Avalanche size distribution in a random walk model

Physica A: Statistical Mechanics and its Applications, 2004

Avalanche dynamics is an indispensable feature of complex systems. Here we study the self-organized critical dynamics of avalanches on scale-free networks with degree exponent γ through the Bak-Tang-Wiesenfeld (BTW) sandpile model. The threshold height of a node i is set as k 1−η i with 0 ≤ η < 1, where k i is the degree of node i. Using the branching process approach, we obtain the avalanche size and the duration distribution of sand toppling, which follow power-laws with exponents τ and δ, respectively. They are given as τ = (γ−2η)/(γ− 1 − η) and δ = (γ − 1 − η)/(γ − 2) for γ < 3 − η, 3/2 and 2 for γ > 3 − η, respectively. The power-law distributions are modified by a logarithmic correction at γ = 3 − η.

Physical Review E, 1995

We study the Abelian sandpile model on decorated one-dimensional chains. We show that there are two types of avalanches, and determine the effects of finite, though large, system size I on the asymptotic form of distributions of avalanche sizes, and show that these differ qualitatively from the behavior on a simple linear chain. For large L, we find that the probability distribution of the total number of topplings 8 is not described by a simple finite-size scaling form, but by a linear combination of two simple scaling forms: ProbL, (s) = z fi(z) + b f2(~z), where fi and f2 are nonuniversal scaling functions of one argument.

EPL (Europhysics Letters), 2020

We base our study on the statistical analysis of the Rigan earthquake 2010 December 20, which consists of estimating the earthquake network by means of virtual seismometer technique, and also considering the avalanche-type dynamics on top of this complex network.The virtual seismometer complex network shows power-law degree distribution with the exponent γ = 2.3 ± 0.2. Our findings show that the seismic activity is strongly intermittent, and have a cyclic shape as is seen in the natural situations, which is main finding of this study. The branching ratio inside and between avalanches reveal that the system is at (or more precisely close to) the critical point with power-law behavior for the distribution function of the size and the mass and the duration of the avalanches, and with some scaling relations between these quantities. The critical exponent of the size of avalanches is τS = 1.45 ± 0.02. We find a considerable correlation between the dynamical Green function and the nodes centralities.

Physical Review Letters, 2004

The sizes of snow slab failure that trigger snow avalanches are power-law distributed. Such a power-law probability distribution function has also been proposed to characterize different landslide types. In order to understand this scaling for gravity driven systems, we introduce a two-threshold 2-d cellular automaton, in which failure occurs irreversibly. Taking snow slab avalanches as a model system, we find that the sizes of the largest avalanches just preceeding the lattice system breakdown are power law distributed. By tuning the maximum value of the ratio of the two failure thresholds our model reproduces the range of power law exponents observed for land-, rock-or snow avalanches. We suggest this control parameter represents the material cohesion anisotropy.

2015

Instead of a linear and smooth evolution, many physical system react to external stimuli in avalanche dynamics. When an out of equilibrium system governed by disorder is externally driven the evolution of internal variables is local and non-homogeneous. This process is a collective behaviour adiabatically quick known as avalanches. Avalanche dynamics are associated to the transformation of spatial domains in different scales: from microscopic, to large catastrophic events such as earthquakes or solar flares. Avalanche dynamics is also involved in interdisiplinar topics such as the return prices of stock markets, the signalling in neuron networks or the biological evolution. Many avalanche dynamics are characterised by scale invariance, trademark of criticality. The physics in a so-called critical point are the same in all observational scales. Some avalanche dynamics share empirical laws and can define Universality Classes, reducing the complexity of systems to simpler mathematical ...

Physical Review E, 1996

Scientific Reports, 2015

Both earthquake size-distributions and aftershock decay rates obey power laws. Recent studies have demonstrated the sensibility of their parameters to faulting properties such as focal mechanism, rupture speed or fault complexity. The faulting style dependence may be related to the magnitude of the differential stress, but no model so far has been able to reproduce this behaviour. Here we investigate the statistical properties of avalanches in a dissipative, bimodal particulate system under slow shear. We find that the event size-distribution obeys a power law only in the proximity of a critical volume fraction, whereas power-law aftershock decay rates are observed at all volume fractions accessible in the model. Then, we show that both the exponent of the event size-distribution and the time delay before the onset of the power-law aftershock decay rate are decreasing functions of the shear stress. These results are consistent with recent seismological observations of earthquake size-distribution and aftershock statistics.

Physical Review Letters, 2000

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

2007

We discuss recent results on a new analysis regarding models showing Self-Organized Criticality (SOC), and in particular on the OFC one. We show that Probability Density Functions (PDFs) for the avalanche size differences at different times have fat tails with a q-Gaussian shape. This behavior does not depend on the time interval adopted and it is also found when considering energy differences between real earthquakes.

Physical Review E, 2007

2002

In the last decade, a considerable number of publications have been dedicated to the occurrence of power-law behavior in systems involving interacting threshold elements driven by slow external input. The dynamics accounts for phenomena occurring in such diverse systems as piles of granular matter 1, earthquakes 2, the game of life 3, friction 4, and sound generated in the lung during breathing 5.

Physical Review Letters, 2002

We numerically investigate the Olami-Feder-Christensen model on a quenched random graph. Contrary to the case of annealed random neighbors, we find that the quenched model exhibits self-organized criticality deep within the nonconservative regime. The probability distribution for avalanche size obeys finite size scaling, with universal critical exponents. In addition, a power law relation between the size and the duration of an avalanche exists. We propose that this may represent the correct mean-field limit of the model rather than the annealed random neighbor version.

Geophysical Research Letters, 2008

1] We calculate the spreading exponents and some geometrical properties of avalanches in a novel avalanche model of solar flares, closely built on Parker's physical picture of coronal heating by nanoflares. The model is based on an idealized representation of a coronal loop as a bundle of magnetic flux strands wrapping around one another, numerically implemented as an anisotropic cellular automaton. We demonstrate that the growth of avalanches in this model exhibits power-laws correlations that are numerically consistent with the behavior of a general class of statistical physical systems in the vicinity of a stationary critical point. This demonstrates that the model indeed operates in a self-organized critical regime. Moreover, we find that the frequency distribution of avalanche peak areas A assumes a power-law form f(A) / A Àa A with an index a A ' 2.45, in excellent agreement with observationallyinferred values. Citation: Morales, L. F., and P. Charbonneau (2008), Scaling laws and frequency distributions of avalanche areas in a self-organized criticality model of solar flares, Geophys.

Europhysics Letters (EPL), 1995

The general conditions for a sandpile system to evolve spontaneously into a critical

Physical Review Letters, 2009

It is a common belief that power-law distributed avalanches are inherently unpredictable. This idea affects phenomena as diverse as evolution, earthquakes, superconducting vortices, stock markets, etc; from atomic to social scales. It mainly comes from the concept of "Self-organized criticality" (SOC), where criticality is interpreted in the way that at any moment, any small avalanche can eventually cascade into a large event. Nevertheless, this work demonstrates experimentally the possibility of avalanche prediction in the classical paradigm of SOC: a sandpile. By knowing the position of every grain in a two-dimensional pile, avalanches of moving grains follow a distinct powerlaw distribution. Large avalanches, although uncorrelated, are preceded by continuous, detectable variations in the internal structure of the pile that are monitored in order to achieve prediction.