Avalanche dynamics in evolution, growth, and depinning models

Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

Volumn 53, Issue 1, 1996, Pages 414-443

  Paczuski, Maya ^a,b   Maslov, Sergei ^a,b   Bak, Per ^a,b  

^a BROOKHAVEN NATIONAL LABORATORY   (United States)

^b STONY BROOK UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0001414241     PISSN: 1063651X     EISSN: None     Source Type: Journal    
DOI: 10.1103/PhysRevE.53.414     Document Type: Article

References (114)

1. Y. C. Zhang, J. Phys. (France) I 1, 971 (1991);
2. H.C. Fogedby, J. Stat. Phys. 69, 411 (1992).
3. B.B. Mandelbrot, The Fractal Geometry of Nature (Freeman, New York, 1983).
4. P.H. Coleman, L. Pietronero and R.H. Sanders, Astron. Astrophys. 200, L32 (1988);
5. L. Pietronero, Physica A 144, 257 (1987).
6. A. Rinaldo, I. Rodriguez Iturbe, R. Rigon, E. Ijjasz Vasquez and R.L. Bras, Phys. Rev. Lett. 70, 822 (1993).
7. Fractals in the Earth Sciences, edited by C.C. Barton and P.R. Lapointe (Plenum, New York, 1994).
8. W.H. Press, Comments Astrophys. 7, 103 (1978).
9. See, for example, the empirical observations of H.E. Hurst described in Refs. cite mandelbrot and cite feder.
10. F. Gruneis, M. Nakao and M. Yamamoto, Biol. Cybernetics 62, 407 (1990).
11. B. Gutenberg and C.F. Richter, Ann. Geofis. 9, 1 (1956).
12. E. Somfai, A. Czirok and T. Vicsek, J. Phys. A 27, L757 (1994).
13. V. Frette, K. Christensen, A. Malthe Sorensen, J. Feder, T. Jossang, and P. Meakin (unpublished).
14. S. Field, J. Witt, F. Nori and X. Ling, Phys. Rev. Lett. 74, 1206 (1995).
15. S.J. Gould and N. Eldredge, Paleobiology 3, 114 (1977);
16. S.J. Gould and N. Eldredge, Nature 366, 223 (1993).
17. J. Feder, Fractals (Plenum, New York, 1989).
18. H. Bai Lin, Chaos (World Scientific, Singapore, 1984).
19. P. Bak, C. Tang and K. Wiesenfeld, Phys. Rev. Lett. 59, 381 (1987);
20. P. Bak, C. Tang and K. Wiesenfeld, Phys. Rev. A 38, 364 (1988)
21. P. Bak and M. Creutz, in Fractals and Disordered Systems, edited by A. Bunde and S. Havlin (Springer, Berlin, 1994), Vol. II;
22. P. Bak and M. Paczuski, Phys. World 6, (12), 39 (1993).
23. C. Tang and P. Bak, Phys. Rev. Lett. 60, 2347 (1988).
24. P. Bak and C. Tang, J. Geophys. Res. B 94, 15635 (1989)
25. Z. Olami, H.J.S. Feder and K. Christensen, Phys. Rev. Lett. 68, 1244 (1992);
26. K. Christensen and Z. Olami, Phys. Rev. A 46, 1829 (1992).
27. J.M. Carlson, J.T. Chayes, E.R. Grannan and G.H. Swindle, Phys. Rev. Lett. 65, 2547 (1990).
28. P. Bak, K. Chen and C. Tang, Phys. Lett. A 147, 297 (1990);
29. B. Drossel and F. Schwabl, Phys. Rev. Lett. 69, 1629 (1992).
30. B. Drossel, S. Clar and F. Schwabl, Phys. Rev. Lett. 71, 3739 (1993).
31. M. Paczuski and P. Bak, Phys. Rev. E 48, R3214 (1993).
32. D. Dhar, Phys. Rev. Lett. 64, 1613 (1990);
33. D. Dhar and S.N. Majumdar, J. Phys. A 23, 4333 (1990);
34. S.N. Majumdar and D. Dhar, Physica A 185, 129 (1992).
35. D. Dhar and R. Ramaswamy, Phys. Rev. Lett. 63, 1659 (1989).
36. L. Pietronero, A. Erzan and C. Evertsz, Phys. Rev. Lett. 61, 861 (1988);
37. L. Pietronero, A. Erzan and C. Evertsz, Physica A 151, 207 (1988)
38. R. Cafiero, L. Pietronero and A. Vespignani, Phys. Rev. Lett. 70, 3939 (1993).
39. T.A. Witten and L.M. Sander, Phys. Rev. Lett. 47, 1400 (1981).
40. L. Pietronero, A. Vespignani and S. Zapperi, Phys. Rev. Lett. 72, 1690 (1994);
41. A. Vespignani, S. Zapperi and L. Pietronero, Phys. Rev. E 51, 1711 (1995).
42. M. Marsili, Europhys. Lett. 28, 385 (1994).
43. S.I. Zaitsev, Physica A 189, 411 (1992).
44. S.L. Miller, W.M. Miller and P.J. McWhorter, J. Appl. Phys. 73, 2617 (1993).
45. P. Bak and K. Sneppen, Phys. Rev. Lett. 71, 4083 (1993).
46. K. Sneppen, Phys. Rev. Lett. 69, 3539 (1992) .
47. D. Wilkinson and J.F. Willemsen, J. Phys. A 16, 3365 (1983).
48. R. Lenormand, C.R. Acad. Sci. Ser. B 291, 279 (1980);
49. R. Chandler, J. Koplik, K. Leman and J. Willemsen, J. Fluid Mech. 119, 249 (1982).
50. S. Roux and E. Guyon, J. Phys. A 22, 3693 (1989).
51. S. Maslov, Phys. Rev. Lett. 74, 562 (1995).
52. M. Paczuski, S. Maslov, and P. Bak, Brookhaven National Laboratory Report No. BNL 49916, 1993 (unpublished);
53. Europhys. Lett. 27, 97 (1994).
54. M. Paczuski, S. Maslov and P. Bak, Europhys. Lett. 28, 295 (1994).
55. M. Paczuski, P. Bak and S. Maslov, Phys. Rev. Lett. 74, 4253 (1995).
56. S. Maslov, M. Paczuski and P. Bak, Phys. Rev. Lett. 73, 2162 (1994).
57. P. Grassberger, Phys. Lett. A 200, 277 (1995)
58. and unpublished.
59. B. Jovanovic, S.V. Buldyrev, S. Havlin and H.E. Stanley, Phys. Rev. E 50, 2403 (1994).
60. P. Sornette, J. Phys. (France) I 4, 209 (1994).
61. L.A.N. Amaral , Phys. Rev. E 51, 4655 (1995).
62. S.V. Buldyrev , Phys. Rev. A 45, R8313 (1992);
63. S.V. Buldyrev , Physica A 200, 200 (1993).
64. M. Kardar, G. Parisi and Y. C. Zhang, Phys. Rev. Lett. 56, 889 (1986).
65. T. Hwa and M. Kardar, Phys. Rev. Lett. 62, 1813 (1989);
66. G. Grinstein, D. H. Lee and S. Sachdev, Phys. Rev. Lett. 64, 1927 (1990).
67. M. Paczuski, Phys. Rev. E 52, 2137 (1995).
68. K. Sneppen, P. Bak, H. Flyvbjerg and M.H. Jensen, Proc. Natl. Acad. Sci. U.S.A. 92, 5209 (1995);
69. P. Bak and M. Paczuski, Proc. Natl. Acad. Sci. U.S.A. 92, 6689 (1995).
70. H. Flyvbjerg, K. Sneppen and P. Bak, Phys. Rev. Lett. 45, 4087 (4087).
71. T.S. Ray and N. Jan, Phys. Rev. Lett. 72, 4045 (1994).
72. J. de Boer, B. Derrida, H. Flyvbjerg, A.D. Jackson and T. Wettig, Phys. Rev. Lett. 73, 906 (1994);
73. J. de Boer, A.D. Jackson and T. Wettig, Phys. Rev. E 51, 1059 (1995).
74. S. Boettcher and M. Paczuski (unpublished).
75. L. H. Tang and H. Leschhorn, Phys. Rev. Lett. 70, 3832 (1993);
76. H. Leschhorn and L. H. Tang, Phys. Rev. E 49, 1238 (1994).
77. S. Maslov and M. Paczuski, Phys. Rev. E 50, R643 (1994).
78. S. Roux and A. Hansen, J. Phys. (France) I 4, 515 (1994).
79. S.F. Edwards and D.R. Wilkinson, Proc. R. Soc. London Ser. A 381, 17 (1982).
80. H. Leschhorn, Physica A 195, 324 (1993).
81. M.A. Rubio, C.A. Edwards, A. Dougherty and J.P. Gollub, Phys. Rev. Lett. 63, 1685 (1989).
82. N. Martys, M.O. Robbins and M. Cieplak, Phys. Rev. B 44, 12294 (1991).
83. R. Bruinsma and G. Aeppli, Phys. Rev. Lett. 52, 1547 (1984);
84. H. Ji and M.O. Robbins, Phys. Rev. B 46, 14519 (1992).
85. L. H. Tang and H. Leschhorn, Phys. Rev. A 45, R8309 (1992).
86. D. Stauffer, Introduction to Percolation Theory (Taylor, London, 1985); G. Grimmett, Percolation (Springer, New York, 1989).
87. I. Jensen and R. Dickman, Phys. Rev. E 48, 1710 (1993).
88. O. Narayan and D.S. Fisher, Phys. Rev. B 48, 7030 (1993).
89. Z. Olami, I. Procaccia and R. Zeitak, Phys. Rev. E 49, 1232 (1994);
90. Z. Olami, I. Procaccia and R. Zeitak, Phys. Rev. E 52, 3402 (1995).
91. K. Sneppen (private communication).
92. The exponent relations derived in this section can be generalized to the case where the set of covered sites forms a fractal of dimension dcovd rather than being compact. This is accomplished by replacing d with dcov in all formulas in this section.
93. J.F. Gouyet, Physica A 168, 581 (1990).
94. P. Grassberger and A. de la Torre, Ann. Phys. (N.Y.) 122, 373 (1979).
95. Note that our definition of η differs from the usual definition of η, which is defined with respect to parallel time, as in Ref. cite Grassb79, and which we refer to as ηt. The quantity which we call η is equal to zηt/D.
96. K. Sneppen and M.H. Jensen, Phys. Rev. Lett. 71, 101 (1993).
97. J. Falk, M.H. Jensen and K. Sneppen, Phys. Rev. E 49, 2804 (1994).
98. L. Furuberg, J. Feder, A. Aharony and T. Jossang, Phys. Rev. Lett. 61, 2117 (1988).
99. S.B. Lowen and M.C. Teich, Phys. Rev. E 47, 992 (1993).
100. K. Ito, Phys. Rev. E 52, 3232 (1995).
101. J.W. Essam, K. De'Bell, J. Adler and F. Bhatti, Phys. Rev. B 33, 1982 (1986).
102. N.P. Ong and P. Monceau, Phys. Rev. B 18, 5272 (1978);
103. R.M. Fleming and C.C. Grimes, Phys. Rev. Lett. 42, 1423 (1979);
104. P.A. Lee and T.M. Rice, Phys. Rev. B 19, 3970 (1979).
105. O. Narayan and A. Middleton, Phys. Rev. B 49, 244 (1994).
106. T. Nattermann, S. Stepanow, L. H. Tang and H. Leschhorn, J. Phys. (France) II 2, 1483 (1992);
107. O. Narayan and D.S. Fisher, Phys. Rev. B 46, 11520 (1992)
108. D. Ertas and M. Kardar, Phys. Rev. E 49, R2532 (1994).
109. S. Galluccio and Y. C Zhang, Phys. Rev. E 51, 1686 (1995).
110. F. Family and T. Viscek, J. Phys. A 18, L75 (1985).
111. T. Halpin Healy and Y. C. Zhang, Phys. Rep. 254, 215 (1995).
112. S. Das Sarma, S.V. Chaisas and J.M. Kim, Phys. Rev. E 49, 122 (1994).
113. H. Saleur and B. Duplantier, Phys. Rev. Lett. 58, 2325 (1987).
114. T. Grossman and A. Aharony, J. Phys. A 19, L745 (1986).