J. Phys. I France
Volume 2, Number 11, November 1992
Page(s) 2065 - 2073
DOI: 10.1051/jp1:1992267
J. Phys. I France 2 (1992) 2065-2073

Critical phase transitions made self-organized : a dynamical system feedback mechanism for self-organized criticality

Didier Sornette

Laboratoire de Physique de la Matière Condensée CNRS URA 190, Université de Nice-Sophia Antipolis, B.P. 71, Parc Valrose 06108 Nice Cedex, France

(Received 10 February 1992, accepted in final form 24 July 1992)

According to Kadanoff, self-organized criticality (SOC) implies the operation of a feedback mechanism that ensures a steady state in which the system is marginally stable against a disturbance. Here, we extend this idea and propose a picture according to which SOC relies on a non-linear feedback of the order parameter on the control parameter(s), the amplitude of this feedback being tuned by the spatial correlation length $\xi$. The self-organized nature of the criticality stems from the fact that the limit $\xi \to +\infty$ is attracting the non-linear feedback dynamics. It is applied to known self-organized critical systems such as "sandpile" models as well as to a simple dynamical generalization of the percolation model. Using this feedback mechanism, it is possible in principle to convert standard "unstable" critical phase transitions into self-organized critical dynamics, thereby enlarging considerably the number of models presenting SOC. These ideas are illustrated on the 2D Ising model and the values of the various "avalanche" exponents are expressed in terms of the static and dynamic Ising critical exponents.

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