Models for self-organized criticality


Self-organized criticality (SOC) is a concept that explains the appearance of scale invariance and power laws without parameter tuning in slowly driven, dissipative systems[1]. Through a series of papers, we addressed SOC-related issues in stick-slip models. It is generally accepted for stick-slip models that criticality is achieved by a marginal phase locking mechanism[2]: Due to inhomogeneities or open boundaries, phase locking is frustrated throughout the system leading to long-range correlations in the phase and a broad range of avalanche sizes, and hence SOC.

We discovered that this mechanism is less general than thought; it does not apply to a class of multiplicatively driven models[3,4]. Instead, phase locking was found to be replaced by coarsening. Surprisingly, contrary to previous understanding boundary conditions become irrelevant. Although non-equilibrium steady state has long been regarded as necessary for the appearance of SOC, our results show that a punctuated approach to equilibrium suffices. In retrospect, there exists a host of experimental systems which have been claimed to exhibit SOC but dismissed on the ground of non-stationarity[5]. Our study urges re-evaluations and re-interpretations of those results.

A spring-block model of earthquakes was introduced in [6]. It incorporates more realistic force-displacement relations than in previous models, especially the well-known OFC model[7]. To our knowledge, this is the first examination of the effects of internal stresses, vectorial forcing and nonlinear force-displacement relationship in those models. We emphasized the key role of internal stresses in the tuning of critical exponents, and pointed out certain pitfalls in the setup of previous models[7].


  1. P. Bak, C. Tang and K. Wiesenfeld, Phys. Rev. A 38, 364 (1988).
  2. A.A. Middleton and C. Tang, Phys. Rev. Lett. 74, 742 (1995).
  3. Self-organized criticality in stick-slip models with periodic boundaries,
    K.-t. Leung, J.V. Andersen, and D. Sornette, Phys. Rev. Lett. 80, 1916 (1998).
  4. Self-organized criticality in an isotropically driven model approaching equilibrium,
    K.-t. Leung, J.V. Andersen, and D. Sornette, Physica A 254, 85 (1998)
  5. D. Sornette, Phys. Rev. Lett. 72, 2306 (1994); J. Phys. I 4, 209 (1994).
  6. Generalization of a two-dimensional Burridge-Knopoff model of earthquakes,
    K.-t. Leung, J. Muller and J.V. Andersen, J. Phys. I 7, 423 (1997)
  7. Z. Olami, J. S. Feder, and K. Christensen, Phys. Rev. Lett. 68, 1244 (1992), and references therein.

Last revised October 7, 2002 ©KtL