Advertisement

Finite size effects at phase transitions

  • K. Binder
Conference paper
Part of the Lecture Notes in Physics book series (LNP, volume 409)

Abstract

For many models of statistical thermodynamics and of lattice gauge theory computer simulation methods have become a valuable tool for the study of critical phenomena, to locate phase transitions, distinguish whether they are of first or second order, and so on. Since simulations always deal with finite systems, analysis of finite size effects by suitable finite size scaling concepts is a key ingredient of such applications. The phenomenological theory of finite size scaling is reviewed with emphasis on the concept of probability distributions of order parameter and/or energy. Attention is also drawn to recent developments concerning anisotropic geometries and anisotropic critical behavior, as well as to crossover phenomena from one “universality class” to another.

Keywords

Ising Model Critical Exponent Linear Dimension Thermodynamic Limit Finite Size 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Binder, K., Heermann, D.W.: Monte Carlo Simulation in Statistical Physics. An Introduction (Springer, Berlin, Heidelberg) (1988)Google Scholar
  2. 2.
    Binder, K.: (ed) Monte Carlo Methods in Statistical Physics, 2nd edn (Springer, Berlin, Heidelberg) (1986)Google Scholar
  3. 3.
    Binder, K.: (ed) Applications of the Monte Carlo Method in Statistical Physics, 2nd edn (Springer, Berlin, Heidelberg) (1987)Google Scholar
  4. 4.
    Binder, K.: (ed) The Monte Carlo Method in Condensed Matter Physics (Springer, Berlin, Heidelberg) (1992)Google Scholar
  5. 5.
    Kalos, M.H., Whitlock, P.: Monte Carlo Methods Vol 1 (Wiley, New York) (1986)Google Scholar
  6. 6.
    Privman, V.: (ed) Finite Size Scaling and Numerical Simulation of Statistical Systems (World Scientific, Singapore) (1990)Google Scholar
  7. 7.
    Stanley, H.E.: An Introduction to Phase Transitions and Critical Phenomena (Oxford University Press, Oxford) (1971)Google Scholar
  8. 8.
    Fisher, M.E.: Rev. Mod. Phys. 46 (1974) 597Google Scholar
  9. 9.
    Fisher, M.E.: In Critical Phenomena, Proc. 1970 E. Fermi Int. School Physics, ed by Green, M.S.: (Academic, New York) (1971) p 1Google Scholar
  10. 10.
    Barber, M.N: In Phase Transitions and Critical Phenomena Vol 8 ed by Domb. C., Lebowitz, J.L.: (Academic, New York) (1983) p 145Google Scholar
  11. 11.
    Binder, K.: Ferroelectrics 73 (1987) 43Google Scholar
  12. 12.
    Cardy, J.L.: (ed) Finite Size Scaling (North-Holland, Amsterdam) (1988)Google Scholar
  13. 13.
    Binder, K.: Ann. Rev. Phys. Chem. 43 (1992) (in press)Google Scholar
  14. 14.
    Stauffer, D.: Introduction to Percolation Theory (Taylor and Francis, London) (1985)Google Scholar
  15. 15.
    Stauffer, D.: Phys. Rep. 54 (1979) 1Google Scholar
  16. 16.
    Kirkpatrick, S.: In III-Condensed Matter, ed by Balian, R., Maynard, R., Toulouse, G., (North-Holland, Amsterdam) (1979) p 321Google Scholar
  17. 17.
    Heermann, D.W., Stauffer D.: Z. Phys. B40 (1980) 133Google Scholar
  18. 18.
    Binder, K.: Z. Physik B43 (1981) 119Google Scholar
  19. 19.
    Stoll, E., Binder, K., Schneider, T.: Phys. Rev. B8 (1973) 3266Google Scholar
  20. 20.
    Binder, K., Landau, D.P.: Phys. Rev. B30 (1984) 1477Google Scholar
  21. 21.
    Binder, K., Rauch, H.: Z. Phys. 219 (1969) 201Google Scholar
  22. 22.
    Stauffer, D., Binder, K.: Z. Phys. B30 (1978 313; B41 (1981) 237Google Scholar
  23. 23.
    Müller-Krumbhaar, H., Binder, K.: Z. Phys. 254 (1972) 269Google Scholar
  24. 24.
    Kasteleyn, P.W., Fortuin, C.M.: J. Phys. Soc. Japan 26, (Suppl.) (1969) 11; Fortuin, C.M., Kasteleyn, P.W.: Physica 57 (1972) 536Google Scholar
  25. 25.
    Potts, R.B.: Proc. Cambridge Philos. Soc. 48 (1952) 106Google Scholar
  26. 26.
    Wu, F.Y.: Rev. Mod. Phys. 54 (1982) 235Google Scholar
  27. 27.
    See also Fisher, M.E.: Physics 3 (1967) 267, and Refs. 28—30 for background on the concept of “physical clusters”.Google Scholar
  28. 28.
    Binder, K.: Ann. Phys. 98 (1976) 390Google Scholar
  29. 29.
    Coniglio, A., Klein, W.: J. Phys. A13 (1980) 2775Google Scholar
  30. 30.
    Hu, C.K.: Phys. Rev. B29 (1984) 5103Google Scholar
  31. 31.
    Swendsen, R.H., Wang, J.S.: Phys. Rev. Lett. 58 (1987) 86Google Scholar
  32. 32.
    For a review, see: Swendsen, R.H., Wang, J.S., Ferrenberg, A.M.: in Ref. 4, Chap.4, and Ref. 33.Google Scholar
  33. 33.
    Wolff, U.: in this volumeGoogle Scholar
  34. 34.
    Wolff, U.: Phys. Rev. Lett. 60 (1988) 1461; Nucl. Phys. B300 [FS22] (1988) 501Google Scholar
  35. 35.
    Edward, R.G., Sokal, A.D.: Phys. Rev. D38 (1988) 2009; D40 (1989) 1374Google Scholar
  36. 36.
    Wolff, U.: Phys. Rev. Lett. 62 (1989) 361; Phys. Lett. B228 (1989) 379; Nucl. Phys. B322 (1989) 759Google Scholar
  37. 37.
    Hu, C.K., Mak, S.K.: Phys. Rev. B40 (1989) 5007Google Scholar
  38. 38.
    D'Onorio, De Meo M., Heermann, D.W., Binder, K.: J. Stat. Phys. 60 (1990) 585Google Scholar
  39. 39.
    Hasenbusch, M.: Nucl. Phys. B333 (1990) 581; Hasenbusch, M., Meyer, S., Phys. Lett. B241 (1990) 238; Phys. Rev. Lett. 66 (1991) 530Google Scholar
  40. 40.
    Wang, J.S., Selke, W., Dotsenko, V.S., Andreichenko, V.B.: Europhys. Lett. 11 (1990) 30; Physica A164 (1990) 221Google Scholar
  41. 41.
    See Hohenberg, P.C., Halperin, B.I.: Rev. Mod. Phys. 49 (1977) 435, for a general review of critical slowing down.Google Scholar
  42. 42.
    Heermann, D.W., Burkitt, A.N.: Physica A162 (1990) 210Google Scholar
  43. 43.
    Li, X.J., Sokal, A.D.: Phys. Rev. Lett. 63 (1989) 827; 67 (1991) 1482Google Scholar
  44. 44.
    Wang, J.S.: Physica A164 (1991) 240Google Scholar
  45. 45.
    Margolina, A., Herrmann, H.J.: Phys. Lett. 104A (1984) 295Google Scholar
  46. 46.
    Onsager, L.: Phys. Rev. 65 (1944) 117Google Scholar
  47. 47.
    Binder, K., Nauenberg, M., Privman, V., Young, A.P.: Phys. Rev. B31 (1985) 1498Google Scholar
  48. 48.
    Binder, K., Wang, J.S.: J. Stat. Phys. 55 (1989) 87; Binder, K.: in Ref 6, p 173Google Scholar
  49. 49.
    Binder, K.: Rept. Progr. Phys. 50 (1987) 783Google Scholar
  50. 50.
    Challa, M.S.S., Landau, D.P., Binder, K.: Phys. Rev. B34 (1986) 1841Google Scholar
  51. 51.
    Landau, L.D., Lifshitz, E.M.: Statistical Physics (Pergamon, Oxford) (1958)Google Scholar
  52. 52.
    Binder, K.: Phys. Rev. A25 (1982) 1699Google Scholar
  53. 53.
    Kaski, K., Binder, K., Gunton, J.D.: Phys. Rev. B29 (1984) 3996Google Scholar
  54. 54.
    Milchev, A., Heermann, D.W., Binder, K.: J. Stat. Phys. 44 (1986) 749Google Scholar
  55. 55.
    Landau, D.P.: J. Magn. Magn. Mater. 31-34 (1983) 11Google Scholar
  56. 56.
    Murty, C.S.S., Landau, D.P.: J. Appl. Phys. 55 (1984) 2429; Challa, M.S.S., Landau, D.P.: Phys. Rev. B33 (1986) 437Google Scholar
  57. 57.
    Landau, D.P., Binder, K.: Phys. Rev. B31 (1985) 5946Google Scholar
  58. 58.
    Binder, K.: Z. Phys. B61 (1985) 13Google Scholar
  59. 59.
    Bhatt, R.N., Young, A.P.: Phys. Rev. Lett. 54 (1985) 925; Phys. Rev. B37 (1988) 5606Google Scholar
  60. 60.
    Scheucher, M.: Reger J D, preprintGoogle Scholar
  61. 61.
    Dünweg, B., Binder, K.: Phys. Rev. B36 (1987) 6935Google Scholar
  62. 62.
    Schmid, F., Binder, K.: (in press) J. Phys. Condensed MatterGoogle Scholar
  63. 63.
    Sariban, A., Binder, K.: J. Chem. Phys. 86 (1987) 5859; Macromolecules 21 (1988) 711Google Scholar
  64. 64.
    Deutsch, H.P., Binder, K.: Europhys. Lett., in press; Macromolecules, submitted; Deutsch, H.P.: J. Stat. Phys., submittedGoogle Scholar
  65. 65.
    Peczak, P., Ferrenberg, A.M., Landau, D.P.: Phys. Rev. B43 (1991) 6087Google Scholar
  66. 66.
    Ferrenberg, A.M., Landau, D.P.: Phys. Rev. B44 (1991) 5081Google Scholar
  67. 67.
    Rovere, M., Heermann, D.W., Binder, K.: J. Phys. Condensed Matter 2 (1990) 7009Google Scholar
  68. 68.
    Novotny, R.M., Binder, K.: Z. Phys. B77 (1989) 287Google Scholar
  69. 69.
    Wansleben, S.: Comp. Phys. Commun. 43 (1987) 9Google Scholar
  70. 70.
    Ferrenberg, A.M., Swendsen, R.H.: Phys. Rev. Lett. 61 (1988) 2635; 63 (1989) 1195Google Scholar
  71. 71.
    Pawley, G.S., Swendsen, R.H., Wallace, D.J., Wilson, K.G.: Phys. Rev. B29 (1984) 4030Google Scholar
  72. 72.
    Blöte, H.W.J., De Bruin, J., Compagner, A., Croolkewit, J.H., Fonk, Y.T.J.C., Heringa, J.R., Hoogland, A., Van Willigen A.L.: Europhys. Lett. 10 (1989) 105Google Scholar
  73. 73.
    Adler, J.: J. Phys. A16 (1983) 3585Google Scholar
  74. 74.
    Liu, A.J., Fisher, M.E.: Physica A165 (1989) 35Google Scholar
  75. 75.
    Landau, D.P.: Phys. Rev. B13 (1976) 2997; Phys. Rev. B14 (1976) 255Google Scholar
  76. 76.
    Binder, K.: Thin Solid Films 20 (1974) 367Google Scholar
  77. 77.
    Binder, K., Landau, D.P.: Phys. Rev. B21 (1980) 1941Google Scholar
  78. 78.
    Landau, D.P.: Phys. Rev. B27 (1983) 5604Google Scholar
  79. 79.
    Krinsky, S., Mukamel, D.: Phys. Rev. B16 (1977) 2313Google Scholar
  80. 80.
    Swendsen, H., Krinsky, S.: Phys. Rev. Lett. 43 (1979) 177Google Scholar
  81. 81.
    Oitmaa, J.: J. Phys. A14 (1981) 1159Google Scholar
  82. 82.
    Nightingale, M.P.: Phys. Lett. A59 (1977) 486Google Scholar
  83. 83.
    Nauenberg, M.P., Nienhuis, B.: Phys. Rev. Lett. 33 (1974) 944Google Scholar
  84. 84.
    Le Guillou, J.C., Zinn-Justin, J.: Phys. Rev. Lett. 39 (1977) 95; Phys. Rev. B21 (1980) 2976Google Scholar
  85. 85.
    Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N., Teller, A.H., Teller, E.: J. Chem. Phys. 21 (1953) 1087Google Scholar
  86. 86.
    Müller-Krumbhaar, H., Binder, K.: J. Stat. Phys. 8 (1973) 1Google Scholar
  87. 87.
    Binder, K.: In Phase Transitions and Critical Phenomena, ed by Domb, C., Green, M.S.: Vol 5b (Academic, New York) (1976) p 1Google Scholar
  88. 88.
    Glauber, R.J.: J. Math. Phys. 4 (1963) 294Google Scholar
  89. 89.
    Hohenberg, P.C., Halperin, B.I.: Rev. Mod. Phys. 49 (1977) 435Google Scholar
  90. 90.
    Wansleben, S., Landau, D.P.: Phys. Rev. B43 (1991) 6006Google Scholar
  91. 91.
    Wansleben, S., Landau, D.P.: J. Appl. Phys. 61 (1987) 3968Google Scholar
  92. 92.
    Ferrenberg, A.M., Landau, D.P., Binder, K.: J. Stat. Phys. 63 (1991) 867Google Scholar
  93. 93.
    Alexander, H.W.:Elements of Mathematical Statistics (Wiley, New York) (1961)Google Scholar
  94. 94.
    Miyashita, S., Takano, H.: Progr. Theor. Phys. 73 (1985) 1122Google Scholar
  95. 95.
    Racz, Z.: Phys. Rev. B13 (1976) 263; Fisher, M.E., Rac, Z.: Phys. Rev. B13 (1976) 5039Google Scholar
  96. 96.
    Kretschmer, R., Binder, K., Stauffer, D.: J. Stat. Phys. 15 (1976) 267Google Scholar
  97. 97.
    For a review, see Binder, K., Heermann, D.W., Milchev, A., Sadiq, A.: In Heidelberg Colloquium on Glassy Dynamics, ed by van Hemmen, L., Morgenstern, I.: Lect. Notes Phys., Vol 275 (Springer, Berlin, Heidelberg) (1987) p 154Google Scholar
  98. 98.
    Sadiq, A., Binder, K.:J. Stat. Phys. 35 (1984) 517Google Scholar
  99. 99.
    Stauffer, D., Binder, K.: Z. Physik B30 (1978) 313; 41 (1981) 237Google Scholar
  100. 100.
    Binder, K.: Z. Physik B45 (1981) 61Google Scholar
  101. 101.
    Binder, K.: Z. Physik B61 (1985) 13Google Scholar
  102. 102.
    Imry, Y.: Phys. Rev. B21 (1980) 2042Google Scholar
  103. 103.
    Fisher, M.E., Berker, A.N.: Phys. Rev. B26 (1982) 2507Google Scholar
  104. 104.
    Blöte, H.W.J., Nightingale, M.P.: Physica A112 (1982) 405Google Scholar
  105. 105.
    Cardy, J.L., Nightingale, M.P.: Phys. Rev. B27 (1983) 4256Google Scholar
  106. 106.
    Privman V, Fisher M E J. Stat. Phys. 33 (1983) 285Google Scholar
  107. 107.
    Fisher, M.E., Privman, V.: Phys. Rev. B32 (1985) 447Google Scholar
  108. 108.
    Privman, V., Rudnick, J.: J. Stat. Phys. 60 (1990) 551Google Scholar
  109. 109.
    Borgs, C., Kotecky, R.: J. Stat. Phys. 61 (1990) 79Google Scholar
  110. 110.
    Borgs, C., Kotecky, R., Miracle-Sole, S.: J. Stat. Phys. 62 (1991) 529Google Scholar
  111. 111.
    Lee, J., Kosterlitz, J.M.: Phys. Rev. B43 (1991) 3265Google Scholar
  112. 112.
    Lee, J., Kosterlitz, J.M.: Phys. Rev. Lett. 65 (1990) 137Google Scholar
  113. 113.
    Fukugita, M., Mino, M., Okawa, M., Ukawa, A.: J. Stat. Phys. 59 (1990) 1397; J. Phys. A23 (1990) L561Google Scholar
  114. 114.
    Peczak, P., Landau, D.P.: Phys. Rev. B39 (1990) 11932Google Scholar
  115. 115.
    Challa, M.S.S., Hetherington, J.H.: Phys. Rev. Lett. 60 (1988) 77; Phys. Rev. A38 (1988) 6324Google Scholar
  116. 116.
    Billoire, A., Gupta, S., Irbäck, A., Lacaze, R., Morel, A., Peterson, B.: Nucl. Phys. B258 (1991) 231; Phys. Rev. B42 (1990) 6743Google Scholar
  117. 117.
    Billoire, A., Lacaze, R., Morel, A.: preprint SPhT-91/122Google Scholar
  118. 118.
    Berg, B.A., Neuhaus, T.: preprintsGoogle Scholar
  119. 119.
    Borgs, C., Janke, W.: preprint FUB-HEP 6/91Google Scholar
  120. 120.
    Cahn, J.W., Hilliard, J.E.: J. Chem. Phys. 28 (1959) 258; 31, 688Google Scholar
  121. 121.
    Lebowitz, J.L., Penrose, 0.: J. Math. Phys. 7 (1966) 98Google Scholar
  122. 122.
    Langer, J.S.: Ann. Phys. (N.Y.) 41 (1967) 108Google Scholar
  123. 123.
    Langer, J.S.: Ann. Phys. (N.Y.) 54 (1969) 258Google Scholar
  124. 124.
    Binder, K., Müller-Krumbhaar, H.: Phys. Rev. B9 (1974) 2328Google Scholar
  125. 125.
    Binder, K., Staufer, D.: Advances of Physics 25 (1976) 343Google Scholar
  126. 126.
    Binder, K.: J. Phys. (Paris) 41, C4–51 (1980)Google Scholar
  127. 127.
    Heermann D W, Klein W, Staufer D (1982) Phys. Rev. Lett. 49, 1262Google Scholar
  128. 128.
    Heermann, D.W., Klein, W.: Phys. Rev. Lett. 50 (1983) 1962; Phys. Rev. B27 (1983) 1732Google Scholar
  129. 129.
    Heermann, D.W., Coniglio, A.W., Klein, W., Stauffer, D.: J. Stat. Phys. 36 (1984) 447Google Scholar
  130. 130.
    Klein, W., Unger, C.: Phys. Rev. B28 (1983) 445Google Scholar
  131. 131.
    Binder, K.: Phys. Rev. A29 (1984) 341Google Scholar
  132. 132.
    Paul, W., Heermann, D.W., Binder, K.: J. Phys. A22 (1989) 3325Google Scholar
  133. 133.
    Challa, M.S.S., Landau, D.P., Binder, K.: Phase Transitions 24-26 (1990) 343Google Scholar
  134. 134.
    Hüller, A.:Univ. of Erlangen preprintGoogle Scholar
  135. 135.
    Baumann, B.: Nucl. Phys. B285 (1987) 391Google Scholar
  136. 136.
    Berg, B.A., Hansmann, U., Neuhaus, T.: preprint FSU-SCRI-91-125Google Scholar
  137. 137.
    Yang, C.N.: Phys. Rev. 85 (1951) 808Google Scholar
  138. 138.
    Binder, K., Landau, D.P.: Surface Science 108 (1981) 503; Binder, K., Landau, D.P.: (1989) in Molecule-Surface Interaction Lawley, K.P., ed) p 91 (J. Wiley, New York)Google Scholar
  139. 139.
    Fisher, M.E., Nakanishi, H.: J. Chem. Phys. 75 (1981) 5857; 78 (1983) 3279Google Scholar
  140. 140.
    Albano, E.V., Binder, K., Heermann, D.W., Paul, W.: J. Chem. Phys. 91 (1989) 3700Google Scholar
  141. 141.
    Binder, K., Landau, D.P.: J. Chem. Phys. (1992) (in press)Google Scholar
  142. 142.
    Kihara, T., Midzuno, Y., Shizume, T.: J. Phys. Soc. Jpn. 9 (1954) 681; Baxter, R.J.: J. Phys. C6 (1973) L445Google Scholar
  143. 143.
    Fisher, M.E., Privman, V.: Commun. Math. Phys. 103 527Google Scholar
  144. 144.
    Singh, S., Pathria, R.K.: Phys. Rev. B37 (1988) 7806Google Scholar
  145. 145.
    Hasenfratz, P., Leutwyler, H.: Nucl. Phys. B343 (1990) 241Google Scholar
  146. 146.
    Chen, K., Landau, D.P.: University of Georgia preprintGoogle Scholar
  147. 147.
    Zinn-Justin, J., Brézin, E.: Nucl. Phys. B257 [FS14] (1985) 867Google Scholar
  148. 148.
    Albano, E.V., Binder, K., Heermann, D.W., Paul, W.: Surface Sci. 223 (1989) 151Google Scholar
  149. 149.
    Albano, E.V., Binder, K., Heermann, D.W., Paul, W.: Z. Physik B77 (1989) 445Google Scholar
  150. 150.
    Cardy, J.L.: in Phase Transitions and Critical Phenomena, Vol 11 (Domb, C., Lebowitz, J.L., eds) p.55 (1987) (Academic Press, New York)Google Scholar
  151. 151.
    Hornreich, R.M., Luban, M., Shtrikman, S.: Phys. Rev. Lett. 35 (1975) 1678Google Scholar
  152. 152.
    Janssen, H.K., Schmittmann, B.: Z. Phys. B64 (1986) 503Google Scholar
  153. 153.
    Leung, K.T., Cardy, J.: J. Stat. Phys. 44 (1986) 567Google Scholar
  154. 154.
    Leung, K.T.: Phys. Rev. Lett. 66 (1991) 453Google Scholar
  155. 155.
    Lawrie, I.D., Sarbach, S.: In Phase Transitions and Critical Phenomena, Vol. 9 (1984) (ed by Domb, C. and Lebowitz, J.L.) p 1 (Academic Press, London)Google Scholar
  156. 156.
    de Gennes, P.G.: Scaling Concepts in Polymer Physics (Cornell University Press, Ithaca) (1979)Google Scholar
  157. 157.
    Binder, K., Deutsch, H.P.: Univ. of Mainz preprintGoogle Scholar
  158. 158.
    Riedel, E., Wegner, F.J.: Z. Phys. 225 (1969) 195Google Scholar
  159. 159.
    Ginzburg, V.L.: Sov.Phys. Solid State 2 (1960) 1824Google Scholar
  160. 160.
    For a review, see Nightingale, M.P.: in Ref. 6, p.287, and Ref.161Google Scholar
  161. 161.
    Henkel, M.: in Ref 6, p 353Google Scholar
  162. 162.
    Bhanot, G. V.: in Ref 6, p 435Google Scholar
  163. 163.
    Rudnick, J.: in Ref 6, p 141Google Scholar
  164. 164.
    Jasnow, D.: in Ref 6, p 99Google Scholar
  165. 165.
    Eisenriegler, E.: Z. Phys. B61 (1985) 299; Nemirowsky, A.M., Freed, K.F.: J. Phys. A19 (1985) L319; J. Phys. A19 (1985) 586; Eisenriegler, E., Tomaschitz, R.: Phys. Rev. B35 (1987) 4876Google Scholar
  166. 166.
    Dohn, V.: Z. Phys. B75 (1989) 109Google Scholar
  167. 167.
    For a review, see Mon, K.K.: in. Ref 6, p 261, and Ref 166Google Scholar
  168. 168.
    Privman, V.: in Ref 6, p 1Google Scholar
  169. 169.
    Krech, M., Dietrich, S.: Phys. Rev. Lett. 66 (1991) 345Google Scholar
  170. 170.
    Privman, V.: Phys. Rev. B38 (1988) 9261Google Scholar
  171. 171.
    Fisher, M.E., de Gennes, P.G.: C. R. Acad. Sci (Paris) Ser. B287 (1978) 207Google Scholar
  172. 172.
    Nightingale, M.P., Blöte, H.W.J.: Phys. Rev. Lett. 60 (1988) 1562Google Scholar
  173. 173.
    Albano, E.V., Binder, K., Heermann, D.W., Paul, W.: J. Stat. Phys. 61 (1990) 161Google Scholar
  174. 174.
    Parry, A.O., Evans, R.: Phys. Rev. Lett. 64 (1990) 439Google Scholar
  175. 175.
    Kroll, D.M., Gompper, G.: Phys. Rev. B39 (1989) 433Google Scholar
  176. 176.
    Nicolaides, D., Bruce, A.D.: J. Phys. A21 (1988) 233Google Scholar
  177. 177.
    Bruce, A.D.: J. Phys. A18 (1985) L873Google Scholar
  178. 178.
    Welding, N.B., Bruce, A.D.: Univ. of Edinburgh preprintGoogle Scholar
  179. 179.
    Desai, R.C., Heermann, D.W., Binder, K.: Phys. Rev. Lett. 49 (1982) 708Google Scholar
  180. 180.
    Mon, K.K., Binder, L.: Univ. of Mainz preprintGoogle Scholar

Copyright information

© Springer-Verlag 1992

Authors and Affiliations

  • K. Binder
    • 1
  1. 1.Institut für PhysikJohannes Gutenberg Universität MainzMainz

Personalised recommendations