Self-organized criticality in a crack-propagation model of earthquakes

Anomalous Self-Organization in Active Piles

Inspired by recent observations on active self-organized critical (SOC) systems, we designed an active pile (or ant pile) model with two ingredients: beyond-threshold toppling and under-threshold active motions. By including the latter component, we were able to replace the typical power-law distribution for geometric observables with a stretched exponential fat-tailed distribution, where the exponent and decay rate are dependent on the activity's strength (ζ). This observation helped us to uncover a hidden connection between active SOC systems and α-stable Levy systems. We demonstrate that one can partially sweep α-stable Levy distributions by changing ζ. The system undergoes a crossover towards Bak-Tang-Weisenfeld (BTW) sandpiles with a power-law behavior (SOC fixed point) below a crossover point ζ<ζ*≈0.1.

Multifractal Properties of Time Series of Synthetic Earthquakes Obtained from a Spring-Block Model

With the spring-block model proposed by Olami, Feder, and Christensen (OFC), we obtained a time series of synthetic earthquakes with different values of the conservation level (β), which measures the fraction of the energy that a relaxing block passes to its neighbors. The time series have multifractal characteristics, and we analyzed them with the Chhabra and Jensen method. We calculated the width, symmetry, and curvature parameters for each spectrum. As the value of conservation level increases, the spectra widen, the symmetric parameter increases, and the curvature around the maximum of the spectra decreases. In a long series of synthetic seismicity, we located earthquakes of the greatest magnitude and built overlapping windows before and after them. For the time series in each window, we performed multifractal analysis to obtain multifractal spectra. We also calculated the width, symmetry, and curvature around the maximum of the multifractal spectrum. We followed the evolution of these parameters before and after large earthquakes. We found that the multifractal spectra had greater widths, were less skewed to the left, and were very pointed around the maximum before rather than after large earthquakes. We studied and calculated the same parameters and found the same results in the analysis of the Southern California seismicity catalog. This suggests that there seems to be a process of preparation for a great earthquake and that its dynamics are different from the one that occurs after this mainshock based on the behavior of the parameters mentioned before.

Review and Update on Some Connections between a Spring-Block SOC Model and Actual Seismicity in the Case of Subduction Zones

The self-organized critical (SOC) spring-block models are accessible and powerful computational tools for the study of seismic subduction. This work aims to highlight some important findings through an integrative approach of several actual seismic properties, reproduced by using the Olami, Feder, and Christensen (OFC) SOC model and some variations of it. A few interesting updates are also included. These results encompass some properties of the power laws present in the model, such as the Gutenberg-Richter (GR) law, the correlation between the parameters a and b of the linear frequency-magnitude relationship, the stepped plots for cumulative seismicity, and the distribution of the recurrence times of large earthquakes. The spring-block model has been related to other relevant properties of seismic phenomena, such as the fractal distribution of fault sizes, and can be combined with the work of Aki, who established an interesting relationship between the fractal dimension and the b-value of the Gutenberg-Richter relationship. Also included is the work incorporating the idea of asperities, which allowed us to incorporate several inhomogeneous models in the spring-block automaton. Finally, the incorporation of a Ruff-Kanamori-type diagram for synthetic seismicity, which is in reasonable accordance with the original Ruff and Kanamori diagram for real seismicity, is discussed.

Quasistatic kinetic avalanches and self-organized criticality in deviatorically loaded granular media

The behavior of granular media under quasistatic loading has recently been shown to attain a stable evolution state corresponding to a manifold in the space of micromechanical variables. This state is characterized by sudden transitions between metastable jammed states, involving the partial micromechanical rearrangement of the granular medium. Using numerical simulations of two-dimensional granular media under quasistatic biaxial compression, we show that the dynamics in the stable evolution state is characterized by scale-free avalanches well before the macromechanical stationary flow regime traditionally linked to a self-organized critical state. This, together with the nonuniqueness and the nonmonotony of macroscopic deformation curves, suggests that the statistical avalanche properties and the susceptibilities of the system cannot be reduced to a function of the macromechanical state. The associated scaling exponents are nonuniversal and depend on the interactions between particles. For stiffer particles (or samples at low confining pressure) we find distributions of avalanche properties compatible with the predictions of mean-field theory. The scaling exponents decrease below the mean-field values for softer interactions between particles. These lower exponents are consistent with observations for amorphous solids at their critical point. We specifically discuss the relationship between microscopic and macroscopic variables, including the relation between the external stress drop and the internal potential energy released during kinetic avalanches.

Waiting-time statistics in magnetic systems

Many complex systems, from earthquakes and financial markets to Barkhausen effect in ferromagnetic materials, respond with a noise consisting of discrete avalanche-like events with broad range of sizes and durations, separated by waiting times. Here we focus on the waiting-time statistics in magnetic systems. By investigating the Barkhausen noise in amorphous and polycrystalline ferromagnetic films having different thicknesses, we uncover the form of the waiting-time distribution in time series recorded from the irregular and irreversible motion of magnetic domain walls. Further, we address the question of if the waiting-time distribution evolves with the threshold level, as well as with the film thickness and structural character of the materials. Our results, besides informing on the temporal avalanche correlations, disclose the waiting-time statistics in magnetic systems also bring fingerprints of the universality classes of Barkhausen avalanches and a dimensional crossover in the domain wall dynamics.

Visibility graph analysis of synthetic earthquakes generated by the Olami-Feder-Christensen spring-block model

In this study, we investigate the relationship between topological and seismological parameters of earthquake sequences generated by the Olami-Feder-Christensen (OFC) [Olami et al., Phys. Rev. Lett. 68(8), 1244 (1992)] spring-block model and converted in undirected graphs by using the visibility graph method [Lacasa et al., Proc. Natl. Acad. Sci. U.S.A. 105(13), 4972-4975 (2008)]. In particular, we study the relationship between the Gutenberg-Richter b-value and the so-called K-M slope, which describes the relationship between magnitudes and connectivity degrees. This relationship was found to follow a rather universal law in observational earthquake sequences, and, thus, in the present work, we aim at verifying such universality also in earthquake sequences generated by the OFC spring-block model. We found that for ⟨b⟩ between approximately 1 and 2, which is nearly the range of variation for most of the real seismicity cases observed worldwide, the relationship between ⟨b⟩ and ⟨K-M slope⟩ does not depend on the lattice size L. Furthermore, the slope of the regression line between ⟨b⟩ and ⟨K-M slope⟩ in the range of ⟨b⟩ between 1 and 2 changes with the definition of magnitude and the length of the earthquake sequence.

A Simple Model to Relate the Elastic Ratio Gamma of a Critically Self-Organized Spring-Block Model with the Age of a Lithospheric Downgoing Plate in a Subduction Zone

In 1980, Ruff and Kanamori (RK) published an article on seismicity and the subduction zones where they reported that the largest characteristic earthquake (Mw) of a subduction zone is correlated with two geophysical quantities: the rate of convergence between the oceanic and continental plates (V) and the age of the corresponding subducting oceanic lithosphere (T). This proposal was synthetized by using an empirical graph (RK-diagram) that includes the variables Mw, V and T. We have recently published an article that reports that there are some common characteristics between real seismicity, sandpaper experiments and a critically self-organized spring-block model. In that paper, among several results we qualitatively recovered a RK-diagram type constructed with equivalent synthetic quantities corresponding to Mw, V and T. In the present paper, we improve that synthetic RK-diagram by means of a simple model relating the elastic ratio γ of a critically self-organized spring-block model with the age of a lithospheric downgoing plate. In addition, we extend the RK-diagram by including some large subduction earthquakes occurred after 1980. Similar behavior to the former RK-diagram is observed and its SOC synthetic counterpart is obtained.

Crackling Dynamics in the Mechanical Response of Knitted Fabrics

Crackling noise, which occurs in a wide range of situations, is characterized by discrete events of various sizes, often correlated in the form of avalanches. We report experimental evidence that the mechanical response of a knitted fabric displays such broadly distributed events both in the force signal and in the deformation field, with statistics analogous to that of earthquakes or soft amorphous materials. A knit consists of a regular network of frictional contacts, linked by the elasticity of the yarn. When deformed, the fabric displays spatially extended avalanchelike yielding events resulting from collective interyarn contact slips. We measure the size distribution of these avalanches, at the stitch level from the analysis of nonelastic displacement fields and externally from force fluctuations. The two measurements yield consistent power law distributions reminiscent of those found in other avalanching systems. Our study shows that a knitted fabric is not only a thread-based metamaterial with highly sought after mechanical properties, but also an original, model system, with topologically protected structural order, where an intermittent, scale-invariant response emerges from minimal ingredients, and thus a significant landmark in the study of out-of-equilibrium universality.

Statistical investigation of avalanches of three-dimensional small-world networks and their boundary and bulk cross-sections

In many situations we are interested in the propagation of energy in some portions of a three-dimensional system with dilute long-range links. In this paper, a sandpile model is defined on the three-dimensional small-world network with real dissipative boundaries and the energy propagation is studied in three dimensions as well as the two-dimensional cross-sections. Two types of cross-sections are defined in the system, one in the bulk and another in the system boundary. The motivation of this is to make clear how the statistics of the avalanches in the bulk cross-section tend to the statistics of the dissipative avalanches, defined in the boundaries as the concentration of long-range links (α) increases. This trend is numerically shown to be a power law in a manner described in the paper. Two regimes of α are considered in this work. For sufficiently small αs the dominant behavior of the system is just like that of the regular BTW, whereas for the intermediate values the behavior is nontrivial with some exponents that are reported in the paper. It is shown that the spatial extent up to which the statistics is similar to the regular BTW model scales with α just like the dissipative BTW model with the dissipation factor (mass in the corresponding ghost model) m^{2}∼α for the three-dimensional system as well as its two-dimensional cross-sections.

Soft modes and strain redistribution in continuous models of amorphous plasticity: the Eshelby paradigm, and beyond?

The deformation of disordered solids relies on swift and localised rearrangements of particles. The inspection of soft vibrational modes can help predict the locations of these rearrangements, while the strain that they actually redistribute mediates collective effects. Here, we study soft modes and strain redistribution in a two-dimensional continuous mesoscopic model based on a Ginzburg-Landau free energy for perfect solids, supplemented with a plastic disorder potential that accounts for shear softening and rearrangements. Regardless of the disorder strength, our numerical simulations show soft modes that are always sharply peaked at the softest point of the material (unlike what happens for the depinning of an elastic interface). Contrary to widespread views, the deformation halo around this peak does not always have a quadrupolar (Eshelby-like) shape. Instead, for finite and narrowly-distributed disorder, it looks like a fracture, with a strain field that concentrates along some easy directions. These findings are rationalised with analytical calculations in the case where the plastic disorder is confined to a point-like 'impurity'. In this case, we unveil a continuous family of elastic propagators, which are identical for the soft modes and for the equilibrium configurations. This family interpolates between the standard quadrupolar propagator and the fracture-like one as the anisotropy of the elastic medium is increased. Therefore, we expect to see a fracture-like propagator when extended regions on the brink of failure have already softened along the shear direction and thus rendered the material anisotropic, but not failed yet. We speculate that this might be the case in carefully aged glasses just before macroscopic failure.

Classification of universality classes for quasideterministic sandpile models

The critical behavior of the two-state rotational sandpile model proposed by Santra et al. [Phys. Rev. E 75, 041122 (2007)PLEEE81539-375510.1103/PhysRevE.75.041122] and the locally deterministic and globally stochastic three-state sandpile model are investigated via Monte Carlo simulations. Through these simulations, we are able to estimate critical exponents that characterize the avalanche properties, i.e., the probability distributions of the avalanche size, area, lifetime, and gyration radius, and the expectation values of the avalanche size and area against time and of the size against area. The results are compared with those of the known universality classes. The two models are found to yield consistent results within the range of statistical error, and appear to be consistent with the stochastic two-state Manna sandpile model; therefore, both models appear to belong to the Manna universality class. Our results contradict the earlier conclusion of Santra et al., which we attribute to the slow convergence of the probability distribution to the asymptotic power-law behavior, particularly for the size and lifetime of avalanches.

Dissipative stochastic sandpile model on small-world networks: Properties of nondissipative and dissipative avalanches

A dissipative stochastic sandpile model is constructed and studied on small-world networks in one and two dimensions with different shortcut densities ϕ, where ϕ=0 represents regular lattice and ϕ=1 represents random network. The effect of dimension, network topology, and specific dissipation mode (bulk or boundary) on the the steady-state critical properties of nondissipative and dissipative avalanches along with all avalanches are analyzed. Though the distributions of all avalanches and nondissipative avalanches display stochastic scaling at ϕ=0 and mean-field scaling at ϕ=1, the dissipative avalanches display nontrivial critical properties at ϕ=0 and 1 in both one and two dimensions. In the small-world regime (2^{-12}≤ϕ≤0.1), the size distributions of different types of avalanches are found to exhibit more than one power-law scaling with different scaling exponents around a crossover toppling size s_{c}. Stochastic scaling is found to occur for s<s_{c} and the mean-field scaling is found to occur for s>s_{c}. As different scaling forms are found to coexist in a single probability distribution, a coexistence scaling theory on small world network is developed and numerically verified.

Self-Organization of Blood Pressure Regulation: Clinical Evidence

The pathogenesis of vasovagal syncope has remained elusive despite many efforts to identify an underlying dysfunction. Catastrophe theory explains the spontaneous occurrence of sudden events in some mathematically complex systems known as self-organized systems poised at criticality. These systems universally exhibit a power law initially described in earthquake occurrence: the Gutenberg Richter law. The magnitude plotted against the total number of earthquakes of at least this magnitude draw a straight line on log-log graph. We hypothesized that vasovagal syncope is a catastrophe occurring spontaneously in the cardiovascular system. We counted the number and magnitude (number of beats) of vasovagal reactions (simultaneous decreases in both blood pressure and heart rate on consecutive beats) in 24 patients with vasovagal symptoms during a head-up tilt test and 24 paired patients with no symptoms during the test. For each patient, we checked whether vasovagal reaction occurrence followed the Gutenberg Richter law. The occurrence followed the Gutenberg Richter law in 43 patients (correlation coefficient |r| = 0.986 ± 0.001, mean ± SEM) out of 48, with no difference between patients with and without symptoms. We demonstrated that vasovagal syncope matches a catastrophe model occurring in a self-organized cardiovascular complex system poised at criticality. This is a new vision of cardiovascular regulation and its related disorders.

Spontaneous instabilities and stick-slip motion in a generalized Hébraud-Lequeux model

We revisit the Hébraud-Lequeux (HL) model for the rheology of jammed materials and argue that a possibly important time scale is missing from HL's initial specification. We show that our generalization of the HL model undergoes interesting oscillating instabilities for a wide range of parameters, which lead to intermittent, stick-slip flows under constant shear rate. The instability we find is akin to the synchronization transition of coupled elements that arises in many different contexts (neurons, fireflies, financial bankruptcies, etc.). We hope that our scenario could shed light on the commonly observed intermittent, serrated flows of glassy materials under shear.

Universal and non-universal features in coarse-grained models of flow in disordered solids

We study the two-dimensional (2D) shear flow of amorphous solids within variants of an elastoplastic model, paying particular attention to spatial correlations and time fluctuations of, e.g., local stresses. The model is based on the local alternation between an elastic regime and plastic events during which the local stress is redistributed. The importance of a fully tensorial description of the stress and of the inclusion of (coarse-grained) convection in the model is investigated; scalar and tensorial models yield similar results, while convection enhances fluctuations and breaks the spurious symmetry between the flow and velocity gradient directions, for instance when shear localisation is observed. Besides, correlation lengths measured with diverse protocols are discussed. One class of such correlation lengths simply scale with the spacing between homogeneously distributed, simultaneous plastic events. This leads to a scaling of the correlation length with the shear rate as γ̇(-1/2) in 2D in the athermal regime, regardless of the details of the model. The radius of the cooperative disk, defined as the near-field region in which plastic events induce a stress redistribution that is not amenable to a mean-field treatment, notably follows this scaling. On the other hand, the cooperative volume measured from the four-point stress susceptibility and its dependence on the system size and the shear rate are model-dependent.

Critical properties of a dissipative sandpile model on small-world networks

A dissipative sandpile model is constructed and studied on small-world networks (SWNs). SWNs are generated by adding extra links between two arbitrary sites of a two-dimensional square lattice with different shortcut densities ϕ. Three regimes are identified: regular lattice (RL) for ϕ≲2(-12), SWN for 2(-12)<ϕ<0.1, and random network (RN) for ϕ≥0.1. In the RL regime, the sandpile dynamics is characterized by the usual Bak, Tang, and Weisenfeld (BTW)-type correlated scaling, whereas in the RN regime it is characterized by mean-field scaling. On SWNs, both scaling behaviors are found to coexist. Small compact avalanches below a certain characteristic size s(c) are found to belong to the BTW universality class, whereas large, sparse avalanches above s(c) are found to belong to the mean-field universality class. A scaling theory for the coexistence of two scaling forms on a SWN is developed and numerically verified. Though finite-size scaling is not valid for the dissipative sandpile model on RLs or on SWNs, it is found to be valid on RNs for the same model. Finite-size scaling on RNs appears to be an outcome of super diffusive sand transport and uncorrelated toppling waves.

Graph Mining Techniques

Graphs appear in several settings, like social networks, recommendation systems, computer communication networks, gene/protein biological networks, among others. A large amount of graph patterns, as well as graph generator models that mimic such patterns have been proposed over the last years. However, a deep and recurring question still remains: “What is a good pattern?” The answer is related to finding a pattern or a tool able to help distinguishing between actual real-world and fake graphs. Here we explore the ability of ShatterPlots, a simple and powerful algorithm to tease out patterns of real graphs, helping us to spot fake/masked graphs. The idea is to force a graph to reach a critical (“Shattering”) point, randomly deleting edges, and study its properties at that point.

A mesoscopic model for the rheology of soft amorphous solids, with application to microchannel flows

We have studied a mesoscopic model for the flow of amorphous solids. The model is based on key features identified at the microscopic level, namely periods of elastic deformation interspersed with localised rearrangements of particles that induce long-range elastic deformations. These long-range deformations are derived following a continuum mechanics approach, in the presence of solid boundaries, and are included in full in the model. Indeed, they mediate spatial cooperativity in the flow, whereby a localised rearrangement may lead a distant region to yield. In particular, we have simulated a channel flow and found manifestations of spatial cooperativity that are consistent with published experimental observations for concentrated emulsions in microchannels. Two categories of effects are distinguished. On the one hand, the coupling of regions subject to different shear rates, for instance, leads to finite shear rate fluctuations in the seemingly unsheared "plug" in the centre of the channel. On the other hand, there is convincing experimental evidence of a specific rheology near rough walls. We discuss the diverse possible physical origins for this effect, and we suggest that it may be associated with the bumps of particles into surface asperities as they slide along the wall.

Minimal integer automaton behind crystal plasticity

Power law fluctuations and scale-free spatial patterns are known to characterize steady state plastic flow in crystalline materials. In this Letter we study the emergence of correlations in a simple Frenkel-Kontorova-type model of 2D plasticity which is largely free of arbitrariness, amenable to analytical study, and is capable of generating critical exponents matching experiments. Our main observation concerns the possibility to reduce continuum plasticity to an integer-valued automaton revealing inherent discreteness of the plastic flow.

Plasticity and dynamical heterogeneity in driven glassy materials

Many amorphous glassy materials exhibit complex spatio-temporal mechanical response and rheology, characterized by an intermittent stress strain response and a fluctuating velocity profile. Under quasistatic and athermal deformation protocols this heterogeneous plastic flow was shown to be composed of plastic events of various sizes, ranging from local quadrupolar plastic rearrangements to system spanning shear bands. In this paper, through numerical study of a 2D Lennard-Jones amorphous solid, we generalize the study of the heterogeneous dynamics of glassy materials to the finite shear rate (gamma not equal to 0) and temperature case (T not equal to 0). In practice, we choose an effectively athermal limit (T approximately 0) and focus on the influence of shear rate on the rheology of the glass. In line with previous works we find that the model Lennard-Jones glass follows the rheological behavior of a yield stress fluid with a Herschel-Bulkley response of the form, sigma = sigmaY + c1gamma(beta). The global mechanical response obtained through the use of Molecular Dynamics is shown to converge in the limit gamma --> 0 to the quasistatic limit obtained with an energy minimization protocol. The detailed analysis of the plastic deformation at different shear rates shows that the glass follows different flow regimes. At sufficiently low shear rates the mechanical response reaches a shear-rate-independent regime that exhibits all the characteristics of the quasistatic response (finite-size effects, cascades of plastic rearrangements, yield stress, ...). At intermediate shear rates the rheological properties are determined by the externally applied shear rate and the response deviates from the quasistatic limit. Finally at higher shear the system reaches a shear-rate-independent homogeneous regime. The existence of these three regimes is also confirmed by the detailed analysis of the atomic motion. The computation of the four-point correlation function shows that the transition from the shear-rate-dominated to the quasistatic regime is accompanied by the growth of a dynamical cooperativity length scale xi that is shown to diverge with shear rate as xi is proportional to gamma(-nu), with nu approximately 0.2 -0.3. This scaling is compared with the prediction of a simple model that assumes the diffusive propagation of plastic events.

Micromechanical model for deformation in solids with universal predictions for stress-strain curves and slip avalanches

A basic micromechanical model for deformation of solids with only one tuning parameter (weakening epsilon) is introduced. The model can reproduce observed stress-strain curves, acoustic emissions and related power spectra, event statistics, and geometrical properties of slip, with a continuous phase transition from brittle to ductile behavior. Exact universal predictions are extracted using mean field theory and renormalization group tools. The results agree with recent experimental observations and simulations of related models for dislocation dynamics, material damage, and earthquake statistics.

Earthquake size-frequency statistics in a forest-fire model of individual faults

The earthquake size-frequency distribution of individual seismic faults commonly differs from the Gutenberg-Richter law of regional seismicity by the presence of an excess of large earthquakes. Here we present a cellular automaton of the forest-fire model type that is able to reproduce several size-frequency distributions depending on the number and location of asperities on the fault plane. The model describes a fault plane as a two-dimensional array of cells where each cell can be either a normal site or a trigger site. Earthquakes start on trigger sites. Asperities appear as the dual entities of the trigger sites. We study the effect that the number and distribution of asperities (the dual of the set of trigger sites), the earthquake rate, and the aspect ratio of the fault have on the size-frequency distribution. Size-frequency distributions have been grouped into subcritical, critical, and supercritical, and the relationship between the model parameters and these three kinds of distributions is presented in the form of phase maps for each of the five asperity types tested. We also study the connection between the model parameters and the aperiodicity of the large earthquakes. For this purpose the concept of aperiodicity spectrum is introduced. The aperiodicity in the recurrence of the large earthquakes in a fault shows an interesting variability that can be potentially useful for prediction purposes.

Driving-induced crossover: from classical criticality to self-organized criticality

We propose a spin model with quenched disorder which exhibits in slow driving two drastically different types of critical nonequilibrium steady states. One of them corresponds to classical criticality requiring fine-tuning of the disorder. The other is a self-organized criticality which is insensitive to disorder. The crossover between the two types of criticality is determined by the mode of driving. As one moves from "soft" to "hard" driving the universality class of the critical point changes from a classical order-disorder to a quenched Edwards-Wilkinson universality class. The model is viewed as prototypical for a broad class of physical phenomena ranging from magnetism to earthquakes.

Asynchronous updating induces order in threshold coupled systems

We study a class of models incorporating threshold-activated coupling on a lattice of chaotic elements, evolving under updating rules incorporating varying degrees of synchronicity. Interestingly, we observe that asynchronous updating, both random and sequential, yields more spatiotemporal order than parallel (synchronous) updating. Further, the order induced by random asynchronous updating is very robust and occurs even for small asynchronicities in the temporal evolution of the local dynamics. So this case study suggests a very different mechanism for inducing regularity in extended systems.

Training-induced criticality in martensites

We propose an explanation for the self-organization towards criticality observed in martensites during the cyclic process known as "training." The scale-free behavior originates from the interplay between the reversible phase transformation and the concurrent activity of lattice defects. The basis of the model is a continuous dynamical system on a rugged energy landscape, which in the quasistatic limit reduces to a sandpile automaton. We reproduce all the principal observations in thermally driven martensites, including power-law statistics, hysteresis shakedown, asymmetric signal shapes, and correlated disorder.

Characteristics of deterministic and stochastic sandpile models in a rotational sandpile model

Rotational constraint representing a local external bias generally has a nontrivial effect on the critical behavior of lattice statistical models in equilibrium critical phenomena. In order to study the effect of rotational bias in an out-of-equilibrium situation like self-organized criticality, a two state "quasideterministic" rotational sandpile model is developed here imposing rotational constraint on the flow of sand grains. An extended set of critical exponents are estimated to characterize the avalanche properties at the nonequilibrium steady state of the model. The probability distribution functions are found to obey usual finite size scaling supported by negative time autocorrelation between the toppling waves. The model exhibits characteristics of both deterministic and stochastic sandpile models.

Space-time clustering and correlations of major earthquakes

Earthquake occurrence in nature is thought to result from correlated elastic stresses, leading to clustering in space and time. We show that the occurrence of major earthquakes in California correlates with time intervals when fluctuations in small earthquakes are suppressed relative to the long term average. We estimate a probability of less than 1% that this coincidence is due to random clustering.

Basic properties of a three-dimensional spring-block model with long-range stress transfer

Spring-block models have been very useful for understanding slip complexity of earthquakes. So far, however, they have been restricted to one or two dimensions or simple nearest-neighbor stress transfer. Here, we set up a three-dimensional spring-block model with long-range stress transfer and, in a second step, implement several simplifications to realize a considerable gain in computational efficiency. Qualitatively, the two versions do not differ and we use the fast version to investigate basic properties of the model and to compare to the Olami Feder Christensen (OFC) model. The spatial distribution of hypocenters is found to be scale free. At the end of a simulation of events 10(7), it takes a value of D2 approximately = 1.8+/-0.1. At this point, however, the spatial slip organization is not stationary yet and the fractal dimension still grows slightly. This does not affect respective frequency size statistics, though, which exhibit a clear power law with a characteristic cutoff that depends on the grid size. The statistics appear smoother than in the OFC model and lack the kink between events of size 1 and 2. In addition, strong periodicities of large events as in the OFC model do not occur in our model. In stark contrast to the OFC model, results remain the same if periodic boundaries are used. Another significant difference is the stableness of results against imposed disorder. Contrary to the OFC model, results do not change if threshold values are randomly distributed in an interval of +/-10% around the mean value. Concluding, the model that we propose shares the main properties of the OFC model, but outreaches the latter in being stable in a larger set of configurations.

Nontrivial temporal scaling in a Galilean stick-slip dynamics

We examine the stick-slip fluctuating response of a rough massive nonrotating cylinder moving on a rough inclined groove which is submitted to weak external perturbations and which is maintained well below the angle of repose. The experiments presented here, which are reminiscent of Galileo's works with rolling objects on inclines, have brought in the last years important insights into the friction between surfaces in relative motion and are of relevance for earthquakes, differing from classical block-spring models by the mechanism of energy input in the system. Robust nontrivial temporal scaling laws appearing in the dynamics of this system are reported, and it is shown that the time-support where dissipation occurs approaches a statistical fractal set with a fixed value of dimension. The distribution of periods of inactivity in the intermittent motion of the cylinder is also studied and found to be closely related to the lacunarity of a random version of the classic triadic Cantor set on the line.

Slow flows of yield stress fluids: Complex spatiotemporal behavior within a simple elastoplastic model

A minimal athermal model for the flow of dense disordered materials is proposed, based on two generic ingredients: local plastic events occuring above a microscopic yield stress, and the nonlocal elastic release of the stress these events induce in the material. A complex spatiotemporal rheological behavior results, with features in line with recent experimental observations. At low shear rates, macroscopic flow actually originates from collective correlated bursts of plastic events, taking place in dynamically generated fragile zones. The related correlation length diverges algebraically at small shear rates. In confined geometries, bursts occur preferentially close to the walls, yielding an intermittent form of flow localization.

Elastic consequences of a single plastic event: a step towards the microscopic modeling of the flow of yield stress fluids

With the eventual aim of describing flowing elasto-plastic materials, we focus here on the elementary process of such a flow, a plastic event, and compute the long-range perturbation it elastically induces in a medium submitted to a global shear strain. We characterize the effect of a nearby wall on this perturbation, and quantify the importance of finite-size effects. Although most of our explicit formulae refer to 2D situations, our statements hold for 3D situations as well.

Local stress relaxation and shear banding in a dry foam under shear

We have developed a realistic simulation of 2D dry foams under quasistatic shear. After a short transient, a shear-banding instability is observed. These results are compared with measurements obtained on real 2D (confined) foams. The numerical model allows us to exhibit the mechanical response of the material to a single plastication event. From the analysis of this elastic propagator, we propose a scenario for the onset and stability of the flow localization process in foams, which should remain valid for most athermal amorphous systems under creep flow.

Extremal model for amorphous media plasticity

An extremal model for the plasticity of amorphous materials is studied in a simple two-dimensional antiplane geometry. The steady state is analyzed through numerical simulations. Long-range spatial and temporal correlations in local slip events are shown to develop, leading to nontrivial and highly anisotropic scaling laws. In particular, the plastic strain is shown to concentrate statistically over a region which tends to align perpendicular to the displacement gradient. By construction, the model can be seen as giving rise to a depinning transition, the threshold of which (i.e., the macroscopic yield stress) also reveals scaling properties reflecting the localization of the activity.

Renormalization-group approach to an Abelian sandpile model on planar lattices

One important step in the renormalization-group (RG) approach to a lattice sandpile model is the exact enumeration of all possible toppling processes of sandpile dynamics inside a cell for RG transformations. Here we propose a computer algorithm to carry out such exact enumeration for cells of planar lattices in the RG approach to the Bak-Tang-Wiesenfeld sandpile model [Phys. Rev. Lett. 59, 381 (1987)] and consider both the reduced-high RG equations proposed by Pietronero, Vespignani, and Zapperi (PVZ) [Phys. Rev. Lett. 72, 1690 (1994)], and the real-height RG equations proposed by Ivashkevich [Phys. Rev. Lett. 76, 3368 (1996)]. Using this algorithm, we are able to carry out RG transformations more quickly with large cell size, e.g., 3x3 cell for the square (SQ) lattice in PVZ RG equations, which is the largest cell size at the present, and find some mistakes in a previous paper [Phys. Rev. E 51, 1711 (1995)]. For SQ and plane triangular (PT) lattices, we obtain the only attractive fixed point for each lattice and calculate the avalanche exponent tau and the dynamical exponent z. Our results suggest that the increase of the cell size in the PVZ RG transformation does not lead to more accurate results. The implication of such result is discussed.

Transition regimes for growing crack populations

Numerous observational papers on crack populations in the material and geological sciences suggest that cracks evolve in such a way as to organize in specific patterns. However, very little is known about how and why the self-organization comes about. We use a model of tensile-like cracks with friction in order to study the time and space evolution of normal faults. The premise of this spring-block analog is that one could model crustal deformation for long time scales assuming a brittle layer coupled to a ductile substrate. The long time-scale physics incorporated into the model are slip-weakening friction, strain-hardening rheology for coupling the two layers, and randomly distributed yield strength of the brittle layer. We investigate how the evolution of populations of cracks depends on these three effects, using linear stability analysis to calculate the stable regimes for the friction as well as numerical simulations to model the nonlinear interactions of the cracks. We find that we can scale the problem to reduce the relevant parameters to a single one, the slip weakening. We show that the distribution of lengths of active cracks makes a transition from an exponential at very low strains, where crack nucleation prevails, to a power law at low to intermediate strains, where crack growth prevails, to an exponential distribution of the largest cracks at higher strains, where coalescence dominates. There is evidence of these different length distributions in continental and oceanic normal faults. For continental deformation the strain is low, and the faults have power-law frequency-size distributions. For mid-ocean ridge flanks the strain is greater, up to an order of magnitude higher than the continental strain, and faults have exponential-like frequency-size distributions. No theory has been offered to explain this difference in the distributions of continental and mid-ocean faults. In this paper we argue that they are indicative of different stages of evolution. The former faults are at an early stage of relatively small deformation, while the latter are at a later stage of the evolution. For high strain the faults reach a saturation regime with system size cracks evenly spaced in proportion to the brittle layer thickness. We asymptotically approximate the time space evolution of faults as a long time-scale phenomenon, thereby avoiding modeling the short time-scale earthquakes. We show that this assumption is valid, which implies that the faults that creep and faults with earthquakes display the same time and space evolutions.

Looking for self-organized critical behavior in avalanches of slightly cohesive powders

We report results from a statistical analysis of avalanches of cohesive powders in a slowly rotated drum. Interparticle adhesion, which diminishes the effect of inertia and whose magnitude strongly fluctuates in a local scale, makes avalanches in slightly cohesive powders eligible for displaying self-organized criticality. However, the results show that avalanche sizes, time interval between avalanches, and maximum stable angle do not follow a power-law distribution. Otherwise, these parameters scale with powder cohesiveness.

Criticality in a dynamic mixed system

We suggest a dynamic generalization of the simplest static hierarchical mixed model introduced by Shnirman and Blanter [Phys. Rev. Lett. 81, 5445 (1998); Phys. Rev. E. 60, 5111 (1998). We show that the stationary solution of the dynamic mixed model (DMM) demonstrates, in general, a linear form of the magnitude-frequency relation and may be considered a self-organized critical system. The dynamic mixed model demonstrates three principal kinds of system behavior: stability, catastrophe, and scale invariance. We show that the catastrophic area exists for all parameters of the mixture, and obtain three analytical expressions for boundary conditions of the stability and the scale invariance domains. As in the static model scale invariance appears as a result of a strong heterogeneity of the mixture. We describe how the magnitude-frequency relation reflects parameters of the heterogeneity and healing conditions for different domains of system behavior. Deviation of the DMM from the static mixed model and possible applications to earthquake prediction are discussed.

Self-organized criticality in a sheared granular stick-slip system

We present an analysis of results obtained from a mechanical apparatus consisting of an annular plate shearing over a granular bed. The size, energy dissipation, and duration of slips in the system exhibit power-law distributions and a 1/f(2) power spectrum, in accordance with self-organized criticality. We draw similarities with earthquakes.

Crackling noise

Crackling noise arises when a system responds to changing external conditions through discrete, impulsive events spanning a broad range of sizes. A wide variety of physical systems exhibiting crackling noise have been studied, from earthquakes on faults to paper crumpling. Because these systems exhibit regular behaviour over a huge range of sizes, their behaviour is likely to be independent of microscopic and macroscopic details, and progress can be made by the use of simple models. The fact that these models and real systems can share the same behaviour on many scales is called universality. We illustrate these ideas by using results for our model of crackling noise in magnets, explaining the use of the renormalization group and scaling collapses, and we highlight some continuing challenges in this still-evolving field.

Spatial-temporal correlations in the process to self-organized criticality

A different type of spatial-temporal correlation in the process approaching the self-organized criticality is investigated for the two simple models for biological evolution. The changed behaviors of the position with minimum barrier are shown to be quantitatively different in the two models. Different results of the correlation are given for the two models. We argue that the correlation can be used, together with the power-law distributions, as criteria for self-organized criticality.

Statistical Properties of SGR 1806-20 Bursts

We present statistics of SGR 1806-20 bursts, combining 290 events detected with the Rossi X-Ray Timing Explorer/Proportional Counter Array, 111 events detected with the Burst and Transient Source Experiment, and 134 events detected with the International Cometary Explorer. We find that the fluence distribution of bursts observed with each instrument are well described by power laws with indices 1.43, 1.76, and 1.67, respectively. The distribution of time intervals between successive bursts from SGR 1806-20 is described by a lognormal function with a peak at 103 s. There is no correlation between the burst intensity and either the waiting times until the next burst or the time elapsed since the previous burst. In all these statistical properties, SGR 1806-20 bursts resemble a self-organized critical system, similar to earthquakes and solar flares. Our results thus support the hypothesis that the energy source for soft gamma repeater bursts is crustquakes due to the evolving, strong magnetic field of the neutron star, rather than any accretion or nuclear power.