Renormalization Group for Directed Sandpile Models

Jeff Hasty; Kurt Wiesenfeld

doi:10.1103/PhysRevLett.81.1722

Access by Xinjiang University

Renormalization Group for Directed Sandpile Models

Jeff Hasty

Kurt Wiesenfeld

Supercomputer Computations Research Institute, Florida State University, Tallahassee, Florida 32310-6046

School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332

PDF

Phys. Rev. Lett. 81, 1722 – Published 24 August, 1998

DOI: https://doi.org/10.1103/PhysRevLett.81.1722

Abstract

Following the lead of earlier researchers, we construct a real space renormalization group map for a class of self-organized critical models. We then test a series of quantitative predictions implied by this map against direct simulations of various models. The theory passes four tests, but apparently fails a fifth. We also find that the map has an interesting nongeneric property, suggesting a deeper structure to the theory than previously appreciated.

References (10)

V. Frette, K. Christensen, A. Malthe-Sorenssen, J. Feder, T. Jossang, and P. Meakin, Nature (London) 379, 49 (1996); H. M. Jaeger, C. H. Liu, and S. R. Nagel, Phys. Rev. Lett. 62, 40 (1989); H. J. S. Feder and J. Feder, 66, 2669 (1991); A. Johansen, P. Dimon, C. Ellegaard, J. S. Larsen, and H. H. Rugh, Phys. Rev. E 48, 4779 (1993).
L. Pietronero, A. Vespignani, and S. Zapperi, Phys. Rev. Lett. 72, 1690 (1994); A. Vespignani, S. Zapperi, and L. Pietronero, Phys. Rev. E 51, 1711 (1995).
D. Dhar, Phys. Rev. Lett. 64, 1613 (1990).
P. Bak and M. Creutz, Fractals and Disordered Systems (Springer-Verlag, Berlin,1994), a ed., Vol. II.
J. M. Carlson, J. T. Chayes, E. R. Grannan, and G. H. Swindle, Phys. Rev. Lett. 65, 2547 (1990).
S. Boettcher and M. Paczuski, Phys. Rev. E 54, 1082 (1996).
A. Ben-Hur, R. Hallgass, and V. Loreto, Phys. Rev. E 54, 1426 (1996).
D. Dhar and R. Ramaswamy, Phys. Rev. Lett. 63, 1659 (1989).
P. Bak, C. Tang, and K. Wiesenfeld, Phys. Rev. Lett. 59, 381 (1987).
Since $x$ is the probability that a site is noncritical, $1 - x$ corresponds to the density of critical sites $ρ$ in the PVZ [[2]] scheme. Our inclusion of $x$ as a probability amplitude leads to a different normalization condition.

[c1] V. Frette, K. Christensen, A. Malthe-Sorenssen, J. Feder, T. Jossang, and P. Meakin, Nature (London) 379, 49 (1996); H. M. Jaeger, C. H. Liu, and S. R. Nagel, Phys. Rev. Lett. 62, 40 (1989); H. J. S. Feder and J. Feder, 66, 2669 (1991); A. Johansen, P. Dimon, C. Ellegaard, J. S. Larsen, and H. H. Rugh, Phys. Rev. E 48, 4779 (1993).

[c2] L. Pietronero, A. Vespignani, and S. Zapperi, Phys. Rev. Lett. 72, 1690 (1994); A. Vespignani, S. Zapperi, and L. Pietronero, Phys. Rev. E 51, 1711 (1995).

[c3] D. Dhar, Phys. Rev. Lett. 64, 1613 (1990).

[c4] P. Bak and M. Creutz, Fractals and Disordered Systems (Springer-Verlag, Berlin,1994), a ed., Vol. II.

[c5] J. M. Carlson, J. T. Chayes, E. R. Grannan, and G. H. Swindle, Phys. Rev. Lett. 65, 2547 (1990).

[c6] S. Boettcher and M. Paczuski, Phys. Rev. E 54, 1082 (1996).

[c7] A. Ben-Hur, R. Hallgass, and V. Loreto, Phys. Rev. E 54, 1426 (1996).

[c8] D. Dhar and R. Ramaswamy, Phys. Rev. Lett. 63, 1659 (1989).

[c9] P. Bak, C. Tang, and K. Wiesenfeld, Phys. Rev. Lett. 59, 381 (1987).

[c10] Since $x$ is the probability that a site is noncritical, $1 - x$ corresponds to the density of critical sites $ρ$ in the PVZ [[2]] scheme. Our inclusion of $x$ as a probability amplitude leads to a different normalization condition.

Physical Review Letters

Export citation

Export citation

Renormalization Group for Directed Sandpile Models

Abstract

References (10)

Outline

Information

Filter

Filter

Article Lookup

Enter a citation