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