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.     

Self-organized criticality with and without conservation

The self-organized sandpile models lose criticality if dissipation is introduced. Recently Christensen et al. have shown that dissipative automata based on the Burridge-Knopoff earthquake model exhibit critical behavior. Criticality is qualitatively different for the cases with and without conservation: A new characteristic length appears for the dissipative case which diverges slower than the system size. For all dissipative models we have found a characteristic frequency in the power spectrum of the released energy, which is absent for the conservative case. The exponents describing criticality change continuously as a function of the strength of dissipation and crossover phenomena occur in the vicinity of conservation. Disorder is irrelevant if conservation is present while it destroys criticality in the dissipative case.     

Self-organized criticality on quasiperiodic graphs

It is shown that quasielastic scattering of ultracold neutrons due to diffusive motion of scatterers at the surface of liquid polymer (fomblin) or adsorbed hydrogenous contaminations of the surface of neutron traps may be possible reason of their energy spreading during long storage time in closed traps, which was observed in the recent experiments. Received 8 July 1998     

Self-organized criticality in the olami-feder-christensen model

A system is in a self-organized critical state if the distribution of some measured events obeys a power law. The finite-size scaling of this distribution with the lattice size is usually enough to assume that the system displays self-organized criticality. This approach, however, can be misleading. In this paper we analyze the behavior of the branching rate sigma of the events to establish whether a system is in a critical state. We apply this method to the Olami-Feder-Christensen model to obtain evidence that, in contrast to previous results, the model is critical in the conservative regime only.     

Self-organized criticality in multi-pulse gamma-ray bursts

The variability in multi-pulse gamma-ray bursts(GRBs)may help to reveal the mechanism of underlying processes from the central engine.To investigate whether the self-organized criticality(SOC)phenomena exist in the prompt phase of GRBs,we statistically study the proper ties of GRBs with more than 3 pulses in each burst by fitting the distributions of several observed physical variables with a Markov Chain Monte Carlo approach,including the isotropic energy E_iso,the duration time T,and the peak count rate P of each pulse.Our sample consists of 454 pulses in 93 GRBs observed by the CGRO/BATSE satellite.The best-fitting values and uncertainties for these power-law indices of the differential frequency distributions are:α_E^d=1.54±0.09,α_T^d=1.82_-0.15^+0.14andα_P^d=2.09_-0.19^0.18,while the power-law indices in the cumulative frequency distributions are:α_E^c=1.44_-0.10^+0.08,α_T^c=1.75_-0.13^0.11andα_P^c=1.99_-0.19^+0.16.We find that these distributions are roughly consistent with the physical framework of a Fractal-Diffusive,Self^Organized Criticality(FD-SOC)system with the spatial dimension S=3 and the classical diffusionβ=1.Our results support that the jet responsible for the GRBs should be magnetically dominated and magnetic instabilities(e.g.,kink model,or tearing-model instability)lead the GRB emission region into the SOC state.     

Self-organized criticality model for brain plasticity

Networks of living neurons exhibit an avalanche mode of activity, experimentally found in organotypic cultures. Here we present a model that is based on self-organized criticality and takes into account brain plasticity, which is able to reproduce the spectrum of electroencephalograms (EEG). The model consists of an electrical network with threshold firing and activity-dependent synapse strengths. The system exhibits an avalanche activity in a power-law distribution. The analysis of the power spectra of the electrical signal reproduces very robustly the power-law behavior with the exponent 0.8, experimentally measured in EEG spectra. The same value of the exponent is found on small-world lattices and for leaky neurons, indicating that universality holds for a wide class of brain models.     

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 criticality in a computer network model

We study the collective behavior of computer network nodes by using a cellular automaton model. The results show that when the load of network is constant, the throughputs and buffer contents of nodes are power-law distributed in both space and time. Also the feature of 1/f noise appears in the power spectrum of the change of the number of nodes that bear a fixed part of the system load. It can be seen as yet another example of self-organized criticality. Power-law decay in the distribution of buffer contents implies that heavy network congestion occurs with small probability. The temporal power-law distribution for throughput might be a reasonable explanation for the observed self-similarity in computer network traffic.     

Magnetic moment of square SIS Josephson arrays: Self-organized criticality

The temperature-and magnetic-field dependences of the magnetic moment of square Josephson arrays with SIS-type junctions are studied experimentally. Two temperature regions are observed with different types of magnetization curves. Magnetic flux avalanches are detected in the low-temperature region. Statistical analysis of avalanche amplitudes A shows that their size distribution varies in accordance with the power law P∝Aⁿ with crossover, when exponent n varies from n=?0.7 for small avalanches to n=?6 for large avalanches, while the frequency spectrum varies in accordance with the law 1/f^α. Such behavior is interpreted as a manifestation of self-organized criticality.     

Growing inhomogeneities in cosmological Goldstone modes

We examine the evolution of initial inhomogeneities in a Goldstone field in an expanding Friedmann-Robertson-Walker universe. We find subhorizon inhomogeneities grow, relative to the homogeneous state. This stems not from growing fluctuations — which simply redshift — but from rapid ( a⁻⁶) decay of the homogeneous state. We show how Goldstone modes escape assumptions — some inapplicable, some ill-founded — underpinning conventional analyses of cosmological fluctuations. Finally, we reconcile our analysis to standard cosmology, noting that the Goldstone evolution is essentially decoupled and dynamical.     

Self-organized criticality in MHD driven plasma edge turbulence

We analyze long-range time correlations and self-similar characteristics of the electrostatic turbulence at the plasma edge and scrape-off layer in the Tokamak Chauffage Alfvén Brésillien (TCABR), with low and high Magnetohydrodynamics (MHD) activity. We find evidence of self-organized criticality (SOC), mainly in the region near the tokamak limiter. Comparative analyses of data before and during the MHD activity reveals that during the high MHD activity the Hurst parameter decreases. Finally, we present a cellular automaton whose parameters are adjusted to simulate the analyzed turbulence SOC change with the MHD activity variation.     

Self-organized criticality in 1D stochastic traffic flow model

Results of computer simulations of a 1D particle hopping model of traffic flow are presented. The model is characterized by parallel update and fully asymmetric stochastic hopping dynamics which allows unbounded series of jumps to empty neighbour sites on the right. The considered case of open boundary conditions can be used to model a “bottleneck” situation in traffic. Evidence for self-organized criticality is found in two aspects: the presence of long-range spatial correlations manifested in the shape of density profiles, and long-time temporal correlations showing up in the low-frequency behaviour of the spectral density of the total particle number and flow. A plausible conjecture is to interpret the observed qualitative changes in these features, as a function of the injection rate and the hopping probability, in terms of a nonequilibrium phase transition between a low-density phase and a maximal current phase. This conjecture is supported by the phase diagram obtained in mean-field approximation.     

Lorentzian Goldstone modes shared among photons and gravitons

It has long been known that photons and gravitons may appear as vector and tensor Goldstone modes caused by spontaneous Lorentz invariance violation (SLIV). Usually this approach is considered for photons and gravitons separately. We develop the emergent electrogravity theory consisting of the ordinary QED and the tensor-field gravity model which mimics the linearized general relativity in Minkowski spacetime. In this theory, Lorentz symmetry appears incorporated into higher global symmetries of the length-fixing constraints put on the vector and tensor fields involved, \(A_{\mu }^{2}=\pm M_{A}^{2}\) and \(H_{\mu \nu }^{2}=\pm M_{H}^{2}\) (\(M_{A}\) and \(M_{H}\) are the proposed symmetry breaking scales). We show that such a SLIV pattern being related to breaking of global symmetries underlying these constraints induces the massless Goldstone and pseudo-Goldstone modes shared by photon and graviton. While for a vector field case the symmetry of the constraint coincides with Lorentz symmetry SO(1, 3) of the electrogravity Lagrangian, the tensor-field constraint itself possesses much higher global symmetry SO(7, 3), whose spontaneous violation provides a sufficient number of zero modes collected in a graviton. Accordingly, while the photon may only contain true Goldstone modes, the graviton appears at least partially to be composed of pseudo-Goldstone modes rather than of pure Goldstone ones. When expressed in terms of these modes, the theory looks essentially nonlinear and contains a variety of Lorentz and CPT violating couplings. However, all SLIV effects turn out to be strictly cancelled in the lowest order processes considered in some detail. How this emergent electrogravity theory could be observationally different from conventional QED and GR theories is also briefly discussed.