelberg, New York: Springer 1988

    Google Scholar 

  6. Binder, K. (ed.): The Monte Carlo method in condensed matter physics. Berlin, Heidelberg, New York: Springer 1992

    Google Scholar 

  7. Stanley, H.E.: An introduction to phase transitions and critical phenomena. Oxford: Oxford University Press 1971

    Google Scholar 

  8. Binder, K.: Rep. Progr. Phys.50, 783 (1987)

    Google Scholar 

  9. Fisher, M.E.: In: Green, M.S. (ed.) Critical phenomena, Proc 1970 Enrico Fermi International School on Physics, p. 1. New York: Academic Press 1971

    Google Scholar 

  10. Barber, M.N.: In: Domb, C., Lebowitz, J.L. (eds.) Phase transitions and critical phenomena, Vol. 8, p. 145. New York: Academic Press 1983

    Google Scholar 

  11. Binder, K.: Ferroelectrics73, 43 (1987)

    Google Scholar 

  12. Cardy, J.L. (ed.) Finite size scaling. Amsterdam: North-Heiland 1988

    Google Scholar 

  13. Privman, V. (ed.): Finite size scaling and numerical simulatic of statistical systems. Singapore: World Scientific 1990

    Google Scholar 

  14. Binder, K.: Ann. Rev. Phys. Chem.43, 33 (1992)

    Google Scholar 

  15. Binder, K.: In: Lang, C.B., Gausterer, A. (eds.) Computation methods in field theory, p. 57. Berlin, Heidelberg, New York: Springer 1992

    Google Scholar 

  16. Binder, K., Rauch, H.: Z. Phys.219, 201 (1969); Binder, K.: Z. Angew. Phys.30, 51 (1970); Binder, K.: Physica62, 508 (1972)

    Google Scholar 

  17. Binder, K.: Z. Phys. B43, 119 (1981); Phys. Rev. Lett.47, 69 (1981)

    Google Scholar 

  18. Imry, Y.: Phys. Rev. B21, 2042 (1980)

    Google Scholar 

  19. Fisher, M.E., Berker, A.N.: Phys. Rev. B26, 2507 (1982)

    Google Scholar 

  20. Blöte, H.W.J., Nightingale, M.P.: Physics A112, 405 (1982); Cardy, J.L., Nightingale, M.P.: Phys. Rev. B27, 4256 (1983)

    Google Scholar 

  21. Privman, V., Fisher, M.E.: J. Stat. Phys.33, 385 (1983)

    Google Scholar 

  22. Binder, K., Landau, D.P.: Phys. Rev. B30, 1477 (1984)

    Google Scholar 

  23. Fisher, M.E., Privman, V.: Phys. Rev. B32, 447 (1985)

    Google Scholar 

  24. Privman, V., Rudnick, J.E.: J. Stat. Phys.60, 551 (1990)

    Google Scholar 

  25. Challa, M.S.S., Landau, D.P., Binder, K.: Phys. Rev. B34, 1841 (1986)

    Google Scholar 

  26. Borgs, C., Kotecky, R.: J. Stat. Phys.61, 79 (1990)

    Google Scholar 

  27. Borgs, C., Kotecky, R., Miracle-Sole, S.: J. Stat. Phys.62, 529 (1991)

    Google Scholar 

  28. Gavai, R.V., Karsch, F., Petersson, B.: Nucl. Phys. B322, 738 (1989)

    Google Scholar 

  29. Fukugita, M., Okawa, M.: Phys. Rev. Lett.63, 13 (1989)

    Google Scholar 

  30. Fukugita, M., Mino, H., Okawa, M., Ukawa, A.: J. Phys. A23, L561 (1990)

    Google Scholar 

  31. Billoire, A., Gupta, S., Irback, A., Lacare, R., Morel, A., Petersson, B.: Phys. Rev. B42, 6743 (1990); Nucl. Phys. B258, 231 (1990)

    Google Scholar 

  32. Peczak, P., Landau, D.P.: Phys. Rev. B39, 11932 (1990)

    Google Scholar 

  33. Lee, J., Kosterlitz, J.M.: Phys. Rev. Lett.65, 137 (1990); Phys. Rev. B43, 3265 (1991)

    Google Scholar 

  34. Billoire, A., Lacare, R., Morel, A.: Nucl. Phys. B370, 773 (1992)

    Google Scholar 

  35. Berg, B.A., Neuhaus, T.: Phys. Lett. L3267, 249 (1991); Phys. Rev. Lett.68, 9 (1992)

    Google Scholar 

  36. Borgs, C., Janke, W.: Phys. Rev. Lett.68, 1738 (1992)

    Google Scholar 

  37. Kikuchi, M., Okabe, Y.: J. Phys. Soc. Jpn.61, 3503 (1992)

    Google Scholar 

  38. Fukugita, M., Miro, H., Okawa, M., Ukawa, A.: J. Stat. Phys.59, 1397 (1990)

    Google Scholar 

  39. Binder, K., Vollmayr, K., Deutsch, H.-P., Reger, J.D., Scheucher, M., Landau, D.P.: Int. J. Mod. Phys. C3, 1025 (1992)

    Google Scholar 

  40. Landau, L.D., Lifshitz, E.M.: Statistical physics. London: Pergamon Press 1984

    Google Scholar 

  41. Potts, R.B.: Proc. Comb. Philos. Soc.48, 106 (1952)

    Google Scholar 

  42. Wu, F.Y.: Rev. Mod. Phys.54, 235 (1982)

    Google Scholar 

  43. Lawrie, I.D., Sarbach, S.: In: Domb, C., Lebowitz, J.L. (eds.) Phase transitions and critical phenomena. Vol. 9, p. 1. New York: Academic Press 1984

    Google Scholar 

  44. Morris, J.R., Gooding, R.J.: Phys. Rev. B43, 6057 (1991); J. Stat. Phys.67, 471 (1992), R.J. Gooding: Scri. Met.25, 105 (1991)

    Google Scholar 

  45. Vollmayr, K.: Diplomarbeit (Universität Mainz, 1992, unpublished)

  46. Stephanov, M.A., Tsypin, M.M.: Nucl. Phys. B366, 420 (1991)

    Google Scholar 

  47. Fisher, M.E.: Rev. Mod. Phys.46, 597 (1974)

    Google Scholar 

  48. Herrmann, H.J.: Z. Phys. B35, 1736 (1979); Knak-Jensen, S.J., Mouritsen, O.G.: Phys. Rev. Lett.43, 1736 (1979)

    Google Scholar 

  49. Blöte, H.W., Swendsen, R.H.: Phys. Rev. Lett.43, 799 (1979)

    Google Scholar 

  50. Milchev, A., Binder, K., Heermann, D.W.: Z. Phys. B63, 521 (1986)

    Google Scholar 

  51. Ferrenberg, A.M., Landau, D.P.: Phys. Rev. B44, 5881 (1991)

    Google Scholar 

  52. Scheucher, M., Reger, J.D., Binder, K., Young, A.P.: Phys. Rev. B42, 6881 (1990)

    Google Scholar 

  53. Wilson, W.G., Vause, C.A.: Phys. Rev. B36, 587 (1987)

    Google Scholar 

  54. Klümper, A., Schadschneider, A., Zittartz, J.: Z. Phys. B76, 247 (1989)

    Google Scholar 

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Authors and Affiliations

  1. Institut für Physik, Johannes Gutenberg Universität Mainz, Staudinger Weg 7, W-6500, Mainz, Germany

    Katharina Vollmayr, Joseph D. Reger, Manfred Scheucher & Kurt Binder

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  1. Katharina Vollmayr
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  2. Joseph D. Reger
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  3. Manfred Scheucher
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  4. Kurt Binder
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Vollmayr, K., Reger, J.D., Scheucher, M. et al. Finite size effects at thermally-driven first order phase transitions: A phenomenological theory of the order parameter distribution. Z. Physik B - Condensed Matter 91, 113–125 (1993). https://doi.org/10.1007/BF01316713

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  • DOI: https://doi.org/10.1007/BF01316713

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