Monte Carlo Methods

Chapter

Abstract

Nature is composed of gross assemblies of huge numbers of atoms and molecules showing a wide variety of phenomena according to the way how they are assembling. The macroscopic behaviors of such systems are rather different from the microscopic laws in the world of atoms and molecules. For example, in the usual cases of macroscopic systems, the motion of the atoms and molecules can be regarded simply as heat. That is, the average kinetic energy of each atom and molecule in a macroscopic system is equal to a quantity measured as the temperature. The other contributor to macroscopic behavior is the cooperative motion of atoms and molecules (or sometimes electrons). Since atoms and molecules interact with each other, their macroscopic assemblies can have cooperative motions. Many examples can be seen in our daily life: spring or rubber elasticity, magnetization in permanent magnets, shape memory alloys, liquid flows, surface tension of liquids, swelling of polymers by water absorption, etc. Sometimes such cooperative motions are frozen as the temperature decreases. In this case, the states which have been realized at higher temperatures become unstable.

References

  1. 1.
    H.E. Stanley, Introduction to Phase Transitions and Critical Phenomena (Oxford University Press, Oxford, 1971)Google Scholar
  2. 2.
    K. Binder (ed.), The Monte Carlo Method in Condensed Matter Physics (Springer, Berlin, 1992)Google Scholar
  3. 3.
    K. Binder, D.W. Heermann, Monte Carlo Simulation in Statistical Physics (Springer, Berlin, 1988)Google Scholar
  4. 4.
    K. Binder (ed.), Monte Carlo Methods in Statistical Physics (Springer, Berlin, 1979)Google Scholar
  5. 5.
    L.D. Landau, E.M. Lifshitz, Statistical Physics (Pergamon, Oxford, 1959, 1980)Google Scholar
  6. 6.
    D.E. Knuth, The Art of Computer Programming, vol. 2, ed. by M.A. Harrison (Addison-Wesley, Reading, Massachusetts, 1981). Chapter 3Google Scholar
  7. 7.
    N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.M. Teller, E. Teller, J. Chem. Phys. 21, 1087 (1953)Google Scholar
  8. 8.
    R.J. Glauber, J. Math. Phys. 4, 294 (1963)Google Scholar
  9. 9.
    K. Kawasaki, in Phase Transitions and Critical Phenomena, vol. 2, ed. by C. Domb, M.S Green (Academic Press, London, 1972). Chapter 11Google Scholar
  10. 10.
    B.I. Halperin, P.C. Hohenberg, S.-K. Ma, Phys. Rev. Lett. 29, 1548 (1972); P.C. Hohenberg, B.I. Halperin. Rev. Mod. Phys. 49, 435 (1977)Google Scholar
  11. 11.
    J.M. Ziman, Models of Disorder (Cambridge University Press, Cambridge, 1979)Google Scholar
  12. 12.
    R. Evans, in Liquids at Interfaces, ed. by J. Charvolin, J.F. Joanny, J. Zinn-Justin (North-Holland, Amsterdam, 1990)Google Scholar
  13. 13.
    D. Frenkel, B. Smit, Understanding Molecular Simulation: From Algorithms to Applications (Academic Press, San Diego, 1996)Google Scholar
  14. 14.
    F. Yonezawa, in Solid State Physics, vol. 45, ed. by H. Ehrenreich, D. Turnbull (Academic Press, San Diego, 1991), p. 179Google Scholar
  15. 15.
    R.H. Jones, W.W. Gerberich (eds.), Modeling Environmental Effects on Crack Growth Processes (Metallurgical Society of the AIME, Warrendale, Pennsylvania, 1986)Google Scholar
  16. 16.
    W. Eckstein, Computer Simulation of Ion-Solid Interactions. Materials Science Series, vol. 10 (Springer, Berlin, 1991). Chapter 4Google Scholar
  17. 17.
    A.R. Leach, Molecular Modeling (Longman, Harlow, 1996)Google Scholar
  18. 18.
    L. Monnerie, U.W. Suter, Atomistic Modeling of Physical Properties, vol. 116 (Advances in Polymer Science (Springer, Berlin, 1994)Google Scholar
  19. 19.
    A. Gavezzotti, Theoretical Aspects and Computer Modeling of the Molecular Solid State (Wiley, New York, 1997)Google Scholar
  20. 20.
    M.-C. Desjonquères, D. Spanjaard, Concepts in Surface Physics, 2nd edn. (Springer, Berlin, 1995)Google Scholar
  21. 21.
    M.N. Rosenbluth, A.W. Rosenbluth, J. Chem. Phys. 22, 881 (1954)Google Scholar
  22. 22.
    B.J. Alder, T.E. Wainwright, J. Chem. Phys. 27, 1208 (1957)Google Scholar
  23. 23.
    W.W. Wood, J.D. Jacobson, J. Chem. Phys. 27, 1207 (1957)Google Scholar
  24. 24.
    W.W. Wood, F.R. Parker, J. Chem. Phys. 27, 720 (1957)Google Scholar
  25. 25.
    T.R. McDonald, K. Singer, Faraday Disc. 43, 40 (1967)Google Scholar
  26. 26.
    A.M. Ferrenberg, R.H. Swendsen, Phys. Rev. Lett. 63, 1195 (1989)Google Scholar
  27. 27.
    S. Kumer, D. Bouzida, R.H. Swendsen, P.A. Kollman, J.M. Rosenberg, J. Comput. Chem. 13, 1011 (1992)Google Scholar
  28. 28.
    E.M. Boczko, C.L. Brooks, III, J. Phys. Chem. 97, 4509 (1993); Science 269, 393 (1995)Google Scholar
  29. 29.
    D.J. Adams, Mol. Phys. 28, 1241 (1974)Google Scholar
  30. 30.
    L.A. Rowley, D. Nicholson, N.G. Parsonage, J. Comput. Phys. 17, 401 (1975)Google Scholar
  31. 31.
    A.Z. Panagiotopoulos, Mol. Phys. 61, 813 (1987); 62, 701 (1987)Google Scholar
  32. 32.
    B. Widom, J. Chem. Phys. 39, 2802 (1963)Google Scholar
  33. 33.
    J.G. Kirkwood, J. Chem. Phys. 3, 300 (1935)Google Scholar
  34. 34.
    D. Frenkel, A.J.C. Ladd, J. Chem. Phys. 81, 3188 (1984)Google Scholar
  35. 35.
    C.H. Bernett, J. Comput. Phys. 22, 245 (1976)Google Scholar
  36. 36.
    G.M. Torrie, J.P. Valleau, J. Comput. Phys. 23, 187 (1977)Google Scholar
  37. 37.
    S.C. Harvey, M. Prabhakaran, J. Phys. Chem. 91, 4799 (1987)Google Scholar
  38. 38.
    D. Chandler, An Introduction to Modern Statistical Mechanics (Oxford Universion Press, New York, 1987)Google Scholar
  39. 39.
    J. Mezei, J. Comput. Phys. 68, 237 (1987)Google Scholar
  40. 40.
    R.W.W. Hooft, J. Chem. Phys. 97, 6690 (1992)Google Scholar
  41. 41.
    C. Bartels, M. Karplus, J. Comput. Chem. 18, 1450 (1997); J. Phys. Chem. B 102, 865 (1998)Google Scholar
  42. 42.
    B.A. Berg, T. Neuhaus, Phys. Lett. B 267, 249 (1991)Google Scholar
  43. 43.
    B.A. Berg, T. Celik, Phys. Rev. Lett. 69, 2292 (1992)Google Scholar
  44. 44.
    U.H.E. Hansmann, Y. Okamoto, J. Comput. Chem. 14, 1333 (1993)Google Scholar
  45. 45.
    Y. Okamaoto, U.H.E. Hansmann, J. Phys. Chem. 99, 11276 (1995)Google Scholar
  46. 46.
    U.H.E. Hansmann, Y. Okamoto, F. Eisenmenger, Chem. Phys. Lett. 259, 321 (1996)Google Scholar
  47. 47.
    N. Nakajima, H. Nakamura, A. Kidera, J. Phys. Chem. B 101, 817 (1997)Google Scholar
  48. 48.
    C. Bartels, M. Karplus, J. Phys. Chem. 102, 865 (1998)Google Scholar
  49. 49.
    D. Stauffer, A. Aharony, Introduction to Percolation Theory, 2nd edn (Taylor and Francis, London, 1992, 1994)Google Scholar
  50. 50.
    P.W. Kasteleyn, P.W. Fortuin, J. Phys. Soc. Jpn. Suppl. 26, 11 (1969); P.W. Fortuin, P.W. Kasteleyn, Physica (Utrecht) 57, 536 (1972)Google Scholar
  51. 51.
    R.J. Baxter, J. Stat. Phys. 15, 485 (1976)Google Scholar
  52. 52.
    R.J. Baxter, 6 Exactly Solved Models in Statistical Mechanics (Academic Press, London, 1982)Google Scholar
  53. 53.
    F. Family, T. Vicsek, Dynamics of Fractal Surfaces (World Scientific, Singapore, 1991)Google Scholar
  54. 54.
    T. Vicsek, Fractal Growth Phenomena, 2nd edn. (World Scientific, Singapore, 1992)Google Scholar
  55. 55.
    R.J. Reynolds, H.E. Stanley, W. Klein, J. Phys. C: Condens. Matt. 10, L167 (1977); Phys. Rev. B 21, 1223 (1980)Google Scholar
  56. 56.
    J. Bernasconi, Phys. Rev. B 18, 2185 (1978)Google Scholar
  57. 57.
    H. Nakanishi, R.J. Reynolds, Phys. Lett. A 71, 252 (1979)Google Scholar
  58. 58.
    P.D. Eschbach, D. Stauffer, H.J. Herrmann, Phys. Rev. B 23, 422 (1981)Google Scholar
  59. 59.
    A. Conglio, Phys. Rev. Lett. 46, 250 (1981)Google Scholar
  60. 60.
    R.M. Ziff, Phys. Rev. Lett. 69, 2670 (1992)Google Scholar
  61. 61.
    T.C. Lubensky, J. Issacson, Phys. Rev. Lett. 41, 829 (1978). Phys. Rev. A 20, 2130 (1979)Google Scholar
  62. 62.
    J. Issacson, T.C. Lubensky, J. Physique 41, L469 (1980)Google Scholar
  63. 63.
    G. Parisi, N. Sourlas, Phys. Rev. Lett. 46, 871 (1981)Google Scholar
  64. 64.
    S. Kumer, in Computer Simulation of Polymers, ed. by E.A. Colbourn (Longman Scientific & Technical, London, 1994). Chapter 7; K. Binder, ibid., Chapter 3Google Scholar
  65. 65.
    K. Binder (ed.), Monte Carlo and Molecular Dynamics Simulations in Polymer Science (Oxford University Press, New York, 1995)Google Scholar
  66. 66.
    K. Ohno, K. Binder, J. Stat. Phys. 64, 781 (1991); K. Ohno, Macromolecular symposia 81, 121 (1994)Google Scholar
  67. 67.
    P.G. de Gennes, Scaling Concepts in Polymer Physics (Cornell University Press, Ithaca, New York, 1979)Google Scholar
  68. 68.
    P.G. de Gennes, Phys. Lett. A 38, 339 (1972)Google Scholar
  69. 69.
    K. Ohno, K. Binder, J. Physique 49, 1329 (1988)Google Scholar
  70. 70.
    J. des Cloizeaux, J. Physique 42, 635 (1981)Google Scholar
  71. 71.
    J.J. Freire, in Advances in Polymer Science, vol. 143 (Springer, Berlin, 1999), p. 37Google Scholar
  72. 72.
    P.J. Flory, J. Chem. Phys. 9, 660 (1941); P.J. Flory, J. Chem. Phys. 10, 51 (1942)Google Scholar
  73. 73.
    M.L. Huggins, J. Chem. Phys. 9, 440 (1941)Google Scholar
  74. 74.
    A.R. Miller, Proc. Camb. Philos. Soc. 39, 54 (1942)Google Scholar
  75. 75.
    E.A. Guggenheim, Proc. R. Soc. Lond. A 183, 213 (1944)Google Scholar
  76. 76.
    J. des Cloizeaux, J. Physique 36, 281 (1975)Google Scholar
  77. 77.
    K. Shida, K. Ohno, M. Kimura, Y. Kawazoe, Comput. Theor. Polymer Sci., in submissionGoogle Scholar
  78. 78.
    K. Ohno, K. Shida, M. Kimura, Y. Kawazoe, Macromolecules 29, 2269 (1996)Google Scholar
  79. 79.
    A. Rubio, J. Freire, Macromolecules 29, 6946 (1996)Google Scholar
  80. 80.
    S. Kumer, I. Szleifer, A. Panagiotopoulos, Phys. Rev. Lett. 66, 2935 (1991)Google Scholar
  81. 81.
    S. Kumer, in Computer Simulation of Polymers, ed. by E.A. Colbourn (Longman Scientific & Technical, London, 1994). Chapter 7Google Scholar
  82. 82.
    B. Widom, J. Chem. Phys. 39, 11 (1963); J. Chem. Phys. 39, 2808 (1963)Google Scholar
  83. 83.
    W.G. Madden, A.I. Pesci, K.F. Freed, Macromolecules 23, 1181 (1990)Google Scholar
  84. 84.
    B. Smit, S. Karaborni, J.I. Siepmann, Macromol. Symp. 81, 343 (1994)Google Scholar
  85. 85.
    H.A. Kramers, J. Chem. Phys. 14, 415 (1946)Google Scholar
  86. 86.
    H.L. Frisch, N. Pistoor, A. Sariban, K. Binder, S. Fesjian, J. Chem. Phys. 89, 5194 (1988)Google Scholar
  87. 87.
    B.H. Zimm, Macromolecues 13, 592 (1980); Macromolecules 17, 795 (1984). Macromolecules 17, 2441 (1984)Google Scholar
  88. 88.
    M.K. Wilkinson, D.S. Gaunt, L.E.G. Lipson, S.G. Wittingon, Macromolecules 21, 1818 (1988)Google Scholar
  89. 89.
    K. Shida, K. Ohno, M. Kimura, Y. Kawazoe, Y. Nakamura, Macromolecules 31, 2343 (1998)Google Scholar
  90. 90.
    A.K. Kron, Vysokomol. Soedin. 7, 1228 (1965) [Polym. Sci. USSR 7, 1361 (1965)]; F.T. Wall, F. Mandel. J. Chem. Phys. 63, 4592 (1975)Google Scholar
  91. 91.
    I. Carmesin, K. Kremer, J. Physique 51, 915 (1990)Google Scholar
  92. 92.
    K. Ohno, M. Schulz, K. Binder, H.L. Frisch, J. Chem. Phys. 101, 4452 (1994)Google Scholar
  93. 93.
    N. Urakami, M. Takasu, J. Phys. Soc. Jpn. 65, 2694 (1996)Google Scholar
  94. 94.
    M. Murat, T.A. Witten, Macromolecules 23, 520 (1990)Google Scholar
  95. 95.
    M. Seul, D. Andelman, Science 267, 476 (1995)Google Scholar
  96. 96.
    M. Ishihara, H. Mizuseki, K. Ohno, Y. Kawazoe, in Proc. Fourth Int. Symp. Phys. Magn. Mater. (1998), to appearGoogle Scholar
  97. 97.
    K. Binder, Phys. Rev. Lett 45, 811 (1980)Google Scholar
  98. 98.
    R. Kikuchi, Phys. Rev. 81, 988 (1951)Google Scholar
  99. 99.
    N.S. Golsov, L.E. Popov, L. Ya. Pudan, J. Phys. Chem. Solids 34, 1149 (1973); J. Phys. Chem. Solids 34, 1157 (1973)Google Scholar
  100. 100.
    R. Kikuchi, H. Sato, Acta Metall. 22, 1099 (1974)Google Scholar
  101. 101.
    T. Matsumiya, H. Sawada, W. Yamada, A. Nogami, in Computer Aided Innovation of New Materials, ed. by M. Doyama, et al. (Elsevier, Amsterdam, 1991), p. 779Google Scholar
  102. 102.
    K. Terakura, T. Oguchi, T. Mohri, K. Watanabe, Phys. Rev. B 35, 2169 (1987)Google Scholar
  103. 103.
    T. Mohri, Bull. JIM 28, 268 (1989)Google Scholar
  104. 104.
    G.J. Ackland, V. Vitek, Phys. Rev. B 41, 10324 (1990)Google Scholar
  105. 105.
    R. Sahara, H. Ichikawa, Master Thesis, Tohoku University (1999). R. Sahara, H. Ichikawa, H. Mizuseki, K. Ohno, H. Kubo, Y. Kawazoe. J. Chem. Phys. 120, 9297 (2004)Google Scholar
  106. 106.
    B.M. McCoy, T.T. Wu, The Two-Dimensional Ising Models (Harbard University Press, Cambridge, 1973)Google Scholar
  107. 107.
    C. Domb, M.F. Sykes, Proc. R. Soc. A 240, 214 (1957). J. Math. Phys. 2, 63 (1961)Google Scholar
  108. 108.
    H.E. Stanley, in Phase Transitions and Critical Phenomena, vol. 3, ed. by C. Domb, M.S. Green (Academic Press, New York, 1974), p. 485Google Scholar
  109. 109.
    M. Suzuki, J. Phys. Soc. Jpn. 55, 4205 (1986); Prog. Theor. Phys. Suppl. 87, 1 (1986)Google Scholar
  110. 110.
    M. Suzuki, M. Katori, X. Hu, J. Phys. Soc. Jpn. 56, 3092 (1987); X. Hu, M. Katori, M. Suzuki, J. Phys. Soc. Jpn. 56, 3865 (1987)Google Scholar
  111. 111.
    E. Stoll, K. Binder, T. Schneider, Phys. Rev. B 8, 3266 (1973)Google Scholar
  112. 112.
    J. Tobochnik, S. Sarker, R. Cordery, Phys. Rev. Lett 46, 1417 (1981)Google Scholar
  113. 113.
    M.C. Yalabik, J.D. Gunton, Phys. Rev. B 25, 534 (1082)Google Scholar
  114. 114.
    S.L. Katz, J.D. Gunton, C.P. Liu, Phys. Rev. B 25, 6008 (1982)Google Scholar
  115. 115.
    M.N. Barber, R.B. Pearson, J.L. Richardson, D. Touissaint, Phys. Rev. B 32, 1720 (1985)Google Scholar
  116. 116.
    A.M. Ferrenberg, D.P. Landau, Phys. Rev. B 44, 5081 (1991)Google Scholar
  117. 117.
    D.S. Gaunt, M.F. Sykes, J. Phys. A Math. Nucl. Gen. 6, 1517 (1973)Google Scholar
  118. 118.
    K.G. Wilson, Phys. Rev. B 4, 3174 (1971); Phys. Rev. B 4, 3184 (1971). Phys. Rev. Lett. 28, 548 (1972)Google Scholar
  119. 119.
    K.G. Wilson, M.E. Fisher, Phys. Rev. Lett. 28, 240 (1972)Google Scholar
  120. 120.
    E. Brézin, J.C. Le Guillou, J. Zinn-Justin, B. Nickel, Phys. Lett. A 44, 227 (1973)Google Scholar
  121. 121.
    G.A. Baker Jr., B.G. Nickel, M.S. Green, P.I. Meiron, Phys. Rev. Lett. 36, 1351 (1976)Google Scholar
  122. 122.
    J.C. Le Guillou, J. Zinn-Justin, Phys. Rev. Lett. 39, 95 (1977)Google Scholar
  123. 123.
    S.K. Ma, Modern Theory of Critical Phenomena, Frontiers in Physics, vol. 46 Lecture Note Series (Benjamin, Reading, 1976)Google Scholar
  124. 124.
    D.P. Landau, in The Monte Carlo Method in Condensed Matter Physics, ed. by K. Binder (Springer, Berlin, 1992, 1995). Chapter 2Google Scholar
  125. 125.
    R.J. Elliot, Phys. Rev. 124, 346 (1961)Google Scholar
  126. 126.
    E.I. Dinaburg, A.E. Mazel, Ya G. Sinai, Sov. Sci. Rev. C 6, 113 (1987)Google Scholar
  127. 127.
    M.E. Fisher, W. Selke, Phys. Rev. Lett. 44, 1502 (1980)Google Scholar
  128. 128.
    W. Selke, Z. Phys, B 29, 133 (1978); Z. Phys. B 43, 335 (1981)Google Scholar
  129. 129.
    W. Selke, M.E. Fisher, Phys. Rev. B 20, 257 (1979). Z. Phys. B 40, 71 (1980)Google Scholar
  130. 130.
    E.B. Rasmussen, S.J. Knak, Jensen. Phys. Rev. 24, 2744 (1989)Google Scholar
  131. 131.
    W. Selke, J. Magn. Mag. Mat. 9, 7 (1978); Sol. State Commun. 27, 1417 (1978); Z. Phys. C: Condens. Matt. 13, L261 (1980)Google Scholar
  132. 132.
    S. Redner, H.E. Stanley, J. Phys. C: Condens. Matt. 10, 4765 (1978); Phys. Rev. B 16, 4901 (1977a, b)Google Scholar
  133. 133.
    W. Selke, in Phase Transitions and Critical Phenomena, vol. 15, ed. by C. Domb, J.L. Lebowitz (Academic Press, London, 1992), p. 1Google Scholar
  134. 134.
    K. Binder, A.P. Young, Rev. Mod. Phys. 58, 801 (1986)Google Scholar
  135. 135.
    M. Mezard, G. Parisi, M.A. Virasoro, Spin Glass Theory and Beyond. Lecture Notes in Physics, vol. 9 (World Scientific, Singapore, 1987)Google Scholar
  136. 136.
    R.H. Swendsen, J.-S. Wang, Phys. Rev. Lett. 57, 2607 (1986)Google Scholar
  137. 137.
    J.S. Wang, R.H. Swendsen, Phys. Rev. B 37, 7745 (1988)Google Scholar
  138. 138.
    B.A. Berg, U.E. Hansmann, T. Celik, Nucl. Phys. B 42, 905 (1995)Google Scholar
  139. 139.
    E. Marinari, G. Parisi, Europhys. Lett. 19, 451 (1992)Google Scholar
  140. 140.
    H. Nakanishi, M.E. Fisher, Phys. Rev. Lett. 49, 1565 (1982)Google Scholar
  141. 141.
    S. Dietrich, in Phase Transitions and Critical Phenomena, vol. 12, ed. by C. Domb, J.L. Lebowitz (Academic Press, London, 1988), p. 1Google Scholar
  142. 142.
    Y. Okabe, K. Ohno, Phys. Rev. 30, 6573 (1984)Google Scholar
  143. 143.
    K. Binder, in Phase Transitions and Critical Phenomena, vol. 8, ed. by C. Domb, J.L. Lebowitz (Academic Press, London, 1983) pp. 1, 467Google Scholar
  144. 144.
    K. Binder, P.C. Hohenberg, Phys. Rev. B 6, 3461 (1972). Phys. Rev. B 9, 2194 (1974)Google Scholar
  145. 145.
    M. Kikuchi, Y. Okabe, Prog. Theor. Phys. 74, 458 (1985)Google Scholar
  146. 146.
    Y. Okabe, M. Kikuchi, K. Ohno, Prog. Theor. Phys. 75, 496 (1986)Google Scholar
  147. 147.
    K. Ohno, Y. Okabe, J. Phys. Soc. Jpn. 55, 2627 (1986)Google Scholar
  148. 148.
    S.G. Whittington, G.M. Torrie, A.J. Guttmann, J. Phys. A. Math. Gen. 12, 2449 (1979)Google Scholar
  149. 149.
    K. Ohno, Y. Okabe, A. Morita, Prog. Theor. Phys. 71, 714 (1984)Google Scholar
  150. 150.
    H.W. Diehl, in Phase Transitions and Critical Phenomena, vol. 10, ed. by C. Domb, J.L. Lebowitz (Academic Press, London, 1986), p. 75Google Scholar
  151. 151.
    K. Ohno, in Trends in Statistical Physics (Research Trends, Poojapura, Trivandrum, 1999). (in press)Google Scholar
  152. 152.
    R. Zorn, H.J. Herrmann, C. Rebbi, Comput. Phys. Commun. 23, 337 (1981)Google Scholar
  153. 153.
    R.H. Swendsen, J.-S. Wang, Phys. Rev. Lett. 58, 86 (1987)Google Scholar
  154. 154.
    U. Wolff, Phys. Rev. Lett. 62, 361 (1989)Google Scholar
  155. 155.
    U. Wolff, Phys. Lett. B 228, 379 (1989). Nucl. Phys. B 322, 759 (1989)Google Scholar
  156. 156.
    R.H. Swendsen, J.-S. Wang, A.M. Ferrenberg, in The Monte Carlo Method in Condensed Matter Physics, ed. by K. Binder (Springer, Berlin, 1992, 1995). Chapter 4Google Scholar
  157. 157.
    H.H. Zhao, Z.Y. Xie, Q.N. Chen, Z.C. Wei, J.W. Cai, T. Xiang, Phys. Rev. B 81, 174411 (2010)Google Scholar
  158. 158.
    M. Levin, C.P. Nave, Phys. Rev. Lett. 99, 120601 (2007)Google Scholar
  159. 159.
    Z.Y. Xie, J. Chen, M.P. Qin, J.W. Zhu, L.P. Yang, T. Xiang, Phys. Rev. B 86, 045139 (2012)Google Scholar
  160. 160.
    T.H. Berlin, M. Kac, Phys. Rev. 86, 821 (1952)Google Scholar
  161. 161.
    H.E. Stanley, Phys. Rev. 176, 718 (1968)Google Scholar
  162. 162.
    N.D. Mermin, H. Wagner, Phys. Rev. Lett. 17, 1133 (1966)Google Scholar
  163. 163.
    J.M. Kosterlitz, D.J. Thouless, J. Phys. C6, 1181 (1973)Google Scholar
  164. 164.
    D.R. Nelson, B.I. Halperin, Phys. Rev. B 19, 2457 (1979)Google Scholar
  165. 165.
    S. Miyashita, H. Nishimori, A. Kuroda, M. Suzuki, Prog. Theor. Phys. 60, 1669 (1978)Google Scholar
  166. 166.
    J. Tobochnik, G.V. Chester, Phys. Rev. B 20, 3761 (1979)Google Scholar
  167. 167.
    J.E. Van Himbergen, S. Chakravarty, Phys. Rev. B 23, 359 (1981)Google Scholar
  168. 168.
    H. Betsuyaku, Physica A 106, 311 (1981)Google Scholar
  169. 169.
    D.P. Landau, J. Mag. Mag. Mat. 31–34, 1115 (1983)Google Scholar
  170. 170.
    M. Hasenbusch, S. Meyer, Phys. Lett. B 241, 238 (1990)Google Scholar
  171. 171.
    W. Janke, Phys. Lett. A 148, 306 (1990)Google Scholar
  172. 172.
    A.P. Gottlob, M. Hasenbusch, Physica A 201, 593 (1993)Google Scholar
  173. 173.
    R.G. Bowers, G.S. Joyce, Phys. Rev. Lett. 19, 630 (1967)Google Scholar
  174. 174.
    M. Ferer, M.A. Moore, M. Wortis, Phys. Rev. B 8, 5205 (1973)Google Scholar
  175. 175.
    P.J. Wood, G.S. Rushbrooke, Phys. Rev. Lett. 17, 307 (1966)Google Scholar
  176. 176.
    M. Ferer, M.A. Moore, M. Wortis, Phys. Rev. B 4, 3954 (1971)Google Scholar
  177. 177.
    R.E. Watson, M. Blume, G.H. Vineyard, Phys. Rev. 181, 811 (1969)Google Scholar
  178. 178.
    K. Binder, H. Rauch, Z. Phys. 219, 201 (1969)Google Scholar
  179. 179.
    K. Binder, H. Rauch, V. Wildpaner, J. Phys. Chem. Solids 31, 391 (1970)Google Scholar
  180. 180.
    P. Peczak, A.M. Ferrenberg, D.P. Landau, Phys. Rev. B 43, 6087 (1991)Google Scholar
  181. 181.
    K. Binder, Phys. Rev. Lett. 47, 693 (1981). Z. Phys. B 43, 119 (1981)Google Scholar
  182. 182.
    G.S. Pawley, R.H. Swendsen, D.J. Wallece, K.G. Wilson, Phys. Rev. B 29, 4030 (1984)Google Scholar
  183. 183.
    L.P. Kadanoff, Physics (New York) 2, 263 (1966)Google Scholar
  184. 184.
    P. Martin, Potts Models and Related Problems in Statistical Mechanics (World Scientific, Singapore, 1991)Google Scholar
  185. 185.
    C. Rabbi, R.H. Swendsen, Phys. Rev. B 21, 4094 (1980)Google Scholar
  186. 186.
    S.Y. Tang, D.P. Landau, J. Appl. Phys. 61, 4409 (1987); Phys. Rev. B 43, 6006 (1991)Google Scholar
  187. 187.
    S.J.K. Jensen, O.G. Mouritsen, Phys. Rev. Lett. 43, 1736 (1979)Google Scholar
  188. 188.
    H.J. Herrmann, Z. Phys. 35, 171 (1979)Google Scholar
  189. 189.
    H.W.J. Blöte and R. H. Swendsen, J. Appl. Phys. 50, 7382 (1979); J. Magn. Mag. Mat. 15-18, 399 (1980)Google Scholar
  190. 190.
    J.R. Banavar, G.S. Grest, D. Jasnow, Phys. Rev. Lett. 45, 1424 (1980). Phys. Rev. B 25, 4639 (1982)Google Scholar
  191. 191.
    B. Hoppe, L.L. Hirst, J. Phys. A Math. Gen. 18, 3375 (1986); Phys. Rev. B 34, 6589 (1986)Google Scholar
  192. 192.
    J.-S. Wang, R.H. Swendsen, R. Kotecký, Phys. Rev. Lett. 63, 109 (1989). Phys. Rev. B 42, 2465 (1990)Google Scholar
  193. 193.
    A.P. Gottlob, Hasenbusch. Physica A 210, 217 (1994)Google Scholar
  194. 194.
    K. Yasumura, T. Oguchi, J. Phys. Soc. Jpn. 53, 515 (1984)Google Scholar
  195. 195.
    Y. Okabe, M. Kikuchi, in Computational Approaches in Condensed-Matter Physics, Proceedings in Physics, vol. 70, ed. by S. Miyashita, M. Imada, H. Takayama (Springer, Berlin, 1992), p. 193Google Scholar
  196. 196.
    K. Ohno, H.-O. Carmesin, H. Kawamura, Y. Okabe, Phys. Rev. B 42, 10360 (1990)Google Scholar
  197. 197.
    P.A. Lebwohl, G. Lasher, Phys. Rev. A 6, 426 (1972)Google Scholar
  198. 198.
    K. Binder, J.D. Reger, Adv. Phys. 41, 547 (1992)Google Scholar
  199. 199.
    H.O. Carmesin, K. Ohno, J. Magn. Magn. Mater. 104–107, 264 (1992)Google Scholar
  200. 200.
    I.V. Markov, Crystal Growth for Beginners (World Scientific, Singapore, 1979)Google Scholar
  201. 201.
    Y. Saito, Statistical Physics of Crystal Growth (World Scientific, Singapore, 1996)Google Scholar
  202. 202.
    E. Mayer, J. Cryst. Growth 74, 425 (1986)Google Scholar
  203. 203.
    K. Ohno, H. Trinkaus, H. Müller-Krumbhaar, J. Cryst. Growth 99, 68 (1990)Google Scholar
  204. 204.
    R. Becker, W. Döring, Ann. Phys. 24, 719 (1935)Google Scholar
  205. 205.
    I.M. Lifshitz, V.V. Slyozov, J. Phys. Chem. Solids 19, 35 (1961)Google Scholar
  206. 206.
    K. Binder, P. Stauffer, Phys. Rev. Lett. 33, 1006 (1974)Google Scholar
  207. 207.
    R. Sahara, H. Mizuseki, K. Ohno, S. Uda, T. Fukuda, Y. Kawazoe, Sci. Rep. Res. Inst. Tohoku Univ. A 43, 23 (1997)Google Scholar
  208. 208.
    I. Steinbach, Annual Rev. Mater. Res. 43, 89 (2013)Google Scholar
  209. 209.
    N. Provatas, K. Elder, Phase-Field Methods in Materials Science and Engineering (Wiley-VCH, Weinheim, 2010)Google Scholar
  210. 210.
    W. Cahn, J.E. Hilliard, J. Chem. Phys. 28, 258 (1958)Google Scholar
  211. 211.
    D. Lee, J.-Y. Huh, D. Jeong, J. Shin, A. Yun, J. Kim, Comput. Mater. Sci. 81, 216 (2014)Google Scholar
  212. 212.
    S. Bhattacharyya, R. Sahara, K. Ohno, to appearGoogle Scholar
  213. 213.
    R. Kubo, Rep. Prog. Phys. 29, Part I, 255 (1966)Google Scholar
  214. 214.
    S. Masatsuji, Y. Misumi, S. Ishii, K. Ohno 49, 2424 (2008)Google Scholar
  215. 215.
    Y. Misumi, S. Masatsuji, R. Sahara, S. Ishii, K. Ohno, J. Chem. Phys. 128, 234702 (2008)Google Scholar
  216. 216.
    I.M. Lifshitz, Sov. Phys. JETP 15, 939 (1962)Google Scholar
  217. 217.
    J.D. Gunton, in Phase Transitions and Critical Phenomena, vol. 8, ed. by C. Domb, J.L. Lebowitz (Academic Press, London, 1983) pp. 267, 479Google Scholar
  218. 218.
    K. Binder, P. Stauffer, Adv. Phys. 25, 343 (1976)Google Scholar
  219. 219.
    G. Gompper, M. Schick, Self-Assembling Amphiphilic Systems. Phase Transitions and Critical Phenomena, vol. 16, ed. by C. Domb, J.L. Lebowitz (Academic Press, London, 1994); K. Kawasaki, T. Kawakatsu. Physica A 164, 549 (1990)Google Scholar
  220. 220.
    B. Widom, J. Chem. Phys. 84, 6943 (1986)Google Scholar
  221. 221.
    S.T. Milner, T.A. Witten, J. Physique 49, 1951 (1988)Google Scholar
  222. 222.
    P. Chandra, S.A. Safran, Europhys. Lett. 17, 691 (1992)Google Scholar
  223. 223.
    D. Wu, D. Chandler, B. Smit, J. Phys. Chem. 96, 4077 (1992)Google Scholar
  224. 224.
    Y. Okabe, Private CommunicationGoogle Scholar
  225. 225.
    W.W. Mullins, R.F. Sekerka, J. Appl. Phys. 34, 323 (1963)Google Scholar
  226. 226.
    W. Kossel, Nach. Ges. Wiss. Göttingen 135 (1927)Google Scholar
  227. 227.
    W. Kossel, Naturwissenschaften 18, 901 (1930)Google Scholar
  228. 228.
    I. Stranski, R. Kaischew, Z. Phys, Chem. B 26, 31 (1934)Google Scholar
  229. 229.
    M. Volmer, Kinetik der Phasenbildung (Leipzig, Leipzig, 1939)Google Scholar
  230. 230.
    R.H. Swendsen, Phys. Rev. B 15, 5421 (1977). Phys. Rev. B 18, 492 (1978)Google Scholar
  231. 231.
    G. Gompper, D.M. Kroll, R. Lipowsky, Phys. Rev. B 42, 961 (1990)Google Scholar
  232. 232.
    W.K. Burton, N. Cabrera, F.C. Frank, Phil. Trans. R. Soc. Lond. A 243, 299 (1950–1)Google Scholar
  233. 233.
    J.J. Harris, B.A. Joyce, P.J. Dobson, Surf. Sci. 103, L90 (1981)Google Scholar
  234. 234.
    C.E.C. Wood, Surf. Sci. 108, L441 (1981)Google Scholar
  235. 235.
    W. Braun, L. Däweritz, K.H. Ploog, Phys. Rev. Lett. 80, 4935 (1998)Google Scholar
  236. 236.
    R.L. Schwoebel, E.J. Shipsey, J. Appl. Phys. 37, 3682 (1966)Google Scholar
  237. 237.
    R.L. Schwoebel, J. Appl. Phys. 40, 614 (1969)Google Scholar
  238. 238.
    M. Uwaha, Y. Saito, Phys. Rev. Lett. 68, 224 (1992)Google Scholar
  239. 239.
    M. Sato, M. Uwaha, Y. Saito, Phys. Rev. Lett. 80, 4233 (1998)Google Scholar
  240. 240.
    P. Meakin, in Phase Transitions and Critical Phenomena, vol. 12, ed. by C. Domb, J.L. Lebowitz (Academic Press, London, 1988), p. 335Google Scholar
  241. 241.
    H.E. Stanley, N. Ostrowsky, Random Fluctuations and Pattern Growth: Experiments and Models (Kluwer, Amsterdam, 1988)Google Scholar
  242. 242.
    B.H. Kaye, A Random Walk Trhough Fractal Dimensions (VCH, Weinheim, 1989)Google Scholar
  243. 243.
    A. Witten, L.M. Sander, Phys. Rev. Lett. 47, 1400 (1981)Google Scholar
  244. 244.
    P. Meakin, Phys. Rev. A 27, 604 (1983). Phys. Rev. A 27, 1495 (1983)Google Scholar
  245. 245.
    K. Ohno, K. Kikuchi, H. Yasuhara, Phys. Rev. A 46, 3400 (1992)Google Scholar
  246. 246.
    M. Muthukumer, Phys. Rev. Lett. 50, 839 (1983); M. Tokuyama, K. Kawasaki. Phys. Lett. A 100, 337 (1984)Google Scholar
  247. 247.
    H. Fujikawa, M. Matsushita, J. Phys. Soc. Jpn. 60, 88 (1991)Google Scholar
  248. 248.
    H.C. Fogedby, J. Phys. Soc. Jpn. 60, 704 (1991)Google Scholar
  249. 249.
    M. Ohgiwari, M. Matsushita, T. Matsuyama, J. Phys. Soc. Jpn. 61, 816 (1992)Google Scholar
  250. 250.
    M. Eden, in Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, vol. IV, ed. by J. Neyman (University of California Press, Berkeley, 1961), p. 223Google Scholar
  251. 251.
    P. Ramanlal, L.M. Sander, Phys. Rev. Lett. 54, 1828 (1985)Google Scholar
  252. 252.
    F. Family, T. Vicsek, J. Phys. A 18, L75 (1985)Google Scholar
  253. 253.
    P. Meakin, J. Phys. A 18, L661 (1985)Google Scholar
  254. 254.
    S. Liang, L.P. Kadanoff, Phys. Rev. A 31, 2628 (1985)Google Scholar
  255. 255.
    P. Meakin, P. Ramanlal, L.M. Sander, R.C. Ball, Phys. Rev. A 34, 5091 (1986)Google Scholar
  256. 256.
    P. Meakin, Phys. Rev. A 38, 994 (1988)Google Scholar
  257. 257.
    M. Itoh, R. Sahara, M. Takahashi, X. Hu, K. Ohno, Y. Kawazoe, Phys. Rev. E 53, 148 (1996)Google Scholar
  258. 258.
    M. Kardar, G. Parisi, Y. Zhang, Phys. Rev. Lett. 56, 889 (1986)Google Scholar
  259. 259.
    C.T. Capraro, Y. Bar-Yam, Comput. Mater. Sci. 1, 169 (1993)Google Scholar
  260. 260.
    P. Meakin, Phys. Rev. Lett. 51, 1119 (1983)Google Scholar
  261. 261.
    M. Kolb, R. Botet, R. Jullien, Phys. Rev. Lett. 51, 1123 (1983)Google Scholar
  262. 262.
    R. Jullien, M. Kolb, J. Phys. A Math. Gen. 17, L639 (1984)Google Scholar
  263. 263.
    Brown, Ball (1985); W.D. Brown, R.C. Ball, J. Phys. A Math. Gen. 18, L517 (1985)Google Scholar
  264. 264.
    F. Family, P. Meakin, T. Vicsek, J. Chem. Phys. 83, 4141 (1985)Google Scholar
  265. 265.
    M. Kudoh, X. Hu, K. Ohno, Y. Kawazoe, J. Cryst. Growth 128, 1162 (1993)Google Scholar
  266. 266.
    R.K. Iler, The Chemistry of Silica (Wiley, New York, 1979)Google Scholar
  267. 267.
    K. Ohno, Y. Kawazoe, Comput. Theor. Polymer Sci., in submissionGoogle Scholar
  268. 268.
    M. Mikami, Master Thesis, Tohoku University (1991)Google Scholar
  269. 269.
    C. Aubert, D.S. Cannel, Phys. Rev. Lett. 56, 738 (1986)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Kaoru Ohno
    • 1
  • Keivan Esfarjani
    • 2
  • Yoshiyuki Kawazoe
    • 3
  1. 1.Department of PhysicsYokohama National UniversityYokohamaJapan
  2. 2.Department of Mechanical and Aerospace Engineering, Materials Science and Engineering and PhysicsUniversity of VirginiaCharlottesvilleUSA
  3. 3.New Industry Creation Hatchery CenterTohoku UniversitySendaiJapan

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