scholarly journals Faults self-organized by repeated earthquakes in a quasi-static antiplane crack model

1996 ◽  
Vol 3 (1) ◽  
pp. 1-12 ◽  
Author(s):  
D. Sornette ◽  
C. Vanneste
Keyword(s):  
Long Range ◽  
Power Law ◽  
Stress Drop ◽  
Quenched Disorder ◽  
Crack Model ◽  
Fast Time ◽  
Power Law Distribution ◽  
Self Organized ◽  
Elastic Forces ◽  
Large Disorder

Abstract. We study a 2D quasi-static discrete crack anti-plane model of a tectonic plate with long range elastic forces and quenched disorder. The plate is driven at its border and the load is transferred to all elements through elastic forces. This model can be considered as belonging to the class of self-organized models which may exhibit spontaneous criticality, with four additional ingredients compared to sandpile models, namely quenched disorder, boundary driving, long range forces and fast time crack rules. In this "crack" model, as in the "dislocation" version previously studied, we find that the occurrence of repeated earthquakes organizes the activity on well-defined fault-like structures. In contrast with the "dislocation" model, after a transient, the time evolution becomes periodic with run-aways ending each cycle. This stems from the "crack" stress transfer rule preventing criticality to organize in favour of cyclic behaviour. For sufficiently large disorder and weak stress drop, these large events are preceded by a complex spacetime history of foreshock activity, characterized by a Gutenberg-Richter power law distribution with universal exponent B = 1±0.05. This is similar to a power law distribution of small nucleating droplets before the nucleation of the macroscopic phase in a first-order phase transition. For large disorder and large stress drop, and for certain specific initial disorder configurations, the stress field becomes frustrated in fast time: out-of-plane deformations (thrust and normal faulting) and/or a genuine dynamics must be introduced to resolve this frustration.

SELF-ORGANIZED CRITICALITY MODEL FOR EARTHQUAKES: QUIESCENCE, FORESHOCKS AND AFTERSHOCKS

1999 ◽  
Vol 09 (12) ◽  
pp. 2249-2255 ◽  
Author(s):  
S. HAINZL ◽  
G. ZÖLLER ◽  
J. KURTHS
Keyword(s):  
Power Law ◽  
B Value ◽  
Power Law Distribution ◽  
Aftershock Activity ◽  
Omori Law ◽  
Self Organized ◽  
Spatiotemporal Clustering ◽  
Self Organized Criticality ◽  
Clustering Of Events ◽  
Foreshock Activity

We introduce a crust relaxation process in a continuous cellular automaton version of the Burridge–Knopoff model. Analogously to the original model, our model displays a robust power law distribution of event sizes (Gutenberg–Richter law). The principal new result obtained with our model is the spatiotemporal clustering of events exhibiting several characteristics of earthquakes in nature. Large events are accompanied by a precursory quiescence and by localized fore- and aftershocks. The increase of foreshock activity as well as the decrease of aftershock activity follows a power law (Omori law) with similar exponents p and q. All empirically observed power law exponents, the Richter B-value, p and q and their variability can be reproduced simultaneously by our model, which depends mainly on the level of conservation and the relaxation time.

scholarly journals Ultra-small scale-free geometric networks

2006 ◽  
Vol 43 (3) ◽  
pp. 665-677 ◽  
Author(s):  
J. E. Yukich
Keyword(s):  
Long Range ◽  
Power Law ◽  
Degree Distribution ◽  
Small Scale ◽  
Graph Distance ◽  
Power Law Distribution ◽  
Scale Free

We consider a family of long-range percolation models (Gp)p>0on ℤdthat allow dependence between edges and have the following connectivity properties forp∈ (1/d, ∞): (i) the degree distribution of vertices inGphas a power-law distribution; (ii) the graph distance between pointsxandyis bounded by a multiple of logpdlogpd|x-y| with probability 1 -o(1); and (iii) an adversary can delete a relatively small number of nodes fromGp(ℤd∩ [0,n]d), resulting in two large, disconnected subgraphs.

scholarly journals Transition from Self-Organized Criticality into Self-Organization during Sliding Si3N4 Balls against Nanocrystalline Diamond Films

Entropy ◽  
2019 ◽  
Vol 21 (11) ◽  
pp. 1055
Author(s):  
Bogatov ◽  
Podgursky ◽  
Vagiström ◽  
Yashin ◽  
Shaikh ◽  
...  
Keyword(s):  
Friction Force ◽  
Power Law ◽  
Early Stage ◽  
Nanocrystalline Diamond ◽  
The Self ◽  
Self Organization ◽  
Power Law Distribution ◽  
Reciprocating Sliding ◽  
Self Organized ◽  
Self Organized Criticality

The paper investigates the variation of friction force (Fx) during reciprocating sliding tests on nanocrystalline diamond (NCD) films. The analysis of the friction behavior during the run-in period is the focus of the study. The NCD films were grown using microwave plasma-enhanced chemical vapor deposition (MW-PECVD) on single-crystalline diamond SCD(110) substrates. Reciprocating sliding tests were conducted under 500 and 2000 g of normal load using Si3N4 balls as a counter body. The friction force permanently varies during the test, namely Fx value can locally increase or decrease in each cycle of sliding. The distribution of friction force drops (dFx) was extracted from the experimental data using a specially developed program. The analysis revealed a power-law distribution f-µ of dFx for the early stage of the run-in with the exponent value (µ) in the range from 0.6 to 2.9. In addition, the frequency power spectrum of Fx time series follows power-law distribution f-α with α value in the range of 1.0–2.0, with the highest values (1.6–2.0) for the initial stage of the run-in. No power-law distribution of dFx was found for the later stage of the run-in and the steady-state periods of sliding with the exception for periods where a relatively extended decrease of coefficient of friction (COF) was observed. The asperity interlocking leads to the stick-slip like sliding at the early stage of the run-in. This tribological behavior can be related to the self-organized criticality (SOC). The emergence of dissipative structures at the later stages of the run-in, namely the formation of ripples, carbonaceous tribolayer, etc., can be associated with the self-organization (SO).

Self-Organized Critical Models

2003 ◽  
Author(s):  
M. E. J. Newman ◽  
R. G. Palmer
Keyword(s):  
Computer Simulations ◽  
Power Law ◽  
Large Scale ◽  
Fitness Landscape ◽  
Fitness Landscapes ◽  
Power Law Distribution ◽  
Edge Of Chaos ◽  
Self Organized ◽  
Self Organized Criticality ◽  
Phenotype Space

The models discussed in the last chapter are intriguing, but present a number of problems. In particular, most of the results about them come from computer simulations, and little is known analytically about their properties. Results such as the power-law distribution of extinction sizes and the system's evolution to the "edge of chaos" are only as accurate as the simulations in which they are observed. Moreover, it is not even clear what the mechanisms responsible for these results are, beyond the rather general arguments that we have already given. In order to address these shortcomings, Bak and Sneppen (1993; Sneppen et al. 1995; Sneppen 1995; Bak 1996) have taken Kauffman's ideas, with some modification, and used them to create a considerably simpler model of large-scale coevolution which also shows a power-law distribution of avalanche sizes and which is simple enough that its properties can, to some extent, be understood analytically. Although the model does not directly address the question of extinction, a number of authors have interpreted it, using arguments similar to those of section 1.2.2.5, as a possible model for extinction by biotic causes. The Bak-Sneppen model is one of a class of models that show "self-organized criticality," which means that regardless of the state in which they start, they always tune themselves to a critical point of the type discussed in section 2.4, where power-law behavior is seen. We describe self-organized criticality in more detail in section 3.2. First, however, we describe the Bak-Sneppen model itself. In the model of Bak and Sneppen there are no explicit fitness landscapes, as there are in NK models. Instead the model attempts to mimic the effects of landscapes in terms of "fitness barriers." Consider figure 3.1, which is a toy representation of a fitness landscape in which there is only one dimension in the genotype (or phenotype) space. If the mutation rate is low compared with the time scale on which selection takes place (as Kauffman assumed), then a population will spend most of its time localized around a peak in the landscape (labeled P in the figure).

scholarly journals Ultra-small scale-free geometric networks

2006 ◽  
Vol 43 (03) ◽  
pp. 665-677
Author(s):  
J. E. Yukich
Keyword(s):  
Long Range ◽  
Power Law ◽  
Degree Distribution ◽  
Small Scale ◽  
Graph Distance ◽  
Power Law Distribution ◽  
Scale Free

We consider a family of long-range percolation models (G p ) p>0 on ℤ d that allow dependence between edges and have the following connectivity properties for p ∈ (1/d, ∞): (i) the degree distribution of vertices in G p has a power-law distribution; (ii) the graph distance between points x and y is bounded by a multiple of log pd log pd | x - y | with probability 1 - o(1); and (iii) an adversary can delete a relatively small number of nodes from G p (ℤ d ∩ [0, n] d ), resulting in two large, disconnected subgraphs.

scholarly journals A simple model for the earthquake cycle combining self-organized complexity with critical point behavior

2002 ◽  
Vol 9 (5/6) ◽  
pp. 453-461 ◽  
Author(s):  
W. I. Newman ◽  
D. L. Turcotte
Keyword(s):  
Forest Fire ◽  
Power Law ◽  
Substantial Improvement ◽  
Earthquake Cycle ◽  
Power Law Distribution ◽  
Time Step ◽  
Fire Model ◽  
Inverse Cascade ◽  
Self Organized ◽  
Forest Fire Model

Abstract. We have studied a hybrid model combining the forest-fire model with the site-percolation model in order to better understand the earthquake cycle. We consider a square array of sites. At each time step, a "tree" is dropped on a randomly chosen site and is planted if the site is unoccupied. When a cluster of "trees" spans the site (a percolating cluster), all the trees in the cluster are removed ("burned") in a "fire." The removal of the cluster is analogous to a characteristic earthquake and planting "trees" is analogous to increasing the regional stress. The clusters are analogous to the metastable regions of a fault over which an earthquake rupture can propagate once triggered. We find that the frequency-area statistics of the metastable regions are power-law with a negative exponent of two (as in the forest-fire model). This is analogous to the Gutenberg-Richter distribution of seismicity. This "self-organized critical behavior" can be explained in terms of an inverse cascade of clusters. Small clusters of "trees" coalesce to form larger clusters. Individual trees move from small to larger clusters until they are destroyed. This inverse cascade of clusters is self-similar and the power-law distribution of cluster sizes has been shown to have an exponent of two. We have quantified the forecasting of the spanning fires using error diagrams. The assumption that "fires" (earthquakes) are quasi-periodic has moderate predictability. The density of trees gives an improved degree of predictability, while the size of the largest cluster of trees provides a substantial improvement in forecasting a "fire."

scholarly journals A STATISTICAL STUDY OF EARTH’S MAGNETIC FIELD REVERSALS SEQUENCES

2013 ◽  
Vol 31 (3) ◽  
pp. 533
Author(s):  
Marco Aurélio Do Espírito Santo ◽  
Douglas Santos Rodrigues Ferreira ◽  
Cosme Ferreira Da Ponte Neto ◽  
Andrés Reinaldo Rodriguez Papa
Keyword(s):  
Magnetic Field ◽  
Power Law ◽  
Critical Phenomenon ◽  
Synthetic Data ◽  
Statistical Characteristics ◽  
Power Law Distribution ◽  
Earth’S Magnetic Field ◽  
Self Organized ◽  
Earth's Magnetic Field ◽  
Distribution Type

ABSTRACT. This paper presents an analysis of the distribution of periods between consecutive reversals of the Earth’s magnetic field through a non-parametric statistics. The study analyzes whether data in different periods of reversal belong to the same distribution, the distribution type and whether the polarity states are equivalent. This analysis was performed for periods from 0 to 40 Ma, 40 to 80 Ma and 120 to 160 Ma. It was found that the data from the three periods show identical statistical characteristics which leads to the symmetry between the states of polarity and to a distribution compatible with a power law, which shows the possibility of a critical phenomenon acting on the geodynamo. The fact that the data obey a power law distribution prompted a comparison with synthetic data generated using two models based on criticality of reversals (one of them self-organized). These simple models reproduce some features of reversals as its temporal evolution and distribution of polarity intervals, and show a similarity with paleomagnetic data.Keywords: geomagnetic reversals, power law, self-organized criticality. RESUMO. Este artigo apresenta uma análise da distribuição de períodos entre reversões consecutivas do campo magnético da Terra através de uma estatística não-paramétrica. O estudo analisa se os dados dos diferentes períodos de reversão pertencem a uma mesma distribuição, o tipo de distribuição que eles obedecem e se os estados de polaridade são equivalentes. Esta análise foi realizada nos períodos de 0 a 40 Ma, de 40 a 80 Ma e de 120 a 160 Ma. Encontrou-se que os dados dos três períodos apresentam características estatísticas idênticas, o que leva à simetria entre os estados de polaridade e a uma distribuição compatível com uma lei de potência, o que mostra a possibilidade de um fenômeno crítico atuando no geodínamo. O fato dos dados obedecerem a uma distribuição de lei de potências motivou uma comparação com dados sintéticos gerados através de dois modelos de reversões baseados em criticalidade (um deles auto-organizado). Estes modelos simples reproduzem algumas características das reversões, como sua evolução temporal e a distribuição de intervalo de polaridade, e mostram uma similaridade com dados paleomagnéticos.Palavras-chave: reversões geomagnéticas, lei de potências, criticalidade auto-organizada.

scholarly journals Noise-Induced Spiral Waves in Astrocyte Syncytia Show Evidence of Self-Organized Criticality

1998 ◽  
Vol 79 (2) ◽  
pp. 1098-1101 ◽  
Author(s):  
Peter Jung ◽  
Ann Cornell-Bell ◽  
Kathleen Shaver Madden ◽  
Frank Moss
Keyword(s):  
Power Law ◽  
Spiral Waves ◽  
Power Law Distribution ◽  
Self Organized ◽  
Show Evidence ◽  
Self Organized Criticality ◽  
Chemical Waves ◽  
Cultured Networks ◽  
The Waves ◽  
Vertebrate Central Nervous System

Jung, Peter, Ann Cornell-Bell, Kathleen Shaver Madden, and Frank Moss. Noise-induced spiral waves in astrocyte syncytia show evidence of self-organized criticality. J. Neurophysiol. 79: 1098–1101, 1998. Long range (a few centimeters), long lived (many seconds), spiral chemical waves of calcium ions (Ca2+) are observed in cultured networks of glial cells for normal concentrations of the neurotransmitter kainate. A new method for quantitatively measuring the spatiotemporal size of the waves is described. This measure results in a power law distribution of wave sizes, meaning that the process that creates the waves has no preferred spatial or temporal (size or lifetime) scale. This power law is one signature of self-organized critical phenomena, a class of behaviors found in many areas of science. The physiological results for glial networks are fully supported by numerical simulations of a simple network of noisy, communicating threshold elements. By contrast, waves observed in astrocytes cultured from human epileptic foci exhibited radically different behavior. The background random activity, or “noise”, of the network is controlled by the kainate concentration. The mean rate of wave nucleation is mediated by the network noise. However, the power law distribution is invariant, within our experimental precision, over the range of noise intensities tested. These observations indicate that spatially and temporally coherent Ca2+ waves, mediated by network noise may play and important role in generating correlated neural activity (waves) over long distances and times in the healthy vertebrate central nervous system.

scholarly journals Synchronization and desynchronization in the Olami-Feder-Christensen earthquake model and potential implications for real seismicity

2011 ◽  
Vol 18 (5) ◽  
pp. 635-642 ◽  
Author(s):  
S. Hergarten ◽  
R. Krenn
Keyword(s):  
Power Law ◽  
Statistical Properties ◽  
Scaling Exponent ◽  
Power Law Distribution ◽  
Scaling Properties ◽  
Temporal Clustering ◽  
Self Organized ◽  
Earthquake Model ◽  
Self Organized Criticality ◽  
Large Earthquakes

Abstract. The Olami-Feder-Christensen model is probably the most studied model in the context of self-organized criticality and reproduces several statistical properties of real earthquakes. We investigate and explain synchronization and desynchronization of earthquakes in this model in the nonconservative regime and its relevance for the power-law distribution of the event sizes (Gutenberg-Richter law) and for temporal clustering of earthquakes. The power-law distribution emerges from synchronization, and its scaling exponent can be derived as τ = 1.775 from the scaling properties of the rupture areas' perimeter. In contrast, the occurrence of foreshocks and aftershocks according to Omori's law is closely related to desynchronization. This mechanism of foreshock and aftershock generation differs strongly from the widespread idea of spontaneous triggering and gives an idea why some even large earthquakes are not preceded by any foreshocks in nature.

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