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/f 2 in all the dimensions considered.
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References
M. B. Weissman,Rev. Mod. Phys. 60:537 (1988).
B. Mandelbrot,The Fractal Geometry of Nature (Freeman, San Francisco, 1982).
P. Bak, C. Tang, and K. Wiesenfeld,Phys. Rev. Lett. 59:381 (1987).
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.
P. Bak,Physica A 163:403 (1990).
P. Bak and K. Chen,Physica D 38:5 (1989).
P. Bak, K. Chen, and M. Creutz,Nature 342:780 (1989).
P. Bak, K. Chen, and C. Tang,Phys. Rev. Lett. A 147:297 (1990).
P. Bak and C. Tang,Physics Today 42:S27 (1989).
P. Bak, C. Tang, and K. Wiesenfeld,Phys. Rev. A 38:3645 (1989).
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.
J. M. Carlson, J. T. Chayes, E. R. Grannan, and G. H. Swindle,Phys. Rev. A 42:2467 (1990).
K. Chen and P. Bak,Phys. Lett. 140:299 (1989).
D. Dhar,Phys. Rev. Lett. 63:1659 (1989).
D. Dhar and R. Ramaswamy,Phys. Rev. Lett. 64:1613 (1990).
G. Grinstein, D.-H. Lee, and S. Sachdev,Phys. Rev. Lett. 64:1927 (1990).
H. J. Jensen,Phys. Rev. Lett. 64:3103 (1990).
H. J. Jensen, K. Christensen, and H. C. Fogedby,Phys. Rev. B 40:7425 (1989).
J. Kertész and L. B. Kiss,J. Phys. A 23:L433 (1990).
B. McNamara and K. Wiesenfeld,Phys. Rev. A 41:1867 (1990).
S. P. Obukhov,Phys. Rev. Lett. 65:1395 (1990).
K. Wiesenfeld, C. Tang, and P. Bak,J. Stat. Phys. 54:1441 (1989).
K. Wiesenfeld, J. Theiler, and B. McNamara,Phys. Rev. Lett. 65:949 (1990).
P. AlstrØm,Phys. Rev. A 38:4905 (1988).
P. Grassberger and S. S. Manna, Some more sandpiles, Physics Department, University of Wuppertal, Preprint, to appear inJ. Phys. (Paris).
T. Hwa and M. Kardar,Phys. Rev. Lett. 62:1813 (1989);Physica D 38:198 (1989).
L. P. Kadanoff, S. Nagel, L. Wu, and S. Zhou,Phys. Rev. A 39:6524 (1989).
L. P. Kadanoff,Physica A 163:1 (1990).
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).
C. Tang, Scalings in avalanches and elsewhere, Institute for Theoretical Physics, University of California, Santa Barbara, preprint, submitted toPhys. Rev. A.
C. Tang and P. Bak,Phys. Rev. Lett. 60:2347 (1988).
C. Tang and P. Bak,J. Stat. Phys. 51:797 (1988).
Y.-C. Zhang,Phys. Rev. Lett. 63:470 (1989).
P. Bak and C. Tang,J. Geophys. Res. B 94:15635 (1989).
P. Bak and K. Chen, Dynamics of earthquakes, inFractals and Their Application to Geophysics (Geological Society of America, Denver, 1990).
J. M. Carlson and J. S. Langer,Phys. Rev. Lett. 62:2632 (1989).
K. Ito and M. Matsuzaki,J. Geophys. Res. 95:6853 (1990).
A. Sonnette and D. Sonnette,Europhys. Lett. 9:192 (1989).
H. Takayasu and M. Matsuzaki,Phys. Lett. 131:244 (1988).
G. W. Baxter and R. P. Behringer,Phys. Rev. A 42:1017 (1990).
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.
P. Evesque and J. Rajchenbach,Phys. Rev. Lett. 62:44 (1989).
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).
H. M. Jaeger, Chu-heng Liu, and Sidney R. Nagel,Phys. Rev. Lett. 62:40 (1989).
H. M. Jaeger, C.-H. Liu, S. R. Nagel, and T. A. Witten, Friction in granular flow, James Franck Institute, Chicago, Preprint.
D. K. C. MacDonald,Noise and Fluctuations: An Introduction (Wiley, New York, 1962).
P. Z. Peebles, Jr.,Probability, Random Variables, and Random Signal Principles (McGraw-Hill, 1987).
A. Papoulis,The Fourier Integral and its Applications (McGraw-Hill, 1962).
A. van der Ziel,Physica 16:359 (1950).
D. Halford,Proc. IEEE 56:251 (1968).
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Christensen, K., Fogedby, H.C. & Jeldtoft Jensen, H. Dynamical and spatial aspects of sandpile cellular automata. J Stat Phys 63, 653–684 (1991). https://doi.org/10.1007/BF01029204
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DOI: https://doi.org/10.1007/BF01029204