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Proceedings of the National Academy of Sciences of the United States of America logoLink to Proceedings of the National Academy of Sciences of the United States of America
. 1995 Jul 18;92(15):6712–6719. doi: 10.1073/pnas.92.15.6712

Self-organized criticality: sandpiles, singularities, and scaling.

J M Carlson 1, G H Swindle 1
PMCID: PMC41399  PMID: 11607564

Abstract

We present an overview of the statistical mechanics of self-organized criticality. We focus on the successes and failures of hydrodynamic description of transport, which consists of singular diffusion equations. When this description applies, it can predict the scaling features associated with these systems. We also identify a hard driving regime where singular diffusion hydrodynamics fails due to fluctuations and give an explicit criterion for when this failure occurs.

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Selected References

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  1. Bak P, Tang C, Wiesenfeld K. Self-organized criticality: An explanation of the 1/f noise. Phys Rev Lett. 1987 Jul 27;59(4):381–384. doi: 10.1103/PhysRevLett.59.381. [DOI] [PubMed] [Google Scholar]
  2. Carlson JM, Chayes JT, Grannan ER, Swindle GH. Self-orgainzed criticality and singular diffusion. Phys Rev Lett. 1990 Nov 12;65(20):2547–2550. doi: 10.1103/PhysRevLett.65.2547. [DOI] [PubMed] [Google Scholar]
  3. Carlson JM, Chayes JT, Grannan ER, Swindle GH. Self-organized criticality in sandpiles: Nature of the critical phenomenon. Phys Rev A. 1990 Aug 15;42(4):2467–2470. doi: 10.1103/physreva.42.2467. [DOI] [PubMed] [Google Scholar]
  4. Carlson JM, Grannan ER, Singh C, Swindle GH. Fluctuations in self-organizing systems. Phys Rev E Stat Phys Plasmas Fluids Relat Interdiscip Topics. 1993 Aug;48(2):688–698. doi: 10.1103/physreve.48.688. [DOI] [PubMed] [Google Scholar]
  5. Carlson JM, Grannan ER, Swindle GH. Self-organizing systems at finite driving rates. Phys Rev E Stat Phys Plasmas Fluids Relat Interdiscip Topics. 1993 Jan;47(1):93–105. doi: 10.1103/physreve.47.93. [DOI] [PubMed] [Google Scholar]
  6. Coppersmith SN, Littlewood PB. Interference phenomena and mode locking in the model of deformable sliding charge-density waves. Phys Rev Lett. 1986 Oct 13;57(15):1927–1930. doi: 10.1103/PhysRevLett.57.1927. [DOI] [PubMed] [Google Scholar]
  7. Coppersmith SN. Overdamped Frenkel-Kontorova model with randomness as a dynamical system. II. Numerical studies of mode locking. Phys Rev A Gen Phys. 1988 Jul 1;38(1):375–381. doi: 10.1103/physreva.38.375. [DOI] [PubMed] [Google Scholar]
  8. Coppersmith SN. Overdamped Frenkel-Kontorova model with randomness as a dynamical system: Mode locking and derivation of discrete maps. Phys Rev A Gen Phys. 1987 Oct 1;36(7):3375–3382. doi: 10.1103/physreva.36.3375. [DOI] [PubMed] [Google Scholar]
  9. Dhar D, Ramaswamy R. Exactly solved model of self-organized critical phenomena. Phys Rev Lett. 1989 Oct 16;63(16):1659–1662. doi: 10.1103/PhysRevLett.63.1659. [DOI] [PubMed] [Google Scholar]
  10. Dhar D. Self-organized critical state of sandpile automaton models. Phys Rev Lett. 1990 Apr 2;64(14):1613–1616. doi: 10.1103/PhysRevLett.64.1613. [DOI] [PubMed] [Google Scholar]
  11. Garrido PL, Lebowitz JL, Maes C, Spohn H. Long-range correlations for conservative dynamics. Phys Rev A. 1990 Aug 15;42(4):1954–1968. doi: 10.1103/physreva.42.1954. [DOI] [PubMed] [Google Scholar]
  12. Grinstein G, Lee D, Sachdev S. Conservation laws, anisotropy, and "self-organized criticality" in noisy nonequilibrium systems. Phys Rev Lett. 1990 Apr 16;64(16):1927–1930. doi: 10.1103/PhysRevLett.64.1927. [DOI] [PubMed] [Google Scholar]
  13. Hwa T, Kardar M. Avalanches, hydrodynamics, and discharge events in models of sandpiles. Phys Rev A. 1992 May 15;45(10):7002–7023. doi: 10.1103/physreva.45.7002. [DOI] [PubMed] [Google Scholar]
  14. Kadanoff LP, Chhabra AB, Kolan AJ, Feigenbaum MJ, Procaccia I., I Critical indices for singular diffusion. Phys Rev A. 1992 Apr 15;45(8):6095–6098. doi: 10.1103/physreva.45.6095. [DOI] [PubMed] [Google Scholar]
  15. Kadanoff LP, Nagel SR, Wu L, Zhou Sm. Scaling and universality in avalanches. Phys Rev A Gen Phys. 1989 Jun 15;39(12):6524–6537. doi: 10.1103/physreva.39.6524. [DOI] [PubMed] [Google Scholar]
  16. Marx R. E., Carlson E. R. Tissue banking safety: caveats and precautions for the oral and maxillofacial surgeon. J Oral Maxillofac Surg. 1993 Dec;51(12):1372–1379. doi: 10.1016/s0278-2391(10)80144-2. [DOI] [PubMed] [Google Scholar]
  17. Middleton AA. Asymptotic uniqueness of the sliding state for charge-density waves. Phys Rev Lett. 1992 Feb 3;68(5):670–673. doi: 10.1103/PhysRevLett.68.670. [DOI] [PubMed] [Google Scholar]
  18. Middleton AA, Biham O, Littlewood PB, Sibani P. Complete mode locking in models of charge-density waves. Phys Rev Lett. 1992 Mar 9;68(10):1586–1589. doi: 10.1103/PhysRevLett.68.1586. [DOI] [PubMed] [Google Scholar]
  19. Middleton AA, Tang C. Self-Organized Criticality in Nonconserved Systems. Phys Rev Lett. 1995 Jan 30;74(5):742–745. doi: 10.1103/PhysRevLett.74.742. [DOI] [PubMed] [Google Scholar]
  20. Myers CR, Sethna JP. Collective dynamics in a model of sliding charge-density waves. I. Critical behavior. Phys Rev B Condens Matter. 1993 May 1;47(17):11171–11193. doi: 10.1103/physrevb.47.11171. [DOI] [PubMed] [Google Scholar]
  21. Narayan O, Fisher DS. Critical behavior of sliding charge-density waves in 4- epsilon dimensions. Phys Rev B Condens Matter. 1992 Nov 1;46(18):11520–11549. doi: 10.1103/physrevb.46.11520. [DOI] [PubMed] [Google Scholar]
  22. Narayan O, Middleton AA. Avalanches and the renormalization group for pinned charge-density waves. Phys Rev B Condens Matter. 1994 Jan 1;49(1):244–256. doi: 10.1103/physrevb.49.244. [DOI] [PubMed] [Google Scholar]
  23. Olami Z, Feder HJ, Christensen K. Self-organized criticality in a continuous, nonconservative cellular automaton modeling earthquakes. Phys Rev Lett. 1992 Feb 24;68(8):1244–1247. doi: 10.1103/PhysRevLett.68.1244. [DOI] [PubMed] [Google Scholar]
  24. Socolar JE, Grinstein G, Jayaprakash C. On self-organized criticality in nonconserving systems. Phys Rev E Stat Phys Plasmas Fluids Relat Interdiscip Topics. 1993 Apr;47(4):2366–2376. doi: 10.1103/physreve.47.2366. [DOI] [PubMed] [Google Scholar]
  25. Tang C, Bak P. Critical exponents and scaling relations for self-organized critical phenomena. Phys Rev Lett. 1988 Jun 6;60(23):2347–2350. doi: 10.1103/PhysRevLett.60.2347. [DOI] [PubMed] [Google Scholar]
  26. Tang C, Wiesenfeld K, Bak P, Coppersmith S, Littlewood P. Phase organization. Phys Rev Lett. 1987 Mar 23;58(12):1161–1164. doi: 10.1103/PhysRevLett.58.1161. [DOI] [PubMed] [Google Scholar]

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