Skip to main content
SpringerLink
Log in
Book cover

The Monte Carlo Method in Condensed Matter Physics pp 93–120Cite as

Simulation of Random Growth Processes

  • Chapter

  • 577 Accesses

Part of the book series: Topics in Applied Physics ((TAP,volume 71))

Abstract

Many phenomena, like the shape of a snowflake, the size distribution of asteroids or the roughness of a crack surface can only be understood as the end product of a dynamical process which has random and irreversible ingredients. In the last ten years much progress has been achieved in understanding this kind of phenomenon, by studying so-called “growth models” [5.1-3]. These models are very difficult to deal with analytically and for this reason most of the results obtained come from numerical simulations.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   74.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. T. Vicsek: Fractal Growth Phenomena ( World Scientific, Singapore 1989 )

    MATH  Google Scholar 

  2. H.J. Herrmann: Phys. Rep. 136, 153 (1986)

    Article  MathSciNet  ADS  Google Scholar 

  3. P. Meakin: In Phase Transitions and Critical Phenomena, Vol. 12, ed. by C. Domb, J.L. Lebowitz ( Academic, New York 1988 )

    Google Scholar 

  4. M. Eden: In Proc. of the Fourth Berkeley Symp. on Mathematical Statistics and Probability, Vol. IV, ed. by F. Neyman (University of California, Berkeley 1961); in Symposium on Information Theory in Biology, ed. by H.P. Hockey ( Pergamon, New York 1959 ) p. 359

    Google Scholar 

  5. T. Vicsek, M. Cserzö, V.K. Horvâth: Physica A 167, 315 (1990)

    Article  ADS  Google Scholar 

  6. P. Meakin: J. Theor. Biol. 118, 101 (1986)

    Article  MathSciNet  Google Scholar 

  7. F. Family, T. Vicsek: J. Phys. A 18, L75 (1985)

    Article  ADS  Google Scholar 

  8. M. Kardar, G. Parisi, Y.C. Zhang: Phys. Rev. Lett. 56, 889 (1986)

    Article  ADS  MATH  Google Scholar 

  9. D.E. Wolf, J. Kertész: Europhys. Lett. 4, 651 (1987)

    Article  ADS  Google Scholar 

  10. D.E. Wolf, J. Kertész: J. Phys. A 20, L257 (1987)

    Article  ADS  Google Scholar 

  11. J.M. Kim, J.M. Kosterlitz: Phys. Rev. Lett. 62, 2289 (1989)

    Article  ADS  Google Scholar 

  12. P. Freche, D. Stauffer, H.E. Stanley: J. Phys. A 18, L1163 (1985)

    Article  ADS  Google Scholar 

  13. J.G. Zabolitzky, D. Stauffer: Phys. Rev. A 34, 1523 (1986)

    Article  ADS  Google Scholar 

  14. R. Hirsch, D.E. Wolf: J. Phys. A 19, L251 (1986)

    Article  ADS  Google Scholar 

  15. M. Plischke, Z. Rácz: Phys. Rev. Lett. 53, 415 (1984)

    Article  ADS  Google Scholar 

  16. M. Plischke, Z. Rácz: Phys. Rev. Lett. 54, 2056 (1985)

    Article  ADS  Google Scholar 

  17. P. Meakin, R. Jullien, R. Botet: Europhys. Lett. 1, 609 (1986)

    Article  ADS  Google Scholar 

  18. R. Jullien, R. Botet: Phys. Rev. Lett. 54, 2055 (1985)

    Article  ADS  Google Scholar 

  19. R. Jullien, R. Botet: J. Phys. A 18, 2279 (1985)

    Article  ADS  Google Scholar 

  20. J. Kertész, D. Wolf: J. Phys. A 21, 747 (1988)

    Article  ADS  Google Scholar 

  21. J. Szép, J. Cserti, J. Kertész: J. Phys. A 18, L413 (1985)

    Article  ADS  Google Scholar 

  22. D. Liu, M. Plischke: Phys. Rev. B 38, 4781 (1988)

    Article  ADS  Google Scholar 

  23. P. Devillard, H.E. Stanley: Physica A 160, 298 (1989)

    Article  ADS  Google Scholar 

  24. L. Tang, B. Forrest: Phys. Rev. Lett. 64, 1405 (1990)

    Article  ADS  Google Scholar 

  25. W. Renz: Private communication

    Google Scholar 

  26. J.S. Langer: Rev. Mod. Phys. 52, 1 (1980)

    Article  ADS  Google Scholar 

  27. D.A. Kessler, J. Koplik, H. Levine: Adv. Phys. 37, 255 (1988)

    Article  ADS  Google Scholar 

  28. L. Niemeyer, L. Pietronero, H.J. Wiesmann: Phys. Rev. Lett. 52, 1033 (1984)

    Article  MathSciNet  ADS  Google Scholar 

  29. W.H. Press, B.P. Flannery, S.A. Teukolsky, W.T. Vetterling: Numerical Recipes, The Art of Scientific Computing ( Cambridge University Press, Cambridge 1986 )

    Google Scholar 

  30. T.C. Halsey, P. Meakin, I. Procaccia: Phys. Rev. Lett. 56, 854 (1986)

    Article  ADS  Google Scholar 

  31. G.G. Batrouni, A. Hansen: J. Stat. Phys. 52, 747 (1988)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  32. C. Evertsz, E. Hafkenscheid, M. Harmsma, L. Pietronero: In Science and Engineering on Cray Supercomputers, ed. by E.J. Petcher ( Cray Research Inc., Minneapolis, 1988 ) p. 85

    Google Scholar 

  33. T.A. Witten, L.M. Sander: Phys. Rev. Lett. 47, 1400 (1981)

    Article  ADS  Google Scholar 

  34. R.C. Ball, R.M. Brady: J. Phys. A 18, L809 (1985)

    Article  ADS  Google Scholar 

  35. P. Meakin: J. Phys. A 18, L661 (1985)

    Article  ADS  Google Scholar 

  36. P. Meakin, R.C. Ball, P. Ramanlal, L.M. Sander: Phys. Rev. A 35, 5233 (1987)

    Article  ADS  Google Scholar 

  37. S. Tolman, P. Meakin: Phys. Rev. A 40, 428 (1989)

    Article  ADS  Google Scholar 

  38. J. Kertész, T. Vicsek: J. Phys. A 19, L257 (1986)

    Article  ADS  Google Scholar 

  39. J. Nittmann, H.E. Stanley: Nature 321, 663 (1986)

    Article  ADS  Google Scholar 

  40. H. Liebowitz (ed.): Fracture, Vols. I-VII (Academic, New York 1986 )

    Google Scholar 

  41. H.J. Herrmann, S. Roux (eds.): Statistical Models for the Fracture of Disordered Media ( Elsevier, Amsterdam 1990 )

    Google Scholar 

  42. S. Roux, E. Guyon: J. de Phys. Lett. 46, L999 (1985)

    Article  Google Scholar 

  43. R.J. Roark, W.C. Young: Formulas for Stress and Strain ( McGraw-Hill, Tokyo 1975 ) p. 89

    Google Scholar 

  44. E. Louis, F. Guinea, F. Flores: In Fractals in Physics, ed. by L. Pietronero, E. Tossatti ( Elsevier, Amsterdam 1986 )

    Google Scholar 

  45. E. Louis, F. Guinea: Europhys. Lett. 3, 871 (1987)

    Article  ADS  Google Scholar 

  46. P. Meakin, G. Li, L.M. Sander, E. Louis, F. Guinea: J. Phys. A 22, 1393 (1989)

    Article  ADS  Google Scholar 

  47. E.L. Hinrichsen, A. Hansen, S. Roux: Europhys. Lett. 8, 1 (1989)

    Article  ADS  Google Scholar 

  48. H. Van Damme, E. Alsac, C. Laroche: C.R. Acad. Sci., Ser. II 309, 11 (1988)

    Google Scholar 

  49. H. Van Damme: In The Fractal Approach to Heterogeneous Chemistry, ed. by D. Avnir ( Wiley, New York 1989 ) p. 199

    Google Scholar 

  50. S. Roux: J. Stat. Phys. 48, 201 (1987)

    Article  MathSciNet  ADS  Google Scholar 

  51. H.J. Herrmann, J. Kertész, L. de Arcangelis: Europhys. Lett. 10, 147 (1989)

    Article  ADS  Google Scholar 

  52. M.J. Blackburn, W.H. Smyrl, J.A. Feeney: In Stress Corrosion in High Strength Steels and in Titanium and Aluminum Alloys, ed. by B.F. Brown ( Naval Res. Lab., Washington, DC 1972 ) p. 344

    Google Scholar 

  53. S.A. Kauffman: J. Theor. Biol. 22, 437 (1969)

    Article  MathSciNet  Google Scholar 

  54. B. Derrida, Y. Pomeau: Europhys. Lett. 1, 45 (1986)

    Article  ADS  Google Scholar 

  55. B. Derrida, D. Stauffer: Europhys. Lett. 2, 739 (1986)

    Article  ADS  Google Scholar 

  56. L.R. da Silva, H.J. Herrmann: J. Stat. Phys. 52, 463 (1988)

    Article  ADS  Google Scholar 

  57. S. Wolfram: Theory and Application of Cellular Automata (World Scientific, Singapore 1986 ); E. Bienenstock, F. Fogelman-Soulié, G. Weisbuch (eds.): Disordered Systems and Biological Organization, NATO ASI Series, Ser. F (Springer, Berlin, Heidelberg 1986 )

    Google Scholar 

  58. H.J. Herrmann: In Nonlinear Phenomena in Complex Systems, ed. by A.N. Proto ( Elsevier, Amsterdam 1989 )

    Google Scholar 

  59. Y. Pomeau: J. Phys. A 17, L415 (1984)

    MathSciNet  Google Scholar 

  60. H.J. Herrmann: J. Stat. Phys. 45, 145 (1986)

    Article  ADS  Google Scholar 

  61. J.G. Zabolitzky, H.J. Herrmann: J. Comput. Phys. 76, 426 (1988)

    Article  ADS  MATH  Google Scholar 

  62. S.C. Glotzer, D. Stauffer, S. Sastry: Physica A 164, 1 (1990)

    Article  ADS  Google Scholar 

  63. D. Stauffer: In Computer Simulation Studies in Condensed Matter Physics II, ed. by D.P. Landau, K.K. Mon, H.-B. Schüttler, Springer Proc. Phys., Vol. 45 ( Springer, Berlin, Heidelberg 1990 )

    Chapter  Google Scholar 

  64. L. de Arcangelis, A. Coniglio: Europhys. Lett. 7, 113 (1988)

    Article  ADS  Google Scholar 

  65. H.J. Hilhorst, M. Nijmeijer: J. de Phys. 48, 185 (1987)

    Article  MathSciNet  Google Scholar 

  66. O. Golinelli, B. Derrida: J. Phys. A 22, L939 (1989)

    Article  ADS  Google Scholar 

  67. A.M. Mariz: J. Phys. A 23, 979 (1990)

    Article  MathSciNet  ADS  Google Scholar 

  68. H.E. Stanley, D. Stauffer, J. Kertész, H.J. Herrmann: Phys. Rev. Lett. 59, 2326 (1987)

    Article  ADS  Google Scholar 

  69. U. Costa: J. Phys. A 20, L583 (1987)

    Article  ADS  Google Scholar 

  70. G. Le Caër: J. Phys. A 22, L647 (1989)

    Article  Google Scholar 

  71. G. Le Caër: Physica A 159, 329 (1989)

    Article  ADS  Google Scholar 

  72. B. Derrida, G. Weisbuch: Europhys. Lett. 4, 657 (1987)

    Article  ADS  Google Scholar 

  73. A. Coniglio, L. de Arcangelis, H.J. Herrmann, N. Jan: Europhys. Lett. 8, 315 (1989)

    Article  ADS  Google Scholar 

  74. J. Ashkin, E. Teller: Phys. Rev. 64, 178 (1943)

    Article  ADS  Google Scholar 

  75. G. Parisi: Nucl. Phys. B 180, 378 (1981)

    Article  MathSciNet  ADS  Google Scholar 

  76. P. Guenoun, F. Perrot, D. Beysens: Phys. Rev. Lett. 63, 1152 (1989)

    Article  ADS  Google Scholar 

  77. L. de Arcangelis, H.J. Herrmann, A. Coniglio: J. Phys. A 23, L265 (1990)

    Article  Google Scholar 

  78. A.U. Neumann, B. Derrida: J. de Phys. 49, 1647 (1988)

    Article  Google Scholar 

  79. G. Paladin, A. Vulpiani: Phys. Rep. 156, 147 (1987)

    Article  MathSciNet  ADS  Google Scholar 

  80. L. de Arcangelis, A. Coniglio, H.J. Herrmann: Europhys. Lett. 9, 749 (1989)

    Article  ADS  Google Scholar 

  81. I.A. Campbell, L. de Arcangelis: Europhys. Lett. 13, 587 (1990)

    Article  ADS  Google Scholar 

  82. L. de Arcangelis, H.J. Herrmann, A. Coniglio: J. Phys. A 22, 4659 (1989)

    Article  ADS  Google Scholar 

  83. N. Boissin, H.J. Herrmann: J. Phys. A 24, L43 (1991)

    Article  ADS  Google Scholar 

  84. L. de Arcangelis, A. Coniglio: Preprint; A. Coniglio: In Correlations and Connectivity in Constrained Geometries, ed. by H.E. Stanley, N. Ostrowsky ( Kluwer, Dordrecht 1990 )

    Google Scholar 

  85. H.R. da Cruz, U.M.S. Costa, E.M.F. Curado: J. Phys. A 22, L651 (1989)

    Article  Google Scholar 

  86. A.M. Mariz, H.J. Herrmann, L. de Arcangelis: J. Stat. Phys. 59, 1043 (1990)

    Article  ADS  Google Scholar 

  87. D. Stauffer: Physica A 162, 27 (1990)

    Article  MathSciNet  ADS  Google Scholar 

  88. A.M. Mariz, H.J. Herrmann: J. Phys. A 22, L1081 (1989)

    Article  ADS  Google Scholar 

  89. O. Golinelli: Physica A 167, 736 (1990)

    Article  ADS  Google Scholar 

  90. O. Golinelli, B. Derrida: J. de Phys. 49, 1663 (1988)

    Article  MathSciNet  Google Scholar 

  91. S.S. Manna: J. de Phys. 51, 1261 (1990)

    Article  Google Scholar 

  92. M.N. Barber, B. Derrida: J. Stat. Phys. 51, 877 (1988)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  93. E.N. Miranda, N. Parga: J. Phys. A 24, 1059 (1991)

    Article  ADS  Google Scholar 

  94. D.N. Sutherland: J. Colloid Interface Sci. 25, 373 (1967)

    Article  Google Scholar 

  95. P. Meakin: J. Colloid Interface Sci. 102, 505 (1984)

    Article  Google Scholar 

  96. E. Burgos, H. Bonadeo: J. Phys. A 20, 1193 (1987)

    Article  ADS  Google Scholar 

  97. E. Hinrichsen, J. Feder, T. JOssang: J. Stat. Phys. 44, 793 (1986)

    Article  ADS  Google Scholar 

  98. P. Schaaf, J. Talbot: Phys. Rev. Lett. 62, 175 (1989)

    Article  ADS  Google Scholar 

  99. R.D. Vipst, R.M. Ziff: J. Chem. Phys. 91, 2599 (1989)

    Article  ADS  Google Scholar 

  100. R.M. Ziff, E. Gulari, Y. Barshad: Phys. Rev. Lett. 56, 2553 (1986)

    Article  ADS  Google Scholar 

  101. I. Jensen, H.C. Fogedby, R. Dickman: Phys. Rev. A 41, 3411 (1990)

    Article  ADS  Google Scholar 

  102. S. Hayward, D.W. Herrmann, K. Binder: J. Stat. Phys. 49, 1053 (1987)

    Article  ADS  Google Scholar 

  103. R. Dickman: Phys. Rev. B 40, 7005 (1989)

    Article  ADS  Google Scholar 

  104. D.B. Abraham, J. Heiniö, K. Kaski: Phys. Rev. B (1990)

    Google Scholar 

  105. P. Ossadnik: Physica A 176, 454 (1991)

    Article  ADS  Google Scholar 

Download references

Authors
  1. Hans J. Herrmann
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institut für Physik, Johannes-Gutenberg-Universität Mainz, Staudingerweg 7, D-6500, Mainz, Fed. Rep. of Germany

    Kurt Binder

Rights and permissions

Reprints and permissions

Copyright information

© 1992 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Herrmann, H.J. (1992). Simulation of Random Growth Processes. In: Binder, K. (eds) The Monte Carlo Method in Condensed Matter Physics. Topics in Applied Physics, vol 71. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-02855-1_5

Download citation

  • DOI: https://doi.org/10.1007/978-3-662-02855-1_5

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-02857-5

  • Online ISBN: 978-3-662-02855-1

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation