Among the many paradigms proposed to explain the ubiquity of self-organization in nature, the relatively recent concept of Self-Organized Criticality (SOC) has certainly attracted a large audience in the scientific community. The main advantages of this idea are the simplicity and the genericity of the mechanisms involved [1, 2]. Consequently, within the past 13 years the notion of Self-Organized Criticality has become a paradigm for the explanation of a huge variety of phenomena in nature and social sciences. Its origin lies in the attempt to explain the widespread appearance of power-law like statistics for 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 [3].
In this paradigm, a local perturbation induces a chain reaction or avalanche in the system, that can be localized or spread. When submitted to a stationary flux of external excitations the system is furthermore able to adapt itself in a optimal way. Namely, it is able to achieve a stationary regime, referred to as the SOC state, where the flux of external perturbations is redistributed in the system and dissipated at the boundaries (or in the bulk) and where the distribution of avalanches follows a power law. There is therefore a scale invariance in the distribution of avalanches and a local perturbation can induce effects at any scale. There are also long-range spatial and time correlations. This results in a very original state sharing some characteristics of living systems like punctuated response, long memory, and contingent history. Furthermore, this state has also some features reminiscent of thermodynamic systems at the critical point, though in this paradigm the system reaches spontaneously a critical state without any fine tuning of some control parameter.
One reason for the great interest in SOC is that it combines two fascinating concepts namely self-organization and critical behavior 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 however still missing. In the research group "Self-Organization and Self-Organized Criticality", in the frame of the ZiF project, we had two main objectives. The first one was to establish a mathematical framework to study SOC models and to develop further the connexions to dynamical system theory and statistical mechanics. The second one was to test the claims associated with SOC on specific physical, biological and economical systems as well as on models.
Though many models have been proposed to mimic the SOC mechanism, the results available are mainly numerical and only a few rigorous results are known [4, 5]. Besides, despite the large number of papers devoted to SOC, the subject is still missing a general mathematical framework and suffers from the lack of precise definitions. The characterization and the classification of the models essentially relies on the following numerical observation. Fix an observable, say n, measuring some property of an avalanche (duration, size, etc ...), and compute the related probability at stationarity for a system of characteristic size L. The graph of exhibits a power law behavior over a finite range with a cut-off, corresponding to finite size effects. As L increases the power law range increases, leading to the conjecture that converges to a power law as . The corresponding exponent is called the critical exponent for the observable n. Guided by the wisdom coming from the renormalization group analysis and phase transitions in equilibrium systems, it seems natural to look for a possible classification of the models into universality classes characterized by a set of critical exponents, for a family of "relevant" avalanche-observables. However, the link between the "criticality" of the "out of equilibrium" SOC models and the usual statistical mechanics of phase transitions in equilibrium systems remains to be clarified [6, 7]. Furthermore, it remains to establish that the commonly studied observables (size, duration, area, giration radius) constitute a complete set allowing one to classify the models. Finally, even the computation of the critical exponents from numerical data is not easy and there is no agreement yet on the type of finite size scaling treatment one has to use [8, 9, 10].
An alternative approach to better understand the behavior of SOC models can also consist of studying the microscopic dynamics and to infer information about the macroscopic behavior from this analysis. A detailed analysis can, at first sight, seem useless since the conventional wisdom from classical statistical mechanics is that microscopic "details" are irrelevant, and only structural properties like conservation laws and symmetries are essential. However, as mentioned above, the theory of SOC has not yet reached the level of understanding of classical critical phenomena. It suffers in particular from the lack of a thermodynamic formalism and notions such as Gibbs measures and free energy. On the other hand, by having a precise description of the dynamics of the finite size system one can expect a better understanding of the thermodynamic limit and decide which components in the models definition are really "relevant", and which information does the usually computed quantities (like critical exponents) actually give us.
This is the essence of a research program initiated in 1996 and fruitfully developed in the frame of the ZiF project. We found that some of the SOC models like the Zhang's model [11] can be studied with the tools of hyperbolic dynamical system theory [12, 13, 14]. This opened new perspectives and unexpected developments. In this approach 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 perturbations dynamics. Several deep results from the theory of hyperbolic dynamical systems can then be used, having interesting implications on the SOC dynamics. One can then give a description of the SOC attractor discussed by some people [1, 2] and of its fractal structure. The Lyapunov exponents, the geometric structure of the support of the invariant measure (Hausdorff dimensions), and the system size are related to the probability distribution of the avalanche size, via the Ledrappier-Young formula [15] interpreted in this setting as a conservation law [14].
In the beginning of the research year at ZiF, we related the Lyapunov exponents, which are microscopic dynamical quantities, to the macroscopic transport [16]. In particular, we gave bounds on the first negative Lyapunov exponent in terms of the flux dissipated at the boundaries per unit of time. We also established an explicit formula for a wide (extensive) part of the Lyapunov spectrum. The modes corresponding the negative Lyapunov exponents are the transport modes of a diffusion equation in a landscape where the metric is given by the probability that a site is active (density of active sites). It has been argued in the SOC community [17] that SOC requires a wide separation between the excitation and relaxation time scales (slow driving). We actually showed, as a consequence of our more general analysis of the Lyapunov spectrum, that the dynamics of the Zhang provides naturally this separation, and that the infinitely slow driving limit is actually reached as the size of the system goes to infinity. We also shown that the Lyapunov spectrum obey a simple scaling with a scaling exponent related to the scaling exponents of avalanche size, and duration. These results were established for the Zhang model. We are now investigating further developments towards other models like the train model [18].
One of the main objective of the research group at ZiF was the foundation of a thermodynamic formalism for SOC models and the development of new methods to analyze SOC systems. We have indeed establish a new and unexpected bridge towards equilibrium statistical mechanics. We have shown that the thermodynamic formalism developed by Sinai, Ruelle and Bowen [19, 20, 21] for hyperbolic dynamical systems applies in the Zhang's model [22]. This allows to define formal Gibbs measure, partition function and pressure characterizing the stationary state. We have furthermore shown that, when the size of the system tends to infinity, a Lee-Yang phenomenon [23] occurs, corresponding to a failure of analyticity for the asymptotic free energy. This unexpected result suggests that SOC systems might be closer as previously believed to classical critical phenomena and opens some effective way to map self-organized criticality to criticality. We also show that the Lee-Yang phenomena is observed only when the energy is locally conserved. Consequently, the local conservation is necessary, in the Zhang model, to reach a critical state in the thermodynamic limit.
In [24] we have further developed the connexions to Lee-Yang zeroes for general SOC models. A scaling theory of the Lee-Yang zeroes has been proposed, with some analogies with the existing scaling theories of the Lee-Yang zeroes in critical phenomena. We have shown in particular that, when the power law exponent is larger that 1, there is a violation of the scaling usually observed in classical critical phenomena [25, 26]; that is there is an anomalous logarithmic dependence on L for the angle that the zeroes do with the real axis in the plane. We also analyze the effect of a numerical cut off, appearing when the sample size is fixed independently on the lattice size, and show that it induces a bias in the empirical probability distribution, particularly visible on the Lee-Yang zeroes.
The forthcoming objective, initiated during the ZiF research year, are the identification of the relevant scaling fields under renormalization [27], the failure of fluctuation-dissipation relations induced by the presence of a singularity set in the dynamics [28] and the proof of ergodicity for the Zhang model [29].
The ZiF project was also an opportunity to investigate the possible application of SOC and self-organization concepts to new fields like biology or sociology.
In November 2000, the workshop DYNN 2000 (DYNN stands for Dynamical Neural Networks and Applications) was organized at ZiF. This was a place where neurobiologists, theoreticians and engineers exchanged ideas and transferred knowledge. This workshop was intended to present the recent developpements in the dynamical aspects of neural networks: Dynamical system approach of neural networks, Mean-field theory of neural networks, Vision, Control, and new trends. In particular an entire session was devoted to the potential applications of SOC to neural networks. We now intend to gather the main ideas which were discussed in this workshop in a self-contained book.
A collaboration with sociologists was also started [30]. The aim is to model the theory of Luman in a mathematical framework, in order to study the problem of innovation in the spirit of complex systems and self-organization. At the moment we are modeling the basic concepts at the microscopic level before studying the emergent properties.