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3  Self-Organization and Self-Organized Criticality

Ph. Blanchard (BiBoS, Bielefeld University)

In ordinary life we speak of organization if each element of a system acts in a well-defined way on given external influences. The observed behaviour is the result of a joint action. The same process will be called self-organized if there are no external causes given but the elementary parts of the system act in concert by some kind of mutual understanding.

Complex spatial or temporal patterns emerge when simple systems are driven from equilibrium in ways that produce instabilities. Spontaneous formation of spirals, stable in time, is an example of self-organization. The temporal stability of these spirals is due to replication: a situation develops which is by itself non-stationary but its characteristics are invariant in time as they replicate in every elementary time step. A deep scientific understanding of this phenomenon requires first that we find the mechanisms controlling the instabilities and moreover that we analyse the interplay between those mechanisms and the driving forces producing the observed patterns.

In the last 10 years the notion of Self-Organized Criticality (SOC) became a new paradigm for the explanation of a huge variety of phenonema in nature and social science. Its origin lies in an attempt to explain the widespread appearance of power-law like statistics of characteristic events in a multitude of examples like the distribution of the size of earthquakes, 1/f noise, amplitudes of solar flares, species extinction to name only a few cases. As a result a lot of literature in Physics has been devoted to the study of systems exhibiting SOC ([1] and [2]).

A collection of electrons, a pile of sand grains, a network of springs, an ecosystem or the community of stock-market dealers can be seen as systems having many components that interact through some kind of exchange of forces or information. In addition to these interactions these systems are driven by some external influence for example gravitation in the case of sand grains.

The complexity of the dynamics in the above mentioned systems is mainly due to the presence of long-range spatial and time correlation, leading to non-trivial effects like anomalous diffusion. In the stationary state the average incoming flux of external perturbations is simply compensated by the average outgoing one which can leave the system at the boundary or by dissipation into the bulk. Therefore, there is a constant flux through the system, leading to a non-equilibrium situation. What is remarkable in this stationary state, called the SOC state, is that the distribution of avalanches appears to follow a power law, namely there is scale-invariance reminiscent of thermodynamical systems at the critical point.

The interaction between the component parts of the system extend only over a short range. In the critical state these interactions propagate a pattern all the way across the system. In other words it induces a kind of global organization in which details of the particular system get obliterated.

This is certainly one central reason why SOC has attracted the physics community: these systems apparently spontaneously reach a critical state without any fine tuning of some control parameter.

Several models have been propagated to mimic these mechanisms like the sandpile model [1] or the abelian sandpile [2]. Numerical simulations on the one hand and theoretical approaches on the other have led to a good description of SOC in particular with respect to the computation of critical exponents which are believed to characterize the universality class the model belongs to, as they do in second order phase transitions. We have studied [3,4,5] SOC from a dynamical system point of view. It is a natural approach (since Boltzmann and Gibbs) to try to access the macroscopic behaviour of large systems starting from the microscopic dynamical evolution. The macroscopic behaviour in the stationary state is characterized by a probability measure, which has to be extracted from the microscopic evolution. One is seeking a "good" measure from a physical point of view. In the SOC model we consider this measure maximizes the entropy. Furthermore it is robust with respect to noise, it admits a density along the unstable directions and it is formally related to the Gibbs measure. The SOC dynamics can be either described in terms of Iteration Function Systems or as a piecewise hyperbolic dynamical system of skew-product type where one coordinate encodes the sequence of activities. With this approach we give a description of the SOC attractor and of its fractal structure. The Lyapunov exponents are related to the power laws appearing in SOC via a conservation law i.e. the Ledrappier-Young formula.

One reason for this great interest in SOC is that it combines two fascinating concepts namely self-organization and critical behaviour to explain a third no less fascinating notion: complexity. A clear-cut definition of what SOC is, necessary conditions for SOC and a well-established mathematical framework are still missing. What about phase transitions and universality? In the ZiF project we will discuss these problems. Moreover we will test the claims associated with SOC on specific physical, biological and economical systems as well as on models.

 

References

  1. P. Bak, "How Nature works", Springer (1996)
  2. H.J. Jensen, "Self-Organized Criticality Emergent Complex Behaviour in Physical and Biological Systems", Cambridge University Press (1998)
  3. Ph. Blanchard, B. Cessac, T. Krüger, "A dynamical system approach for SOC models of Zhang's type", Jour. Stat. Phys. 88 (1997) 307-318
  4. Ph. Blanchard, B. Cessac, T. Krüger, "A dynamical system approach of Self-Organized Criticality", "Mathematical Results in Statistical Mechanics", Eds. S. Miracle-Solé, J. Ruiz, V. Zagrebnov, World Scientific (1999) 399-410
  5. Ph. Blanchard, B. Cessac, T. Krüger, "What can one learn about Self-Organized Criticality from Dynamical Systems Theory", to appear in Journal of Statistical Physics

 

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