Self-organized criticality in a nutshell

In order to gain insight into the nature of self-organized criticality (SOC), we present a minimal model exhibiting this phenomenon. In this analytically solvable model, the state of the system is fully described by a single-integer variable. The system organizes in its critical state without external tuning. We derive analytically the probability distribution of durations of disturbances propagating through the system. As required by SOC, this distribution is scale invariant and follows a power law over several orders of magnitude. Our solution also reproduces the exponential tail of the distribution due to finite size effects. Moreover, we show that large avalanches are suppressed when stabilizing the system in its critical state. Interestingly, avalanches are affected in a similar way when driving the system away from the critical state. With this model, we have reduced SOC dynamics to a leveling process as described by Ehrenfest's famous flea model.     

Complex scaling behavior of nonconserved self-organized critical systems

The Olami-Feder-Christensen earthquake model is often considered the prototype dissipative self-organized critical model. It is shown that the size distribution of events in this model results from a complex interplay of several different phenomena, including limited floating-point precision. Parallels between the dynamics of synchronized regions and those of a system with periodic boundary conditions are pointed out, and the asymptotic avalanche size distribution is conjectured to be dominated by avalanches of size 1, with the weight of larger avalanches converging towards zero as the system size increases.     

Self-organization of the critical state in granular superconductors

The critical state in granular superconductors is studied using two mathematical models: systems of differential equations for the gauge-invariant phase difference and a simplified model that is described by a system of coupled mappings and in many cases is equivalent to the standard models used for studying self-organized criticality. It is shown that the critical state of granular superconductors is self-organized in all cases studied. In addition, it is shown that the models employed are essentially equivalent, i.e., they demonstrate not only the same critical behavior, but they also lead to the same noncritical phenomena. The first demonstration of the existence of self-organized criticality in a system of nonlinear differential equations and its equivalence to self-organized criticality in standard models is given in this paper.     

Onset of self-organized critical state in a one-dimensional multijunction SQUID as a result of random arrangement of junctions

The critical state of a one-dimensional multijunction SQUID with randomly distributed junctions exposed to a slowly varying magnetic field is studied. It is shown that a small scatter of interjunction distances is sufficient for the critical state of the system to become self-organized. A simplified and basically new model is proposed for studying the self-organization in a system where this phenomenon occurs in a fully deterministic situation.     

Self-organized model of cascade spreading

We study simultaneous price drops of real stocks and show that for high drop thresholds they follow a power-law distribution. To reproduce these collective downturns, we propose a minimal self-organized model of cascade spreading based on a probabilistic response of the system elements to stress conditions. This model is solvable using the theory of branching processes and the mean-field approximation. For a wide range of parameters, the system is in a critical state and displays a power-law cascade-size distribution similar to the empirically observed one. We further generalize the model to reproduce volatility clustering and other observed properties of real stocks.     

Bubbling and large-scale structures in avalanche dynamics

Using a simple lattice model for granular media, we present a scenario of self-organization that we term self-organized structuring where the steady state has several unusual features: (1) large-scale spatial and/or temporal inhomogeneities and (2) the occurrence of a nontrivial peaked distribution of large events which propagate like "bubbles" and have a well-defined frequency of occurrence. We discuss the applicability of such a scenario for other models introduced in the framework of self-organized criticality.     

Evolution of avalanche conducting states in electrorheological liquids

Charge transport in electrorheological fluids is studied experimentally under strongly nonequilibrium conditions. By injecting an electrical current into a suspension of conducting nanoparticles we are able to initiate a process of self-organization which leads, in certain cases, to formation of a stable pattern which consists of continuous conducting chains of particles. The evolution of the dissipative state in such a system is a complex process. It starts as an avalanche process characterized by nucleation, growth, and thermal destruction of such dissipative elements as continuous conducting chains of particles as well as electroconvective vortices. A power-law distribution of avalanche sizes and durations, observed at this stage of the evolution, indicates that the system is in a self-organized critical state. A sharp transition into an avalanche-free state with a stable pattern of conducting chains is observed when the power dissipated in the fluid reaches its maximum. We propose a simple evolution model which obeys the maximum power condition and also shows a power-law distribution of the avalanche sizes.     

Self-organized criticality and stock market dynamics: an empirical study

The stock market is a complex self-interacting system, characterized by intermittent behaviour. Periods of high activity alternate with periods of relative calm. In the present work we investigate empirically the possibility that the market is in a self-organized critical state (SOC). A wavelet transform method is used in order to separate high activity periods, related to the avalanches found in sandpile models, from quiescent. A statistical analysis of the filtered data shows a power law behaviour in the avalanche size, duration and laminar times. The memory process, implied by the power law distribution of the laminar times, is not consistent with classical conservative models for self-organized criticality. We argue that a “near-SOC” state or a time dependence in the driver, which may be chaotic, can explain this behaviour.     

Self-organized critical limit of autocatalytic surface reactions

We argue that the autocatalytic surface reaction 2CO+O₂→2CO₂ on Pt(110) may show self-organized critical behavior if the appropriate range of parameter values is investigated. Such a self-organized critical state is characterized by a power-law distribution of reconstructed surface regions.     

Olami-Feder-Christensen model on different networks

We investigate numerically the Self Organized Criticality (SOC) properties of the dissipative Olami-Feder-Christensen model on small-world and scale-free networks. We find that the small-world OFC model exhibits self-organized criticality. Indeed, in this case we observe power law behavior of earthquakes size distribution with finite size scaling for the cut-off region. In the scale-free OFC model, instead, the strength of disorder hinders synchronization and does not allow to reach a critical state.     

Avalanche dynamics of magnetic flux in a two-dimensional discrete superconductor

The critical state of a two-dimensional discrete superconductor in an external magnetic field is studied. This state is found to be self-organized in the generalized sense, i.e., is a set of metastable states that transform to each other by means of avalanches. An avalanche is characterized by the penetration of a magnetic flux to the system. The sizes of the occurring avalanches, i.e., changes in the magnetic flux, exhibit the power-law distribution. It is also shown that the size of the avalanche occurring in the critical state and the external magnetic field causing its change are statistically independent quantities.     

Universality and self-similarity of an energy-constrained sandpile model with random neighbors

We study an energy-constrained sandpile model with random neighbors. The critical behavior of the model is in the same universality class as the mean-field self-organized criticality sandpile. The critical energy E(c) depends on the number of neighbors n for each site, but the various exponents are independent of n. A self-similar structure with n-1 major peaks is developed for the energy distribution p(E) when the system approaches its stationary state. The avalanche dynamics contributes to the major peaks appearing at E(p(k))=2k/(2n-1) with k=1,2,...,n-1, while the fine self-similar structure is a natural result of the way the system is disturbed.     

Attaining and maintaining criticality in a neuronal network model

We propose a cellular automaton model for neuronal networks that combines short-term synaptic plasticity with long-term metaplasticity. We investigate how these two mechanisms contribute to attaining and maintaining operation at the critical point. We find that short-term plasticity, represented in the model by synaptic depression and synaptic recovery, is sufficient to allow the system to attain the critical state, if the level of plasticity is properly chosen. However, it is not sufficient to maintain the criticality if the system is perturbed. But the long time scale change in the short-term plasticity, a change in the way synaptic efficacy is modified, allows the system to recover from perturbation. Working together, these two time scales of plasticity could help the system to attain and maintain criticality, leading to a self-organized critical state.     

Structure of the critical state in a discrete superconductor for different values of the SQUID parameter

The critical state of a 1D multijunction SQUID with intrinsic spatial randomness has been studied. It is shown that the system behavior is independent of the SQUID parameter and the critical state under consideration is self-organized.     

Finite-size effects in the self-organized critical forest-fire model

We study finite-size effects in the self-organized critical forest-fire model by numerically evaluating the tree density and the fire size distribution. The results show that this model does not display the finite-size scaling seen in conventional critical systems. Rather, the system is composed of relatively homogeneous patches of different tree densities, leading to two qualitatively different types of fires: those that span an entire patch and those that do not. As the system size becomes smaller, the system contains less patches, and finally becomes homogeneous, with large density fluctuations in time. Received 24 April 1999 and Received in final form 26 October 1999     

A sharp transition between a trivial 1D BTW model and self-organized critical rice-pile model

A one-dimensional model of a rice-pile is numerically studied for different driving mechanisms. We found that for a sufficiently large system, there is a sharp transition between the trivial behaviour of a 1D BTW model and self-organized critical (SOC) behaviour. Depending on the driving mechanism, the self-organized critical rice-pile model belongs to two different universality classes. Received 18 December 1998     

Self-organized criticality in the El Niño Southern oscillation

The Fourier analysis has been applied to observational sea surface temperature (SST) data from the Southern Pacific. The spectral response of the daily temperature fluctuations indicates that the so-called El Niño Southern Oscillation (ENSO) belongs to the class of dynamical phenomena which are in a self-organized critical state. This has implications on the predictability of the significant events in the ocean-atmosphere interaction process. A toy model is used to point out the similarities of this system with other large scale phenomena, as earthquakes and volcanic eruptions, which are considered to be consistent with the hypothesis of self-organized criticality. The qualitative agreement between observational data and results of numerical simulations demonstrates the validity extent of the theoretical approach.     

Criticality in a simple model for brain functioning

We introduce a simple model for a set of interacting idealized neurons. The model presents a self-organized state in which avalanches of all sizes are observed and activity is detected in the whole extension of the simulated system without a typical length scale. The basic elements of the model are endowed with the main features of a neuron function. On this basis it is speculated that the collective system that they form, i.e., the brain, could display self-organized criticality in some situations.     

Self-organization of the critical state in a chain of SQUIDs

The critical state of a 1D granular superconductor (modeled as a chain of SQUIDs or, in other words, a 1D Josephson-junction array) is studied on the basis of a system of differential equations for the gauge-invariant phase difference. It is established that the critical state is self-organized. It is shown that the problems of self-organization in the sandpile model and in a granular superconductor belong to the same universality class. Pis’ma Zh. éksp. Teor. Fiz. 68, No. 9, 688–694 (10 November 1998)     

Kinetic theory of strength and a self-organized critical state in the process of fracture of materials

The basic principles of the Zhurkov kinetic approach are analyzed in light of the modern idea of self-organization of nonlinear systems under nonequilibrium conditions. It is indicated that the final state before fracture can be described in terms of the model of a self-organized critical state.