Are there “dragon-kings” events (i.e. genuine outliers) among extreme avalanches?

Predicting the occurrence and spatial extent of extreme avalanches is a longstanding issue. Using field data pooled from various sites within the same mountain range, authors showed that the avalanche size distribution can be described using either an extreme value distribution or a thick-tailed distribution, which implies that although they are much larger than common avalanches, extreme avalanches belong to the same population of events as “small” avalanches. Yet, when looking at historical records of catastrophic avalanches, archives reveal that a few avalanches had features that made them “extra-ordinary.” Applying avalanche-dynamics or statistical models to simulate these past events runs into considerable difficulty since the model parameters or the statical properties are very different from the values usually set to model extreme avalanches. Were these events genuine outliers (also called “dragon-kings”)? What were their distinctive features? This paper reviews some of the concepts in use to model extreme events, gives examples of processes that were at play in extreme avalanches, and shows that the concept of dragon-king avalanches is of particular relevance to describing some extreme avalanches.     

Finite-size effects on the statistics of extreme events in the BTW model

The BTW Abelian sandpile model is a prominent example of systems showing self-organised criticality (SOC) in the infinite size limit. We study finite-size effects with special focus on the statistics of extreme events, i.e., of particularly large avalanches. Not only the avalanche size probability distribution, but also the mutual independence of large avalanches in the critical state is affected by finite-size effects. Instead of a Poissonian recurrencetime distribution, in the finite system we find a repulsion of extreme events that depends on the avalanche size and not on the respective probability. The dependence of these effects on the system size is investigated and some data collapse is found. Our results imply that SOC is an unsuitable mechanism for the explanation of extreme events which occur in clusters.     

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.     

Large-scale simulation of avalanche cluster distribution in sand pile model

The avalanche cluster distribution of the sand pile model of self-organized criticality is studied on the square lattice. A vectorized multispin coding algorithm is developed for this study with three bits per site. The exponents characterizing the size and the lifetime of the avalanches are slightly different from the previous estimates.     

Extremal dynamics and the approach to the critical state: experiments on a three dimensional pile of rice

The evolution of the growth of a ricepile is studied in three dimensions. With time, the pile approaches a critical state with a certain slope. Assuming extremal dynamics in the evolution of the pile, the way the critical state is approached is dictated by the scaling properties of the critical state itself. Experimentally, we determine the envelope of the maximal slope, which is a measure for the distance from the critical state, as well as the growth of the average avalanche size with time. These quantities obey power-law scaling, where the experimental exponents are in good agreement with those obtained from an earlier determination of the critical state properties and extremal dynamics. Furthermore, we discuss the influence of the transient state on the avalanche size distribution, which may have applications in the prevention of large avalanches in natural systems.     

Stabilization of avalanche processes on dynamical networks

The stabilization of avalanches on dynamical networks has been studied. Dynamical networks are networks where the structure of links varies in time owing to the presence of the individual “activity” of each site, which determines the probability of establishing links with other sites per unit time. An interesting case where the times of existence of links in a network are equal to the avalanche development times has been examined. A new mathematical model of a system with the avalanche dynamics has been constructed including changes in the network on which avalanches are developed. A square lattice with a variable structure of links has been considered as a dynamical network within this model. Avalanche processes on it have been simulated using the modified Abelian sandpile model and fixed-energy sandpile model. It has been shown that avalanche processes on the dynamical lattice under study are more stable than a static lattice with respect to the appearance of catastrophic events. In particular, this is manifested in a decrease in the maximum size of an avalanche in the Abelian sandpile model on the dynamical lattice as compared to that on the static lattice. For the fixed-energy sandpile model, it has been shown that, in contrast to the static lattice, where an avalanche process becomes infinite in time, the existence of avalanches finite in time is always possible.     

Scaling of a Slope: The Erosion of Tilted Landscapes

We formulate a stochastic equation to model the erosion of a surface with fixed inclination. Because the inclination imposes a preferred direction for material transport, the problem is intrinsically anisotropic. At zeroth order, the anisotropy manifests itself in a linear equation that predicts that the prefactor of the surface height–height correlations depends on direction. The first higher order nonlinear contribution from the anisotropy is studied by applying the dynamic renormalization group. Assuming an inhomogeneous distribution of soil substrate that is modeled by a source of static noise, we estimate the scaling exponents at first order in an ε-expansion. These exponents also depend on direction. We compare these predictions with empirical measurements made from real landscapes and find good agreement. We propose that our anisotropic theory applies principally to small scales and that a previously proposed isotropic theory applies principally to larger scales. Lastly, by considering our model as a transport equation for a driven diffusive system, we construct scaling arguments for the size distribution of erosion “events” or “avalanches.” We derive a relationship between the exponents characterizing the surface anisotropy and the avalanche size distribution, and indicate how this result may be used to interpret previous findings of power-law size distributions in real submarine avalanches.     

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.     

Universal phenomenon of self-organized criticality in magnetic flux dynamics

Magnetization curves of square arrays of Josephson junctions of two basic types were investigated: superconductor–insulator–superconductor (SIS) and superconductor–normal metal–superconductor (SNS).

Magnetic flux avalanches were observed in SIS arrays. A statistical analysis of flux avalanches showed that their size distribution can be described by a power law with a crossover where the exponent n varies from −1.2 for small avalanches to −3.5 for the large ones. Such a behavior of avalanches is interpreted as the self-organized criticality (SOC) manifestation. In SNS arrays, the flux avalanches were not observed, but a considerable asymmetry of a hysteresis curve was revealed.     

Crossover behavior in burst avalanches: signature of imminent failure

The statistics of damage avalanches during a failure process typically follows a power law. When these avalanches are recorded only near the point at which the system fails catastrophically, one finds that the power law has an exponent which is different from that one finds if the recording of events starts away from the vicinity of catastrophic failure. We demonstrate this analytically for bundles of many fibers, with statistically distributed breakdown thresholds for the individual fibers and where the load is uniformly distributed among the surviving fibers. In this case the distribution D(Delta) of the avalanches (Delta) follows the power law Delta-xi with xi=3/2 near catastrophic failure and xi=5/2 away from it. We also study numerically square networks of electrical fuses and find xi=2.0 near catastrophic failure and xi=3.0 away from it. We propose that this crossover in xi may be used as a signal of imminent failure.     

Effect of the link lifetime in a dynamical lattice on the properties of the avalanche processes on it

The properties of the avalanche processes that develop on a dynamical lattice, the structure of links in which changes due to a specific characteristic of each lattice node, namely, its “activity,” which determines the probability of connection of a certain node with neighboring nodes in one step of lattice evolution. The statistics of the sizes of the avalanches appearing in the lattice system is studied as a function of the node activity and the link lifetime (the lifetime of the links formed in the system). It is analytically and numerically shows that the type of avalanche dynamics in the system changes as a function of these parameters. The following three regimes can take place in the system: (1) avalanches of any sizes, from small to catastrophic, can appear, which is reflected in the power-law behavior of the probability density function of the appearance of avalanches of certain sizes; (2) avalanches of a certain average size mainly appear in the system, and the probability density is close to that of a normal distribution; and (3) transient regime, where the probability density function of the appearance of avalanches of certain sizes is close to an exponential function. These results open up the possibilities of controlling the behavior of a complex system; in particular, they can be used to prevent catastrophic avalanches by changing the link lifetime and the average node activity.     

Subsampling effects in neuronal avalanche distributions recorded in vivo

Background  

Many systems in nature are characterized by complex behaviour where large cascades of events, or avalanches, unpredictably alternate with periods of little activity. Snow avalanches are an example. Often the size distribution f(s) of a system's avalanches follows a power law, and the branching parameter sigma, the average number of events triggered by a single preceding event, is unity. A power law for f(s), and sigma = 1, are hallmark features of self-organized critical (SOC) systems, and both have been found for neuronal activity in vitro. Therefore, and since SOC systems and neuronal activity both show large variability, long-term stability and memory capabilities, SOC has been proposed to govern neuronal dynamics in vivo. Testing this hypothesis is difficult because neuronal activity is spatially or temporally subsampled, while theories of SOC systems assume full sampling. To close this gap, we investigated how subsampling affects f(s) and sigma by imposing subsampling on three different SOC models. We then compared f(s) and sigma of the subsampled models with those of multielectrode local field potential (LFP) activity recorded in three macaque monkeys performing a short term memory task.     

Monte Carlo simulation of electron avalanches and avalanche size distributions in methane

The simulation of electron avalanches and avalanche size distributions in methane is presented in this paper. A model for electron transport under the influence of a constant electric field based on the Monte Carlo method is described in detail. The model is verified and then used to simulate the avalanche development, to calculate the number of electrons in the avalanche (avalanche size), and to determine the avalanche size distribution. The simulated avalanche size distributions in methane are compared with the experimental results, and a good agreement is observed. The influence of inter‐electrode distance, pressure, and reduced electric field on the shape of the avalanche size distribution is discussed. The assumption from the literature that for a constant reduced electric field the shape of the reduced avalanche size distribution is independent of the mean size of the avalanche is confirmed for a wide range of experimental conditions. The simulations have shown that avalanche size distributions depend only on the reduced electric field, confirming the similarity principle.     

Area Distribution for Directed Random Walks

We study the probability distribution for the area under a directed random walk in the plane. The walk can serve as a simple model for avalanches based on the idea that the front of an avalanche can be described by a random walk and the size is given by the area enclosed. This model captures some of the qualitative features of earthquakes, avalanches, and other self-organized critical phenomena in one dimension. By finding nonlinear functional relations for the generating functions we calculate directly the exponent in the size distribution law and find it to be 4/3.     

Statistics of avalanches in the self-organized criticality state of a Josephson junction

Magnetic flux avalanches in Josephson junctions that include superconductor-insulator-superconductor (SIS) tunnel junctions and are magnetized at temperatures lower than approximately 5 K have been studied in detail. Avalanches are of stochastic character and appear when the magnetic field penetration depth λ into a junction becomes equal to the length a of the Josephson junction with a decrease in the temperature. The statistical properties of such avalanches are presented. The size distribution of the avalanches is a power law with a negative noninteger exponent about unity, indicating the self-organized criticality state. The self-organized criticality state is not observed in Josephson junctions with a superconductor-normal metal-superconductor (SNS) junction.     

Magnetic properties of two-dimensional Josephson networks: Self-organized criticality in magnetic flux dynamics

The field dependence of the magnetic moment of square (100×100) Josephson networks was examined with the use of a SQUID magnetometer. The field dependence of the magnetic moment was found to be regular with features corresponding to integer and half-integer numbers of flux quanta per cell. At temperatures below 5.8 K, jumps in the magnetization curves associated with the entry and exit of avalanches of tens and hundreds of fluxons were observed. It was shown that the probability distribution of these processes corresponded to the theory of self-organized criticality. An avalanche character of flux motion was observed at temperatures at which the size of the fluxons did not exceed the size of the cell, that is, when a discrete vortex structure occurred.     

Scaling and complex avalanche dynamics in the Abelian sandpile model

We study the two-dimensional Abelian Sandpile Model on a squarelattice of linear size L. We introduce the notion of avalanche’sfine structure and compare the behavior of avalanches and waves oftoppling. We show that according to the degree of complexity inthe fine structure of avalanches, which is a direct consequence ofthe intricate superposition of the boundaries of successive waves,avalanches fall into two different categories. We propose scalingansätz for these avalanche types and verify them numerically.We find that while the first type of avalanches (α) has a simplescaling behavior, the second complex type (β) is characterized by anavalanche-size dependent scaling exponent. In particular, we define an exponent γto characterize the conditional probability distribution functions for these typesof avalanches and show that γ _α = 0.42, while 0.7 ≤ γ _β ≤ 1.0depending on the avalanche size. This distinction provides aframework within which one can understand the lack of aconsistent scaling behavior in this model, and directly addresses thelong-standing puzzle of finite-size scaling in the Abelian sandpile model.     

Size distribution of different types of sites in Abelian sandpile avalanches

The avalanches in the Abelian sandpile model have a compact structure and consist of shells. The different shells consist solely of sites that have toppled a fixed number of times. For the outermost shell the number is one and increases successively by one from an outer shell to the next inner shell. In this paper, we determine the size-distribution exponents of various categories of sites: sites that have toppled at least once, at least twice, at least thrice, only once and only twice. A particular subclass of avalanches is identified which have no shell structure and consist solely of sites which have toppled only once. The size distribution exponent of this type of clusters is determined.     

From the stress response function (back) to the sand pile “dip”

We relate the pressure dip observed at the bottom of a sand pile prepared by successive avalanches to the stress profile obtained on sheared granular layers in response to a localized vertical overload. We show that, within a simple anisotropic elastic analysis, the skewness and the tilt of the response profile caused by shearing provide a qualitative agreement with the sand pile dip effect. We conclude that the texture anisotropy produced by the avalanches is in essence similar to that induced by a simple shearing --albeit tilted by the angle of repose of the pile. This work also shows that this response function technique could be very well adapted to probe the texture of static granular packing.     

Diffusion dominated dynamics of avalanching systems

A surprisingly large number of systems in nature are thought to be governed by internal dynamics which causes avalanches of various sizes. In such systems energy, which is delivered from outside, is redistributed as a result of the occurrence of localized avalanches. Starting an avalanche requires that some threshold condition be satisfied. Random driving (energy input) brings the system into a strongly inhomogeneous state, so that the probability of triggering an avalanche in a large part of the system is small. In most physical systems energy redistribution may occur due to diffusive processes without avalanches. Diffusion also makes the system more uniform, making large avalanche triggering more probable. The observed behavior of a such system may crucially depend on the competition between diffusion and driving. In this paper, the effects of diffusive processes are investigated using a dissipative, isotropic one-dimensional model, in which avalanches can propagate in both directions. It is shown that the system behavior changes progressively as the diffusion rate increases. In the absence of diffusion, many small avalanches are triggered. Increasing the diffusion rate gradually suppresses these small avalanches and leads to the development of large, quasi-periodic bursts.