Scale Invariant Avalanches: A Critical Confusion (original) (raw)

Abstract

The "Self-organized criticality" (SOC), introduced in 1987 by Bak, Tang and Wiesenfeld, was an attempt to explain the 1/f noise, but it rapidly evolved towards a more ambitious scope: explaining scale invariant avalanches. In two decades, phenomena as diverse as earthquakes, granular piles, snow avalanches, solar flares, superconducting vortices, sub-critical fracture, evolution, and even stock market crashes have been reported to evolve through scale invariant avalanches. The theory, based on the key axiom that a critical state is an attractor of the dynamics, presented an exponent close to -1 (in two dimensions) for the power-law distribution of avalanche sizes. However, the majority of real phenomena classified as SOC present smaller exponents, i.e., larger absolute values of negative exponents, a situation that has provoked a lot of confusion in the field of scale invariant avalanches. The main goal of this chapter is to shed light on this issue. The essential role of ...

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References (98)

  1. Olami, Z.; Feder, H. J. S.; Christensen, K. Phys. Rev. Lett. 1992, 68, 1244.
  2. Bak, P.; Tang C. J. Geophys. Res. 1989, 94 (15) 635.
  3. Jagla, E. A. Phys. Rev. E 2010, 81, 046117.
  4. Held, A. et al. Phys. Rev. Lett. 1990, 65, 1120.
  5. Frette, V. et al. Nature 1996, 379, 49.
  6. Altshuler, E.; Ramos, O.; Martínez, C.; Flores, L. E.; Noda, C. Phys. Rev. Lett. 2001, 86, 5490.
  7. Aegerter, C. M.; Günther, R.; Wijngaarden, R. J. Phys. Rev. E 2003, 67, 051306.
  8. Costello, R. M. et al. Phys. Rev. E 2003, 67, 041304.
  9. Nerone, N.; Aguirre, M. A.; Calvo, A.; Bideau, D.; Ippolito, I. Phys. Rev. E 2003, 67, 011302.
  10. Aegerter, C. M.; Lõrincz, K. A.; Welling, M. S.; Wijngaarden, R. J. Phys. Rev. Lett. 2004, 92, 058702.
  11. Lõrincz, K. A.; Wijngaarden, R. J. Phys. Rev. E 2007, 76, 040301(R).
  12. Ramos, O.; Altshuler, E.; Måløy, K. J. Phys. Rev. Lett. 2009, 102, 078701.
  13. Birkeland, K. W.; Landry, C. C. Geophys. Res. Lett. 2002, 29 (11), 1554.
  14. Hamon, D.; Nicodemi, M.; Jensen, H. J. Astron. Astrophys. 2002, 387, 326334.
  15. Bassler, K.; Paczuski, M. Phys. Rev. Lett. 1998, 81, 37613764.
  16. Altshuler, E.; Johansen, T. H. Rev. Mod. Phys. 2004, 76, 471487.
  17. Altshuler, E. et al. Phys. Rev. B 2004, 70, 140505(R).
  18. Altshuler, E. et al. Physica C 2004, 501, 408-410.
  19. Santucci, S.; Vanel, L.; Ciliberto, S. Phys. Rev. Lett. 2004, 93, 095505
  20. Sneppen, K.; Bak, P.; Flyvbjerg, H.; Jensen, M. H. PNAS 1995, 92, 5209.
  21. Lee, Y.; Nunes Amaral, L. A.; Canning, D.; Meyer, M.; Stanley, H. E. Phys. Rev. Lett. 1998, 81, 3275.
  22. Bak, P.; Tang C.; Wiesenfeld K. Phys. Rev. Lett. 1987, 59, 381-384.
  23. Jensen, H. J. Self-Organized Criticality, Emergent Complex Behavior in Physical and Biological Systems; Cambridge Univ. Press: Cambridge, 1998.
  24. Bak, P. How Nature works-The Science of Self-organized Criticality; Oxford Univ. Press: Oxford, 1997.
  25. Alstrøm, P. Phys. Rev. A 1988, 38, 4905-4906.
  26. Pietronero, L.; Vespignani, A.; Zapperi, S. Phys. Rev. Lett. 1994, 72, 1690.
  27. Vespignani, A.; Zapperi, S. Phys. Rev. Lett. 1997, 78, 4793-4796.
  28. Mandelbrot, B. B. Les Objets fractals, forme, hasard et dimension; Flammarion: Paris, 1975.
  29. Mandelbrot, B. B. The Fractal Geometry of Nature; W. H. Freeman & Co: San Fran- cisco, CA, 1982.
  30. Koch, H. V. In Classics on Fractals; Edgar, G. A.; Ed.; Addison-Wesley, Reading MA, 1993.
  31. Gutenberg, B.; Richter, C. F. Ann. Geophys. 1956, 9, 1-15.
  32. Peek, J. B. H.; Bergmans, J. W. M.; Haaren, J. A. M. M.; Toolenaar, F.; Stan, S. G. Origins and Successors of the Compact Disc; Eindhoven, 2009; Philips Research Book Series, Vol. 11.
  33. Russ, J. C. The image processing handbook; CRS Press: Boca Raton, FL, 2002; pp 23.
  34. Pogson, N. R. Mon. Not. R. Astron. Soc. 1856, 17, 12-15.
  35. Ptolemy. First printed in Almagestum Cl. Ptolemei; Venice, 1515.
  36. Toomer, G. T. In Dictionary of Scientific Biography; Gillespie, C. C.; Ed.; Charles Scribner, New York, 1978; Vol. 15, pp 207-224.
  37. Fechner, G. T. Elemente der Psychophysik; Breitkopf und Härtel Leipzig, 1860.
  38. Stevens, S. S. Science 1961, 133, 8086.
  39. Richter, C. F. Bull. Seismol. Soc. Am. 1935, 25, 1-32.
  40. Los Angeles Times. (October 01, 1985)
  41. Charles F. Richter Dies; Earthquake Scale Pioneer.
  42. Richter, C. F. Elementary Seismology; W. H. Freeman & Co: San Francisco, CA, 1958; pp 135-149; 650-653.
  43. Sethna, J. P. Statistical Mechanics: Entropy, Order Parameters, and Complexity; Clarendon press: Oxford, 2009; pp 265.
  44. Bak, P.; Christensen, K.; Danon, L.; Scanlon, T. Phys. Rev. Lett. 2002, 88, 178501.
  45. Main, I. Nature Debate 25 February to 8 April (1999). http://www.nature.com/nature/debates/earthquake/equake frameset.html
  46. U.S. Geological Survey (2010). Earthquake Facts and Statistics. http://earthquake.usgs.gov/earthquakes/eqarchives/year/eqstats.php
  47. Ramos, O. Tectonophysics 2010, 485 321-326.
  48. Pruesner, G.; Peters, O. Phys. Rev. E 2006, 73, 025106(R).
  49. Stanley, H. E. Introduction to Phase Transitions and Critical Phenomena; Oxford Univ. Press: New York, 1987.
  50. Clusel, M.; Fortin, J.-Y.; Holdsworth, P. C. W. Phys. Rev. E 2004, 70, 046112.
  51. Newman, M. E. J.; Barkema, G. T. Monte Carlo Methods in Statistical Physics; Ox- ford Univ. Press: New York, 1999.
  52. Onsager, L. Phys. Rev. 1944, 65, 117.
  53. Stauffer, D. Int. J. Mod. Phys. C 1998, 9, 562.
  54. Bramwell, S. T. et al. Phys. Rev. Lett. 2000, 84, 3744.
  55. Bramwell, S. T.; Holdsworth, P. C. W.; Pinton, J.-F. Nature 1998, 396, 552.
  56. Metropolis, N. et al. J. Chem. Phys. 1953, 21, 1087.
  57. Wolff, U. Phys. Rev. Lett. 1989, 62, 361.
  58. Joubaud, S.; Petrosyan, A.; Ciliberto, S.; Garnier, N. B. Phys. Rev. Lett. 2008, 100, 180601.
  59. Planet, R.; Santucci, S.; Ortín, J. Phys. Rev. Lett. 2009, 102, 094502.
  60. Dunlavy, M. J.; Venus, D. Phys. Rev. B 2005, 71, 144406.
  61. Ciliberto, S. Private communication.
  62. Sethna, J. P.; Dahmen, K. A.; Myers, C. R. In Scale Invariance and Beyond; Dubrulle, B.; Graner, F.; Sornette, D.; Ed.; Springer, Berlin, 1997, Les Houches Workshop, March 10-14, pp 87.
  63. Sethna, J. P.; Dahmen, K. A.; Myers, C. R. Nature 2001, 410, 242.
  64. Zapperi, S.; Cizeau, P.; Durin, G.; Stanley, H. E. Phys. Rev. B 1998, 58, 63536366.
  65. Barkhausen, H. Z Phys. 1919, 20, 401.
  66. Labbé, R.; Pinton, J.-F.; Fauve, S. J. Phys. II France 1996, 6, 1099-1110.
  67. Stauffer, D; Aharony, A. Introduction to Percolation Theory; Taylor & Francis: Lon- don, 1994.
  68. Harris, T. E. The theory of branching processes; Springer-Verlag: Berlin, 1963.
  69. Obukhov, S. P. In Random Fluctuations and Pattern Growth; Stanley, H. E.; Os- trowsky, N.; Ed.; Kluwer, Dordrecht, 1988.
  70. Díaz-Guilera, A. Europhys. Lett. 1994, 26, 177.
  71. Le Doussal, P.; Wiese, K. J. Phys. Rev. E 2009, 79, 051106.
  72. Lauritsen, K. B.; Zapperi, S.; Stanley, H. E. Phys. Rev. E 1996, 54, 2483-2488.
  73. Manna, S. S. J. Stat. Phys. 1990, 59, 509.
  74. Chessa, A.; Stanley, H. E.; Vespignani, A.; Zapperi, S. Phys. Rev. E 1999, 59, R12- R15.
  75. Lübeck, S.; Usadel, K. D. Phys. Rev. E 1997, 56, 5138-5143.
  76. Tang. C.; Bak, P. Phys. Rev. Lett. 1998, 60, 2347-2350.
  77. Carvalho, J. X. de; Prado, C. P. C. Phys. Rev. Lett. 2000, 84, 4006.
  78. Miller, G.; Boulter, C. J. Phys. Rev. E 2002, 66, 016123.
  79. Xia, J.; Gould, H.; Klein, W.; Rundle, J. B. Phys. Rev. E 2008, 77, 031132.
  80. Bonachela, J. A.; Muñoz, M. A. J. Stat. Mech. 2009, P09009, 1742-5468.
  81. R. J. Geller, R. J.; Jackson, D. D.; Kagan, Y. Y.; Mulargia, F. Science 1997, 275, 1616.
  82. Ramos, O.; Altshuler, E.; Måløy, K. J. Phys. Rev. Lett. 2006, 96, 098501.
  83. Burridge, R.; Knopoff, L. J. Geophys. Res. Bull. Seismol. Soc. Am. 1967, 57, 341.
  84. Christensen, K.; Olami, Z. J. Geophys. Res. 1992, 97, B6, 8729-8735.
  85. Scheffer, M. et al. Nature 2009, 461, 53-59.
  86. Manshour, P. et al. Phys. Rev. Lett. 2009, 102, 014101.
  87. Crampin, S.; Gao, Y.; Peacock, S. Geology 2008, 36, 427-430.
  88. Crampin, S.; Gao, Y Geophysical Journal International 2010, 180, 1124.
  89. Wyart, M.; Silbert. L. E.; Nagel, S. R.; Witten, T. A. Phys. Rev. E 2005, 72, 051306.
  90. Brenguier, F. et al. Science 2008, 321, 1478.
  91. Brenguier, F. et al. Nature Geoscience 2008, 1, 126.
  92. Vespignani, A.; Zapperi, S. Phys. Rev. E 1998, 57, 6345.
  93. Reed, W. J.; Hughes, B. D. Phys. Rev. E 2002, 66, 067103.
  94. Daerr, A.; Douady, S. Nature 1999, 399, 241.
  95. Jaeger, H. M.; Liu, C.; Nagel.; S. R. Phys. Rev. Lett. 1989, 62, 4043.
  96. Rosendahl, J.; Vekic, M.; Kelley J. Phys. Rev. E 1993, 47, 14011404.
  97. Rosendahl, J.; Vekic, M.; Rutledge J. E. Phys. Rev. Lett. 1994, 73, 537540.
  98. Cajueiro, D. O.; Andrade, R. F. S. Phys. Rev. E 2010, 81, 015102(R).