Journal of Statistical Physics

, Volume 63, Issue 3–4, pp 653–684 | Cite as

Dynamical and spatial aspects of sandpile cellular automata

  • Kim Christensen
  • Hans C. Fogedby
  • Henrik Jeldtoft Jensen
Articles

Abstract

The Bak, Tang, and Wiesenfeld cellular automaton is simulated in 1, 2, 3, 4, and 5 dimensions. We define a (new) set of scaling exponents by introducing the concept of conditional expectation values. Scaling relations are derived and checked numerically and the critical dimension is discussed. We address the problem of the mass dimension of the avalanches and find that the avalanches are noncompact for dimensions larger than 2. The scaling of the power spectrum derives from the assumption that the instantaneous dissipation rate of the individual avalanches obeys a simple scaling relation. Primarily, the results of our work show that the flow of sand down the slope does not have a 1/f power spectrum in any dimension, although the model does show clear critical behavior with scaling exponents depending on the dimension. The power spectrum behaves as 1/f2 in all the dimensions considered.

Key words

Self-organized critical behavior sandpiles scaling relations power spectra 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    M. B. Weissman,Rev. Mod. Phys. 60:537 (1988).Google Scholar
  2. 2.
    B. Mandelbrot,The Fractal Geometry of Nature (Freeman, San Francisco, 1982).Google Scholar
  3. 3.
    P. Bak, C. Tang, and K. Wiesenfeld,Phys. Rev. Lett. 59:381 (1987).Google Scholar
  4. 4.
    P. AlstrØm, P. Trunfio, and H. E. Stanley, inRandom Fluctuations and Pattern Growth: Experiments and Models, H. H. Stanley and N. Ostrowsky, eds. (Kluwer, Dordrecht, 1989), p. 340.Google Scholar
  5. 5.
    P. Bak,Physica A 163:403 (1990).Google Scholar
  6. 6.
    P. Bak and K. Chen,Physica D 38:5 (1989).Google Scholar
  7. 7.
    P. Bak, K. Chen, and M. Creutz,Nature 342:780 (1989).Google Scholar
  8. 8.
    P. Bak, K. Chen, and C. Tang,Phys. Rev. Lett. A 147:297 (1990).Google Scholar
  9. 9.
    P. Bak and C. Tang,Physics Today 42:S27 (1989).Google Scholar
  10. 10.
    P. Bak, C. Tang, and K. Wiesenfeld,Phys. Rev. A 38:3645 (1989).Google Scholar
  11. 11.
    P. Bak, C. Tang, and K. Wiesenfeld, inRandom Fluctuations and Pattern Growth: Experiments and Models, H. E. Stanley and N. Ostrowsky, eds. (Kluwer, Dordrecht, 1989), p. 329.Google Scholar
  12. 12.
    J. M. Carlson, J. T. Chayes, E. R. Grannan, and G. H. Swindle,Phys. Rev. A 42:2467 (1990).Google Scholar
  13. 13.
    K. Chen and P. Bak,Phys. Lett. 140:299 (1989).Google Scholar
  14. 14.
    D. Dhar,Phys. Rev. Lett. 63:1659 (1989).Google Scholar
  15. 15.
    D. Dhar and R. Ramaswamy,Phys. Rev. Lett. 64:1613 (1990).Google Scholar
  16. 16.
    G. Grinstein, D.-H. Lee, and S. Sachdev,Phys. Rev. Lett. 64:1927 (1990).Google Scholar
  17. 17.
    H. J. Jensen,Phys. Rev. Lett. 64:3103 (1990).Google Scholar
  18. 18.
    H. J. Jensen, K. Christensen, and H. C. Fogedby,Phys. Rev. B 40:7425 (1989).Google Scholar
  19. 19.
    J. Kertész and L. B. Kiss,J. Phys. A 23:L433 (1990).Google Scholar
  20. 20.
    B. McNamara and K. Wiesenfeld,Phys. Rev. A 41:1867 (1990).Google Scholar
  21. 21.
    S. P. Obukhov,Phys. Rev. Lett. 65:1395 (1990).Google Scholar
  22. 22.
    K. Wiesenfeld, C. Tang, and P. Bak,J. Stat. Phys. 54:1441 (1989).Google Scholar
  23. 23.
    K. Wiesenfeld, J. Theiler, and B. McNamara,Phys. Rev. Lett. 65:949 (1990).Google Scholar
  24. 24.
    P. AlstrØm,Phys. Rev. A 38:4905 (1988).Google Scholar
  25. 25.
    P. Grassberger and S. S. Manna, Some more sandpiles, Physics Department, University of Wuppertal, Preprint, to appear inJ. Phys. (Paris).Google Scholar
  26. 26.
    T. Hwa and M. Kardar,Phys. Rev. Lett. 62:1813 (1989);Physica D 38:198 (1989).Google Scholar
  27. 27.
    L. P. Kadanoff, S. Nagel, L. Wu, and S. Zhou,Phys. Rev. A 39:6524 (1989).Google Scholar
  28. 28.
    L. P. Kadanoff,Physica A 163:1 (1990).Google Scholar
  29. 29.
    S. P. Obukhov, inRandom Fluctuations and Pattern Growth: Experiments and Models, H. E. Stanley and N. Ostrowsky, eds. (Kluwer, Dordrecht, 1989), p. 336; see also J. Honkonen,Phys. Lett. A 145:87 (1990).Google Scholar
  30. 30.
    C. Tang, Scalings in avalanches and elsewhere, Institute for Theoretical Physics, University of California, Santa Barbara, preprint, submitted toPhys. Rev. A. Google Scholar
  31. 31.
    C. Tang and P. Bak,Phys. Rev. Lett. 60:2347 (1988).Google Scholar
  32. 32.
    C. Tang and P. Bak,J. Stat. Phys. 51:797 (1988).Google Scholar
  33. 33.
    Y.-C. Zhang,Phys. Rev. Lett. 63:470 (1989).Google Scholar
  34. 34.
    P. Bak and C. Tang,J. Geophys. Res. B 94:15635 (1989).Google Scholar
  35. 35.
    P. Bak and K. Chen, Dynamics of earthquakes, inFractals and Their Application to Geophysics (Geological Society of America, Denver, 1990).Google Scholar
  36. 36.
    J. M. Carlson and J. S. Langer,Phys. Rev. Lett. 62:2632 (1989).Google Scholar
  37. 37.
    K. Ito and M. Matsuzaki,J. Geophys. Res. 95:6853 (1990).Google Scholar
  38. 38.
    A. Sonnette and D. Sonnette,Europhys. Lett. 9:192 (1989).Google Scholar
  39. 39.
    H. Takayasu and M. Matsuzaki,Phys. Lett. 131:244 (1988).Google Scholar
  40. 40.
    G. W. Baxter and R. P. Behringer,Phys. Rev. A 42:1017 (1990).Google Scholar
  41. 41.
    W. Clauss, A. Kittel, U. Rau, J. Parisi, J. Peinke, and R. P. Huebener, Self-organized critical behavior in low-temperature impact ionization breakdown ofp-Ge, Physikalisches Institut, Tübingen, Preprint.Google Scholar
  42. 42.
    P. Evesque and J. Rajchenbach,Phys. Rev. Lett. 62:44 (1989).Google Scholar
  43. 43.
    G. A. Held, D. H. Solina, II, D. T. Keane, W. J. Haag, P. M. Horn, and G. Grinstein,Phys. Rev. Lett. 65:1120 (1990).Google Scholar
  44. 44.
    H. M. Jaeger, Chu-heng Liu, and Sidney R. Nagel,Phys. Rev. Lett. 62:40 (1989).Google Scholar
  45. 45.
    H. M. Jaeger, C.-H. Liu, S. R. Nagel, and T. A. Witten, Friction in granular flow, James Franck Institute, Chicago, Preprint.Google Scholar
  46. 46.
    D. K. C. MacDonald,Noise and Fluctuations: An Introduction (Wiley, New York, 1962).Google Scholar
  47. 47.
    P. Z. Peebles, Jr.,Probability, Random Variables, and Random Signal Principles (McGraw-Hill, 1987).Google Scholar
  48. 48.
    A. Papoulis,The Fourier Integral and its Applications (McGraw-Hill, 1962).Google Scholar
  49. 49.
    A. van der Ziel,Physica 16:359 (1950).Google Scholar
  50. 50.
    D. Halford,Proc. IEEE 56:251 (1968).Google Scholar

Copyright information

© Plenum Publishing Corporation 1991

Authors and Affiliations

  • Kim Christensen
    • 1
    • 2
  • Hans C. Fogedby
    • 1
    • 2
  • Henrik Jeldtoft Jensen
    • 1
    • 2
  1. 1.Institute of PhysicsUniversity of AarhusAarhus CDenmark
  2. 2.NORDITACopenhagen ØDenmark

Personalised recommendations