SELF-ORGANIZED
CRITICALITY AND DYNAMICAL SYSTEM

Collaborators

Ph. Blanchard, Bielefeld university.

J.L. Meunier, Nice University, INLN.

T. Krueger, Bielefeld.

D. Volchenkov, Bielefeld.

Self-Organized Criticality, dynamical systems with singularities, thermodynamics formalism, critical phenonena.

PresentationSelf-Organized criticality models situations in which constraints are locally accumulated until a break point : the relaxation of constraints induces generally an avalanche phenomenon propagating over a variable scale. Moreover, provided that the constraints are applied on a sufficiently low time scale, one observes that avalanches size is statistically distributed according to a truncated power law. Many phenomena in nature exhibit a similar behaviour : earthquakes, market cracks, forest fire, epidemics, etc... There is therefore an important challenge in understanding this phenomena, at least to avoid situations where a system is driven spontaneously in such a critical state.

We have proposed a new approach to self-organized criticality, using methods from dynamical systems, ergodic theory and statistical physics. We have shown that some canonical models of SOC are hyperbolic dynamical systems with singularities. In this context, we have analysed the transport dynamics and related them to the Lyapunov spectrum. This establishes an unexpected relation between the structure of the (fractal) attractor on which lives the dynamics and transport properties. We have also shown that one can construct Gibbs measures (in the sense of Sinai-Ruelle-Bowen) which are directly related to abalanches distributions. It is then possible to use technics from statistical physics of critical phenomena (Lee Yang zeros) to analyse the behaviour of avalanches distribution when the system size tends to infinity. We have shown that this method allows to detect bias in the numerics leading to spurious critical exponents. Finally, with methods from quantum field theory we have studied a stochastic partial differential equation modelling transport in SOC models. We have shown that a perturbative method requires to handle all terms in the series and we have been able to extract two free parameters in the theory, that we can relate to the transport and the scale behaviour of Lyapunov exponents.

- Cessac B., Blanchard Ph.,Krüger T., Meunier J.L., ``Self-Organized Criticality and Thermodynamic formalism’’, Journal of Statistical Physics, Vol. 115, No 516, 1283-1326 (2004).

- Cessac B., "Some fractal aspects of Self-Organized Criticality.", Proceedings de la conférence "Fractales en progrčs" , for the 80th birthday of Benoit Mandelbrot,12-13 Novembre 2004.

- Cessac B., Meunier J.L., "Anomalous scaling and Lee-Yang zeroes in Self-Organized Criticality.", Phys. Rev. E, Vol 65, 036131, 1-18 (2002).

- D. Volchenkov, Ph. Blanchard, B.Cessac,"Quantum field theory renormalization group approach to self-organized criticality : the case of random boundaries.", International Journal of Modern Physics B, Vol. 16, No.8 ; 1171-1204, (2002) .

- Cessac B., Blanchard Ph., Volchenkov D., « Does renormalisation group help very much in Self-Organized criticality », Proceedings of « The science of complexity : from mathématics to technology to a sustainable world », Bielefeld 2002.

- Cessac B., Blanchard Ph.,Krüger T., "Lyapunov exponents and transport in the Zhang model of Self-Organized Criticality.’’, Phys. Rev. E, Vol. 64, 016133, (2001).

- Blanchard Ph., Cessac B., Krüger T., "What can one learn about Self-Organized Criticality from Dynamical System theory", Jour. of Stat. Phys., 98, 375-404 (2000).

- Cessac B., Blanchard Ph., Krüger T., "A dynamical system approach to Self-Organized Criticality", "Proceedings of the International Conference on Mathematical Results in Statistical Mechanics", 27-31 Juillet 1998, Marseille.

- Blanchard Ph., Cessac B. Krüger T., "A dynamical system approach to SOC models of Zhang’s type." Journ. of Stat. Phys., 88 (1997), 307-318.

- "A dynamical system approach to self-organized criticality", Séminaire du Département de Mathématiques, Université Queen Mary, Londres 2004.

- Self-Organized Criticality and Dynamical Systems. Congrčs international en l’honneur de B. Mandelbrot. 12 & 13 NOVEMBRE 2004. Paris.

- Cessac B. "What can one learn about
Self-Organized Criticality from Dynamical System theory ?",
International workshop on ``modeling of complex systems’’, 24-27 Juin
2002, Bielefeld, Allemagne.
**Invited.**