Invasion percolation with temperature and the nature of self-organized criticality in real systems

Andrea Gabrielli; Guido Caldarelli; Luciano Pietronero

doi:10.1103/PhysRevE.62.7638

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Invasion percolation with temperature and the nature of self-organized criticality in real systems

Andrea Gabrielli^1,2, Guido Caldarelli², and Luciano Pietronero²

¹Laboratoire de Physique de la Matière Condensée, Ecole Polytechnique, 91128-Palaiseau Cedex, France
²INFM-Unità di Roma 1 “La Sapienza,” Piazzale Aldo Moro 2, 00185 Roma, Italy

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Phys. Rev. E 62, 7638 – Published 1 December, 2000

DOI: https://doi.org/10.1103/PhysRevE.62.7638

Abstract

In this paper we present a theoretical approach that allows us to describe the transition between critical and noncritical behavior when stocastic noise is introduced in extremal models with disorder. Namely, we show that the introduction of thermal noise in invasion percolation (IP) brings the system outside the critical point. This result suggests a possible definition of self-organized criticality systems as ordinary critical systems where the critical point corresponds to set to $0$ one of the parameters. We recover both the IP and Eden models for $\vec{T} 0$ and $\vec{T} \infty,$ respectively. For small T we find a dynamical second-order transition with correlation length diverging when $\vec{T} 0.$

References (17)

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A. Vespignani, S. Zapperi, and V. Loreto, Phys. Rev. Lett. 77, 4560 (1997); ibid.A. Vespignani and S. Zapperi, 78, 4793 (1997).
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M. Eden (unpublished)
A. Gabrielli, R. Cafiero, M. Marsili, and L. Pietronero, Europhys. Lett. 38, 491 (1997).
A. Gabrielli (unpublished).
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J.T. Chayes, L. Chayes, and C.M. Newman, Commun. Math. Phys. 101, 383 (1985).

[c1] P. Bak, C. Tang, and K. Weisenfeld, Phys. Rev. Lett. 59, 381 (1987).

[c2] A. Vespignani, S. Zapperi, and V. Loreto, Phys. Rev. Lett. 77, 4560 (1997); ibid.A. Vespignani and S. Zapperi, 78, 4793 (1997).

[c3] S. Maslov, Phys. Rev. Lett. 77, 1182 (1996); ibid.M. Marsili, P. De Los Rios, and S. Maslov, 80, 1457 (1998).

[c4] M. Marsili, J. Stat. Phys. 77, 733 (1994); ibid.A. Gabrielli, M. Marsili, R. Cafiero, and L. Pietronero, 84, 889 (1996).

[c5] D. Wilkinson and J.F. Willemsen, J. Phys. A 16, 3365 (1983).

[c6] M. Vergeles, Phys. Rev. Lett. 75, 1969 (1995).

[c7] G. Caldarelli, M. Vendruscolo, and A. Maritan, Europhys. Lett. 35, 481 (1996); G. Caldarelli, Physica A 252, 295 (1998).

[c8] R. Dickman, A. Vespignani, and S. Zapperi, Phys. Rev. E 57, 5095 (1998).

[c9] T.A. Witten and L.M. Sander, Phys. Rev. Lett. 47, 1400 (1981).

[c10] P. Bak and K. Sneppen, Phys. Rev. Lett. 71, 4083 (1993).

[c11] M. Paczuski, P. Bak, and S. Maslov, Phys. Rev. E 53, 414 (1996).

[c12] M. Eden (unpublished)

[c13] A. Gabrielli, R. Cafiero, M. Marsili, and L. Pietronero, Europhys. Lett. 38, 491 (1997).

[c14] A. Gabrielli (unpublished).

[c15] R. Cafiero, A. Gabrielli, and M. Marsili, Phys. Rev. E 55, 7745 (1997).

[c16] R. Cafiero, G. Caldarelli, and A. Gabrielli, Phys. Rev. E 56, R1291 (1997).

[c17] J.T. Chayes, L. Chayes, and C.M. Newman, Commun. Math. Phys. 101, 383 (1985).

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Invasion percolation with temperature and the nature of self-organized criticality in real systems

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