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Rev. Mod. Phys. 68, 963–983 (1996)

[Issue 4 – October 1996 ]

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Stochastic growth equations and reparametrization invariance

Matteo Marsili
Department of Physics, Shuster Laboratory, The University of Manchester, M13 9PL Manchester, United Kingdom
Amos Maritan
Istituto Nazionale Fisica della Materia and International School for Advanced Studies (S.I.S.S.A.) via Beirut 2-4, Trieste 34014 and sezione INFN di Trieste, Italy
Flavio Toigo
Istituto Nazionale Fisica della Materia and Dipartimento di Fisica, Padova University, via Marzolo 8, 35100 Padova, Italy
Jayanth R. Banavar
Department of Physics and Center for Material Physics, 104 Davey Laboratory, The Pennsylvania State University, University Park, Pennsylvania 16802

This article reviews the role of reparametrization invariance (the invariance of the properties of a system with respect to the choice of the co-ordinate system used to describe it) in deriving stochastic equations that describe the growth of surfaces. By imposing reparametrization invariance on a system, the authors identify the physical origin of many of the terms in its growth equations. Both continuum-growth equations for interfaces and equations for the coarse-grained evolution of discrete-lattice models are derived with this method. A detailed analysis of the discrete-lattice case and its small-gradient expansion provides a physical basis for terms found in commonly studied growth equations. The reparametrization-invariant formulation of growth processes also has the advantage of allowing one to model shadowing effects that are lost in the no-overhang approximation and to conserve underlying symmetries of the system that are lost in a small-gradient expansion. Finally, a knowledge of the full equation of motion, beyond the lowest-order gradient expansion, may be relevant in problems where the usual perturbative renormalization methods fail. [S0034-6861(96)00104-3]

©1996 The American Physical Society

URL: http://link.aps.org/abstract/RMP/v68/p963
DOI: 10.1103/RevModPhys.68.963

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References

(Reference links marked with dot may require a separate subscription.)
  1. Amar, J. G., and F. Family, 1993, Phys. Rev. E 47, 1595 .
  2. Bales, G. S., R. Bruinsma, E. A. Eklund, R. P. U. Karunasiri, J. Rudnick, and A. Zangwill, 1990, Science 249, 264 [dot INSPEC].
  3. Barabási, A.-L., and H. E. Stanley, 1995, Fractal Concepts in Surface Growth (Cambridge University, New York).
  4. Bausch, R., V. Dohm, H. K. Janssen, and R. K. P. Zia, 1981, Phys. Rev. Lett. 47, 1837 .
  5. Bruinsma, R., and G. Aeppli, 1984, Phys. Rev. Lett. 52, 1547 .
  6. Buldyrev, S. V., A.-L. Barabási, F. Casetra, S. Havlin, H. E. Stanley, and T. Vicsek, 1992, Phys. Rev. A 44, R8313.
  7. Cieplak, M., and M. O. Robbins, 1988, Phys. Rev. Lett. 60, 2042 .
  8. Cook, J., and B. Derrida, 1991, J. Stat. Phys. 63, 505 [dot INSPEC].
  9. Das Sarma, S., 1994, Fractals 1, 784 [dot INSPEC].
  10. Das Sarma, S., and S. V. Ghaisas, 1992, Phys. Rev. Lett. 69, 3762 .
  11. Das Sarma, S., S. V. Ghaisas, and J. M. Kim, 1994, Phys. Rev. E 49, 122 .
  12. Das Sarma, S., and R. Kotlyar, 1994, Phys. Rev. E 50, R4275 .
  13. Das Sarma, S., and P. Tamborenea, 1991, Phys. Rev. Lett. 66, 325 .
  14. Das Sarma, S., C. I. Lanczycki, R. Kotlyar, and S. V. Ghaisas, 1996, Phys. Rev. E 53, 359 .
  15. Deker, U., and F. Haake, 1975, Phys. Rev. A 11, 2043 .
  16. Delker, T., D. B. Pengra, and P.-z. Wong, 1996, Phys. Rev. Lett. 76, 2902 .
  17. Derrida, B., and H. Spohn, 1988, J. Stat. Phys. 51, 817 [dot INSPEC].
  18. Doherty, J. P., M. A. Moore, J. M. Kim, and A. J. Bray, 1994, Phys. Rev. Lett. 72, 2041 .
  19. Eden M., 1958, in Symposium on Information Theory in Biology, edited by H. P. Yockey (Pergamon, New York), p. 359.
  20. Edwards, S. F., and D. R. Wilkinson, 1982, Proc. R. Soc. London, Ser. A 381, 17 [dot INSPEC].
  21. Family, F., and T. Vicsek, 1985, J. Phys. A 18, L75.
  22. Family, F., and T. Vicsek, 1991, Dynamics of Fractal Surfaces (World Scientific, Singapore).
  23. Fisher, D. S., 1986, Phys. Rev. Lett. 56, 1964 .
  24. Fisher, D. S., and D. A. Huse, 1991, Phys. Rev. B 43, 10728 .
  25. Forster, D., D. R. Nelson, and M. J. Stephen, 1977, Phys. Rev. A 16, 732 .
  26. Frey, E., and U. C. Täuber, 1994, Phys. Rev. E 50, 1024 .
  27. Galluccio, S., and Y.-C. Zhang, 1995, Phys. Rev. E 51, 1686 .
  28. Giugliarelli, G., and A. L. Stella, 1991, Phys. Scr. Vol. T 35, 34 [dot INSPEC].
  29. Giugliarelli, G., and A. L. Stella, 1994, Physica A 212, 12 [dot INSPEC].
  30. Golubovic, L., and R. P. U. Karunasiri, 1991, Phys. Rev. Lett. 66, 3156 .
  31. Grossmann, B., H. Guo, and M. Grant, 1991, Phys. Rev. A 43, 1727 .
  32. Halpin-Healy, T., and Y.-C. Zhang, 1995, Phys. Rep. 254.
  33. He, S., G. L. M. K. S. Kahanda, and P.-z. Wong, 1992, Phys. Rev. Lett. 69, 3731 .
  34. Herrasti, P., P. Ocón, L. Vasquez, R. C. Salvarezza, J. M. Vara, and A. J. Arvia, 1992, Phys. Rev. A 45, 7440 .
  35. Hohenberg, P. C., and B. I. Halperin, 1977, Rev. Mod. Phys. 49, 435  [SPIRES].
  36. Horváth, V. K., F. Family, and T. Vicsek, 1991, Phys. Rev. Lett. 67, 3207 .
  37. Hunt, A. W., C. Orme, D. R. M. Williams, B. G. Orr, and L. M. Sander, 1994, Europhys. Lett. 27, 611 [dot INSPEC].
  38. Huse, D. A., J. G. Amar, and F. Family, 1990, Phys. Rev. A 41, 7075 .
  39. Hwa, T., and M. Kardar, 1989, Phys. Rev. Lett. 62, 1813 .
  40. Jensen, M. H., and I. Procaccia, 1991, J. Phys. II, 1, 1139 [dot INSPEC].
  41. Ji, H., and M. O. Robbins, 1991, Phys. Rev. B 46, 14519 .
  42. Jiang, Z., and H. G. E. Hentschel, 1992, Phys. Rev. A 45, 4196.
  43. Kapral, R., R. Livi, G. Oppo, and A. Politi, 1994, Phys. Rev. E 49, 2009 .
  44. Kardar, M., 1994, Turkish J. of Phys. 18, 221.
  45. Kardar, M., G. Parisi, and Y.-C. Zhang, 1986, Phys. Rev. Lett. 56, 889  [SPIRES].
  46. Kardar, M., and Y.-C. Zhang, 1987, Phys. Rev. Lett. 58, 2087 .
  47. Keblinski, P., A. Maritan, F. Toigo, and J. R. Banavar, 1994, Phys. Rev. E 49, R4795 .
  48. Keblinski, P., A. Maritan, F. Toigo, and J. R. Banavar, 1995, Phys. Rev. Lett. 74, 1783 .
  49. Keblinski, P., A. Maritan, F. Toigo, J. Koplik, and J. R. Banavar, 1994, Phys. Rev. E 49, R937 .
  50. Keblinski, P., A. Maritan, F. Toigo, R. Messier, and J. R. Banavar, 1996, Phys. Rev. E 53, 759 .
  51. Kessler, D. A., H. Levine, and L. M. Sander, 1992, Phys. Rev. Lett. 69, 100 .
  52. Kim, J. M., and S. Das Sarma, 1994, Phys. Rev. Lett. 72, 2903 .
  53. Kim, J. M., and S. Das Sarma, 1995, Phys. Rev. B 51, 1889.
  54. Kim, J. M., and J. M. Kosterlitz, 1989, Phys. Rev. Lett. 62, 2289 .
  55. Kim, J. M., M. A. Moore, and A. J. Bray, 1991, Phys. Rev. A 44, 2345 .
  56. Ko, D. Y. K., and F. Seno, 1994, Phys. Rev. E 50, R1741 .
  57. Koplik, J., and Levine H., 1985, Phys. Rev. B 32, 280 .
  58. Kotrla, M., A. C. Levi, and P. Smilauer, 1992, Europhys. Lett. 20, 25 [dot INSPEC].
  59. Krug, J., 1989, J. Phys. A 22, L769 [dot INSPEC].
  60. Krug, J., 1991, J. Phys. I 1, 9 [dot INSPEC].
  61. Krug, J., 1994, Phys. Rev. Lett. 72, 2907 .
  62. Krug, J., and P. Meakin, 1989, Phys. Rev. A 40, 2064 .
  63. Krug, J., and P. Meakin, 1991, Phys. Rev. A 43, 900 .
  64. Krug, J., M. Plischke, and M. Siegert, 1993, Phys. Rev. Lett. 70, 3271 .
  65. Krug, J., and H. Spohn, 1990, in Solids Far From Equilibrium: Growth, Morphology and Defects, edited by C. Godréche (Cambridge University, New York), p. 479.
  66. Lai, Z. W., and S. Das Sarma, 1991, Phys. Rev. Lett. 66, 2348 .
  67. Lam, C.-H., and L. M. Sander, 1993, Phys. Rev. E 48, 979 .
  68. Lam, C.-H., L. M. Sander, and D. E. Wolf, 1992, Phys. Rev. A 46, R6128 .
  69. Leschhorn, H., 1993, Physica A 200, 136 [dot INSPEC].
  70. Liu, D., and M. Plischke, 1988, Phys. Rev. B 38, 4781 .
  71. Ma, S. K., and G. F. Mazenko, 1975, Phys. Rev. B 11, 4077 .
  72. Makse, H. A., 1995, Phys. Rev. E 52, 4080 .
  73. Mandelbrot, B. B., 1986, in Fractals in Physics, edited by L. Pietronero and E. Tosatti (Elsevier Science, Amsterdam), pp. 3, 17, 21.
  74. Maritan, A., F. Toigo, J. Koplik, and J. R. Banavar, 1992, Phys. Rev. Lett. 69, 3193 .
  75. Marsili, M., and A. J. Bray, 1996, Phys. Rev. Lett. 73, 2750.
  76. Marsili, M., A. Maritan, F. Toigo, and J. R. Banavar, 1996, Europhys. Lett. 35, 171 [dot INSPEC].
  77. Martys, N., M. O. Robbins, and M. Cieplak, 1991, Phys. Rev. B 44, 12294 .
  78. Mazor, A., D. J. Srolovitz, P. S. Hagan, and B. G. Bukiet, 1988, Phys. Rev. Lett. 60, 424 .
  79. Meakin, P., 1988a, in Phase Transition and Critical Phenomena, edited by C. Domb and J. L. Lebowitz (Academic, London), vol. 12, p. 335.
  80. Meakin, P., 1988b, Phys. Rev. A 38, 994 .
  81. Meakin, P., 1993, Phys. Rep. 235, 189 [dot INSPEC].
  82. Meakin, P., A. Coniglio, H. E. Stanley, and T. A. Witten, 1986, Phys. Rev. A 34, 3325 .
  83. Meakin, P., P. Ramanlal, L. M. Sander, and R. C. Ball, 1986, Phys. Rev. A 34, 5091 .
  84. Medina, E., T. Hwa, M. Kardar, and Y.-C. Zhang, 1989, Phys. Rev. A 39, 3053 .
  85. Messier, R., and J. E. Yehoda, 1985, J. Appl. Phys. 58, 3739 [dot SPIN][dot INSPEC].
  86. Narayan, O., and D. S. Fisher, 1993, Phys. Rev. B 48, 7030 .
  87. Nattermann, T., 1985, J. Phys. C 18, 6661 [dot INSPEC].
  88. Nattermann, T., S. Stepanov, L.-H. Tang, and H. Leschhorn, 1992, J. Phys. II 2, 1483 [dot INSPEC].
  89. Niemeyer, L., L. Pietronero, and H. J. Wiesmann, 1984, Phys. Rev. Lett. 52, 1033 .
  90. Nittmann, J., G. Daccord, and H. E. Stanley, 1985, Nature 314, 141 [dot INSPEC].
  91. Nolle, C. S., B. Koiller, N. Martys, and M. O. Robbins, 1993, Phys. Rev. Lett. 71, 2074 .
  92. Olami, Z., I. Procaccia, and R. Zeitak, 1994, Phys. Rev. E 49, 1232 .
  93. Pal, S., and D. P. Landau, 1994, Phys. Rev. B 49, 10597 .
  94. Parisi, G., 1992, Europhys. Lett. 17, 673 [dot INSPEC][SPIRES].
  95. Peng, C.-K., S. Havlin, M. Schwartz, and H. E. Stanley, 1991, Phys. Rev. A 44, R2239 .
  96. Pfeifer, P., Y. J. Wu, M. W. Cole, and J. Krim, 1989, Phys. Rev. Lett. 62, 1997 .
  97. Pietronero, L., and E. Tosatti, 1985, Fractals in Physics (North-Holland, Amsterdam).
  98. Plischke, M., and Z. Rácz, 1985, Phys. Rev. A 32, 3825 .
  99. Rácz, Z., M. Siegert, D. Liu, and M. Plischke, 1991, Phys. Rev. A 43, 5275 .
  100. Salvarezza, R. C., L. Vasquez, P. Herrasti, P. Ocón, J. M. Vara, and A. J. Arvia, 1992, Europhys. Lett. 20, 727 [dot INSPEC].
  101. Schroeder, M., M. Siegert, D. E. Wolf, J. D. Shore, and M. Plischke, 1993, Europhys. Lett. 24, 563 [dot INSPEC].
  102. Siegert, M., and M. Plischke, 1992, Phys. Rev. Lett. 68, 2035 .
  103. Siegert, M., and M. Plischke, 1994, Phys. Rev. Lett. 73, 1517 .
  104. Sivashinsky, G. I., 1977, Acta Astron. 4, 1177.
  105. Sivashinsky, G. I., 1979, Acta Astron. 6, 560.
  106. Sneppen, K., 1992, Phys. Rev. Lett. 69, 3539 .
  107. Stokes, J. P., A. P. Kushnick, and M. O. Robbins, 1988, Phys. Rev. Lett. 60, 1386 .
  108. Sun, T., H. Guo, and M. Grant, 1989, Phys. Rev. A 40, 6763 .
  109. Sun, T., and M. Plischke, 1993, Phys. Rev. Lett. 71, 3174 .
  110. Sun, T., and M. Plischke, 1994a, Phys. Rev. E 50, 3370 .
  111. Sun, T., and M. Plischke, 1994b, Phys. Rev. E 49, 5406.
  112. Tang, C., S. Alexander, and R. Bruinsma, 1990, Phys. Rev. Lett. 64, 772  [SPIRES].
  113. Tang, C., and S. Liang, 1993, Phys. Rev. Lett. 71, 2769 .
  114. Tang, L.-H., B. M. Forrest, and D. E. Wolf, 1992, Phys. Rev. A 45, 7162 .
  115. Tang, L.-H., and H. Leschhorn, 1992, Phys. Rev. A 45, R8309 .
  116. Tang, L.-H., and H. Leschhorn, 1993, Phys. Rev. Lett. 70, 3832 .
  117. Tang, L.-H., and T. Nattermann, 1991, Phys. Rev. Lett. 66, 2899 .
  118. Tu, C. W., and J. S. Harris, Jr., 1991, J. Cryst. Growth 111, R9.
  119. Villain, J., 1991, J. Phys. I 1, 19 [dot INSPEC].
  120. Villain, J., A. Pimpinelli, and D. Wolf, 1992, Comments Condens. Matter Phys. 15, 1.
  121. Vvedensky, D. D., A. Zangwill, C. N. Luse, and M. R. Wilby, 1993, Phys. Rev. E 48, 852 .
  122. Witten, T. A., and L. M. Sander, 1981, Phys. Rev. Lett. 47, 1400 .
  123. Wolf, D. E., 1991, Phys. Rev. Lett. 67, 1783 .
  124. Zhang, Y.-C., 1990, J. Phys. 51, 2129.
  125. Zinn-Justin, J., 1993, Quantum Field Theory and Critical Phenomena (Oxford Science, New York).

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