I have proposed a\textit{non-Abelian}and stochastic self-organized criticality model in which each avalanche contains one stochastic site and all remaining sites in the avalanche are deterministic with a constant threshold $E_{c}^{I}$. Studies of avalanche structures, waves and autocorrelations, size moments and probability distribution functions of avalanche size, for the thresholds $4\leq E_{c}^{I}\leq256$, were performed. The shell-like avalanche structures, correlated waves within avalanches, complex size moments and probability distribution functions show multifractal scaling like the \textit{Abelian} and deterministic BTW model despite the fact that the model is \textit{non-Abelian}and stochastic with unbalanced relaxation rules at each stochastic site.
Barkhausen effect in ferromagnetic materials provides an excellent area for investigating scaling phenomena found in disordered systems exhibiting crackling noise. The critical dynamics is characterized by random pulses or avalanches with scale-invariant properties, power-law distributions, and universal features. However, the traditional Barkhausen avalanches statistics may not be sufficient to fully characterize the complex temporal correlation of the magnetic domain walls dynamics. Here we go beyond power laws and focus on the multifractal scenario to quantify the temporal scaling character…
An analysis of moments and spectra shows that, while the distribution of avalanche areas obeys finite size scaling, that of toppling numbers is universally characterized by a full, nonlinear multifractal spectrum. Rare, large avalanches dissipating at the border influence the statistics very sensibly. Only once they are excluded from the sample, the conditional toppling distribution for given area simplifies enough to show also a well defined, multifractal scaling. The resulting picture brings to light unsuspected, novel physics in the model.
The average avalanche size can be calculated exactly in a number of models of self-organised criticality (SOC). While the calculation is straight-forward in one dimension, it is more involved in higher dimensions and further complicated by the presence of different boundary conditions and different forms of external driving. Amplitudes of the leading order are determined analytically and evaluated to obtain analytical references for numerical work. A subtle link exists between the procedure to calculate the average avalanche size and the field theory of SOC.
Multifractal scaling of critical wave functions at a disorder-driven (Anderson) localization transition is modified near boundaries of a sample. Here this effect is studied for the example of the spin quantum Hall plateau transition using the supersymmetry technique for disorder averaging. Upon mapping of the spin quantum Hall transition to the classical percolation problem with reflecting boundaries, a number of multifractal exponents governing wave function scaling near a boundary are obtained exactly. Moreover, additional exact boundary scaling exponents of the localization problem are extr…
We present an experimental study of the influence of quenched disorder on the distribution of flux avalanches in type-II superconductors. In the presence of much quenched disorder, the avalanche sizes are power-law distributed and show finite size scaling, as expected from self-organized criticality (SOC). Furthermore, the shape of the avalanches is observed to be fractal. In the absence of quenched disorder, a preferred size of avalanches is observed and avalanches are smooth. These observations indicate that a certain minimum amount of disorder is necessary for SOC behavior. We relate these …
We review the use of superconductors as a playground for the experimental study of front roughening and avalanches. Using the magneto-optical technique, the spatial distribution of the vortex density in the sample is monitored as a function of time. The roughness and growth exponents corresponding to the vortex landscape are determined and compared to the exponents that characterize the avalanches in the framework of Self-Organized Criticality. For those situations where a thermo-magnetic instability arises, an analytical non-linear and non-local model is discussed, which is found to be consis…
The aim here is to study the concept of pairing multifractality between time series possessing non-Gaussian distributions. The increasing number of rare events creates "criticality". We show how the pairing between two series is affected by rare events, which we call "coupled criticality". A method is proposed for studying the coupled criticality born out of the interaction between two series, using the bivariate multifractal random walk (BiMRW). This method allows studying dependence of the coupled criticality on the criticality of each individual system. This approach is applied to data sets…
We show that large, slowly driven systems can evolve to a self-organized critical state where long range temporal correlations between bursts or avalanches produce low frequency $1/f^α$ noise. The avalanches can occur instantaneously in the external time scale of the slow drive, and their event statistics are described by power law distributions. A specific example of this behavior is provided by numerical simulations of a deterministic ``sandpile'' model.
We show using numerical simulations that slowly driven skyrmions interacting with random pinning move via correlated jumps or avalanches. The avalanches exhibit power law distributions in their duration and size, and the average avalanche shape for different avalanche durations can be scaled to a universal function, in agreement with theoretical predictions for systems in a nonequilibrium critical state. A distinctive feature of skyrmions is the influence of the non-dissipative Magnus term. When we increase the ratio of the Magnus term to the damping term, a change in the universality class of…
The aim of this study is to investigate a wave dynamics and size scaling of avalanches which were created by the mathematical model {[}J. Černák Phys. Rev. E \textbf{65}, 046141 (2002)]. Numerical simulations were carried out on a two dimensional lattice $L\times L$ in which two constant thresholds $E_{c}^{I}=4$ and $E_{c}^{II}>E_{c}^{I}$ were randomly distributed. A density of sites $c$ with the threshold $E_{c}^{II}$ and threshold $E_{c}^{II}$ are parameters of the model. I have determined autocorrelations of avalanche size waves, Hurst exponents, avalanche structures and avalanche size mome…
Plastic events in sheared glasses are considered an example of so-called avalanches, whose sizes obey a power-law probability distribution with the avalanche critical exponent $τ$. Although mean-field theory predicts a universal value of this exponent, $τ_{\rm MF}=1.5$, numerical simulations have reported different values depending on the literature. Moreover, in the elastic regime, it has been noted that the critical exponent can be different from that in the steady state, and even criticality itself is a matter of debate. Because these confusingly varying results were reported under differen…
Zero-temperature Avalanche Criticality Governing Dynamical Heterogeneity in Supercooled Liquids
Toyota Central R&D Labs., Inc.（丰田中央研究所）；D3 Center, The University of Osaka（大阪大学D3中心）；Department of Physics, The University of Osaka（大阪大学物理系）；Department of Physics, Nagoya University（名古屋大学物理系）；Division of Chemical Engineering, Graduate School of Engineering Science, The University of Osaka（大阪大学工学研究科化学工程专修）
We present a two-dimensional system which exhibits features of self-organized criticality. The avalanches which occur on the surface of a pile of rice are found to exhibit finite size scaling in their probability distribution. The critical exponents are $τ$ = 1.21(2) for the avalanche size distribution and $D$ = 1.99(2) for the cut-off size. Furthermore the geometry of the avalanches is studied leading to a fractal dimension of the active sites of $d_B$ = 1.58(2). Using a set of scaling relations, we can calculate the roughness exponent $α= D - d_B$ = 0.41(3) and the dynamic exponent $z = D(2 …
In 1993 Tang proposed [1] that vortex avalanches should produce a self organized critical state in superconductors, but conclusive evidence for this has heretofore been lacking. In the present paper, we report extensive micro-Hall probe data from the vortex dynamics in superconducting niobium, where a broad distribution of avalanche sizes scaling as a power-law for more than two decades is found. The measurements are combined with magneto-optical imaging, and show that over a widely varying magnetic landscape the scaling behaviour does not change, hence establishing that the dynamics of superc…
We show that both square and kagome artificial spin ice systems exhibit disorder-induced nonequilibrium phase transitions, with power law avalanche distributions at the critical disorder level. The different nature of geometrical frustration in the two lattices produces distinct types of critical avalanche behavior. For the square ice, the avalanches involve the propagation of locally stable domain walls separating the two polarized ground states, and the scaling collapse agrees with an interface depinning mechanism. In contrast, avalanches in the fully frustrated kagome ice exhibit pronounced…
Certain systems with slow driving and avalanche-like dissipation events are naturally close to a critical point when the ratio of two energy scales is large. The first energy scale is the threshold above which an avalanche is triggered, the second scale is the threshold above which a site is affected by an avalanche. I present results of computer simulations, and a mean-field theory.
Time series resulting from wave decomposition show the existence of different correlation patterns for avalanche dynamics. For the d=2 Bak-Tang-Wiesenfeld model, long range correlations determine a modification of the wave size distribution under coarse graining in time, and multifractal scaling for avalanches. In the Manna model, the distribution of avalanches coincides with that of waves, which are uncorrelated and obey finite size scaling, a result expected also for the d=3 Bak et al. model.
We present generic scaling laws relating spreading critical exponents and avalanche exponents (in the sense of self-organized criticality) in general systems with absorbing states. Using these scaling laws we present a collection of the state-of-the-art exponents for directed percolation, dynamical percolation and other universality classes. This collection of results should help to elucidate the connections of self-organized criticality and systems with absorbing states. In particular, some non-universality in avalanche exponents is predicted for systems with many absorbing states.
We study an inhomogeneous sandpile model in which two different toppling rules are defined. For any site only one rule is applied corresponding to either the Bak, Tang and Wiesenfeld model {[}P.Bak, C. Tang, and K. Wiesenfeld, Phys. Rev. Lett. \textbf{59}, 381 (1987){]} or the Manna two-state sandpile model {[}S. S. Manna, J. Phys. A \textbf{24}, L363 (1991){]}. A parameter $c$ is introduced which describes a density of sites which are randomly deployed and where the stochastic Manna rules are applied. The results show that the avalanche area exponent $τ_{a}$, avalanche size exponent $τ_{s}$, …
The penetration of magnetic flux into a thin film of YBa$_2$Cu$_3$O$_{7-x}$ is studied when the external field is ramped slowly. In this case the flux penetrates in bursts or avalanches. The size of these avalanches is distributed according to a power law with an exponent of $τ$ = 1.29(2). The additional observation of finite-size scaling of the avalanche distributions, with an avalanche dimension D = 1.89(3), gives strong indications towards self-organized criticality in this system. Furthermore we determine exponents governing the roughening dynamics of the flux surface using some universal …
Potential energy landscape picture of zero-temperature avalanche criticality governing dynamics in supercooled liquids
Toyota Central R&D Labs., Inc.（丰田中央研究所）；D3 Center, The University of Osaka（大阪大学D3中心）；Department of Physics, The University of Osaka（大阪大学物理系）；Department of Physics, Nagoya University（名古屋大学物理系）；Division of Chemical Engineering, Graduate School of Engineering Science, The University of Osaka（大阪大学工学研究科化学工程专攻）
We investigate, by numerical simulations, how the avalanche dynamics of the Bak-Tang-Wiesenfeld (BTW) sandpile model can induce emergence of scale-free (SF) networks and how this emerging structure affects dynamics of the system. We also discuss how the observed phenomenon can be used to explain evolution of scientific collaboration.
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 $γ=2.3\pm 0.2$. Our findings show that the seismic activity is strongly intermittent, and have a \textit{cyclic shape} as is seen in the natural situations, which is main finding of this study. The branching ratio inside and between avalanches…
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.
An interbank market lets participants pool the risk arising from the combination of illiquid investments and random withdrawals by depositors. But it also creates the potential for one bank's failure to trigger off avalanches of further failures. We simulate a model of interbank lending to study the interplay of these two effects. We show that when banks are similar in size and exposure to risk, avalanche effects are small so that widening the interbank market leads to more stability. But as heterogeneity increases, avalanche effects become more important. By varying the heterogeneity and conn…
The sun provides an explosive, heavenly example of self-organized criticality. Sudden bursts of intense radiation emanate from rapid rearrangements of the magnetic field network in the corona. Avalanches are triggered by loops of flux that reconnect or snap into lower energy configurations when they are overly stressed. Our recent analysis of observational data reveals that the loops (links) and footpoints (nodes), where they attach on the photosphere, embody a scale free network. The statistics of the avalanches and of the network structure are unified through a simple dynamical model where t…
An improved version of the Olami-Feder-Christensen model has been introduced to consider avalanche size differences. Our model well demonstrates the power-law behavior and finite size scaling of avalanche size distribution in any range of the adding parameter $p_{add}$ of the model. The probability density functions (PDFs) for the avalanche size differences at consecutive time steps (defined as returns) appear to be well approached, in the thermodynamic limit, by q-Gaussian shape with appropriate q values which can be obtained a priori from the avalanche size exponent $τ$. For the small system…
The dynamics of a fibre-bundle type model with equal load sharing rule is numerically studied. The system, formed by N elements, is driven by a slow increase of the load upon it which is removed in a novel way through internal transfers to the elements broken during avalanches. When an avalanche ends, failed elements are regenerated with strengths taken from a probability distribution. For a large enough N and certain restrictions on the distribution of individual strengths, the system reaches a self-organized critical state where the spectrum of avalanche sizes is a power law with an exponent…
A simple random-neighbor SOC model that combines properties of the Bak-Sneppen and the relaxation oscillators (slip-stick) models is introduced. The analysis in terms of branching processes is transparent and gives insight about the development of large but finite mean avalanche sizes in dissipative models. In the thermodynamic limit, the distribution of states has a simple analytical form and the mean avalanche size, as a function of the coupling parameter strength, is exactly calculable.
We perform a new analysis on the dissipative Olami-Feder-Christensen model on a small world topology considering avalanche size differences. We show that when criticality appears the Probability Density Functions (PDFs) for the avalanche size differences at different times have fat tails with a q-Gaussian shape. This behaviour does not depend on the time interval adopted and is found also when considering energy differences between real earthquakes. Such a result can be analytically understood if the sizes (released energies) of the avalanches (earthquakes) have no correlations. Our findings s…
We show that deterministic systems with strong nonlinearities seem to be more appropriate to model sandpiles than stochastic systems or deterministic systems in which discontinuities are the only nonlinearity. In particular, we are able to reproduce the breakdown of Self-Organized Criticality found in two well known experiments, that is, a centrally fueled pile [Held et al. Phys. Rev. Lett. 65 (1990) 1120] and sand in a rotating tray [Bretz et al. Phys. Rev. Lett. 69 (1992) 2431]. By varying the parameters of the model we recover Self-Organized Criticality, in agreement with other experiments …
We introduce an external control to reduce the size of avalanches in some sandpile models exhibiting self organized criticality. This rather intuitive approach seems to be missing in the vast literature on such systems. The control action, which amounts to triggering avalanches in sites that are near to be come critical, reduces the probability of very large events, so that energy dissipation occurs most locally. The control is applied to a directed Abelian sandpile model driven by both uncorrelated and correlated deposition. The latter is essential to design an efficient and simple control he…
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 eve…
The origin of power-law distributions in self-organized criticality is investigated by treating the variation of the number of active sites in the system as a stochastic process. An avalanche is then regarded as a first-return random walk process in a one-dimensional lattice. Power law distributions of the lifetime and spatial size are found when the random walk is unbiased with equal probability to move in opposite directions. This shows that power-law distributions in self-organized criticality may be caused by the balance of competitive interactions. At the mean time, the mean spatial size …
The theory of a flux steady-state related to avalanche formation is presented for the simplest model of a sand pile within framework of the Lorenz approach. The stationary values of sand velocity and sand pile slope are derived as functions of control parameter (driven sand pile slope). The additive noises of above values are introduced to build the phase diagram, where the noise intensities determine both avalanche and non-avalanche domains, as well as mixed one. Being corresponded to the SOC regime, the last domain is crucial to affect of the noise intensities of vertical component of sand v…
The concept of Self-Organized Criticality (SOC) was proposed in an attempt to explain the widespread appearance of power-law in nature. It describes a mechanism in which a system reaches spontaneously a state where the characteristic events (avalanches) are distributed according to a power law. We present a dynamical systems approach to Self-Organized Criticality where the dynamics is described either in terms of Iterated Function Systems, or as a piecewise hyperbolic dynamical system of skew-product type. Some results linking the structure of the attractor and some characteristic properties o…
The branching process is the minimal model for propagation dynamics, avalanches and criticality, broadly used in neuroscience. A simple extension of it, adding inhibitory nodes, induces a much-richer phenomenology, including, an intermediate phase, between quiescence and saturation, that exhibits the key features of "asynchronous states" in cortical networks. Remarkably, in the inhibition-dominated case, it exhibits an extremely-rich phase diagram, that captures a wealth of non-trivial features of spontaneous brain activity, such as collective excitability, hysteresis, tilted avalanche shapes,…
The avalanche properties of models that exhibit 'self-organized criticality' (SOC) are still mostly awaiting theoretical explanations. A recent mapping (Europhys. Lett.~53, 569) of many sandpile models to interface depinning is presented first, to understand how to reach the SOC ensemble and the differences of this ensemble with the usual depinning scenario. In order to derive the SOC avalanche exponents from those of the depinning critical point, a geometric description is discussed, of the quenched landscape in which the 'interface' measuring the integrated activity moves. It turns out that …
We investigate a one-dimensional rice-pile model. We show that the distribution of dissipated potential energy decays as a power law with an exponent $α=1.53$. The system thus provides a one-dimensional example of self-organized criticality. Different driving conditions are examined in order to allow for comparison with experiments.
Motivated by recent experiments in neuroscience which indicate that neuronal avalanches exhibit scale invariant behavior similar to self-organized critical systems, we study the role of noisy (non-conservative) local dynamics on the critical behavior of a sandpile model which can be taken to mimic the dynamics of neuronal avalanches. We find that despite the fact that noise breaks the strict local conservation required to attain criticality, our system exhibit true criticality for a wide range of noise in various dimensions, given that conservation is respected \textit{on the average}. Althoug…
A new quantity, average fitness, is introduced in Bak-Sneppen evolution model. Through this new quantity, a new hierarchy of avalanches is observed in the evolution of Bak-Sneppen model. An exact gap equation, governing the self-organization of the model, is presented. It is found that the self-organized threshold of the new quantity can be exactly obtained. Two basic exponents, avalanche distribution and avalanche dimension are given through simulations of one- and two-dimensional Bak-Sneppen models. It is suggested that this new quantity may be a good quantity in determining the emergence of…
We study the properties of time sequences extracted from a self-organized critical system, within the framework of the mathematical multifractal analysis. To this end, we propose a fixed-mass algorithm, well suited to deal with highly inhomogeneous one dimensional multifractal measures. We find that the fixed mass (dual) spectrum of generalized dimensions depends on both the system size L and the length N of the sequence considered, being however stable when these two parameters are kept fixed. A finite-size scaling relation is proposed, allowing us to define a renormalized spectrum, independe…
We show that the generating functions of avalanche observables in SOC models exhibits a Lee-Yang phenomenon. This establishes a new link between the classical theory of critical phenomena and SOC. A scaling theory of the Lee-Yang zeroes is proposed including finite sampling effects.
We show that a network of spiking neurons exhibits robust self-organized criticality if the synaptic efficacies follow realistic dynamics. Deriving analytical expressions for the average coupling strengths and inter-spike intervals, we demonstrate that networks with dynamical synapses exhibit critical avalanche dynamics for a wide range of interaction parameters. We prove that in the thermodynamical limit the network becomes critical for all large enough coupling parameters. We thereby explain experimental observations in which cortical neurons show avalanche activity with the total intensity …
We consider the directed percolation process as a prototype of systems displaying a nonequilibrium phase transition into an absorbing state. The model is in a critical state when the activation probability is adjusted at some precise value p_c. Criticality is lost as soon as the probability to activate sites at the first attempt, p1, is changed. We show here that criticality can be restored by "compensating" the change in p1 by an appropriate change of the second time activation probability p2 in the opposite direction. At compensation, we observe that the bulk exponents of the process coincid…
In a recent Letter (EPL 105, 40006; arXiv:1309.7107), Liu and Hu presented a model of toppling-coupled sandpiles, where they found that the avalanche exponents for two toppling-coupled sandpiles are the same as those for a single uncoupled sandpile. In this Comment we provide a proof of this observation for the case when there is conservation of grains in the bulk.
Self-organizing system is studied whose behavior is governed by field of an order parameter, a fluctuation amplitude of conjugate field and a couple of Grassmannian conjugated fields that define the entropy as a control parameter. Within the framework of self-consistent approach the dependencies of macro- and microscopic susceptibilities as well as memory and nonergodicity parameters are determined as a functions of the intensities of thermal and quenched disorders. Making use of the sandpile model shows that proposed scheme determines the conditions of avalanches formation in self-organized c…
In the prototype sandpile model of self-organized criticality time series obtained by decomposing avalanches into waves of toppling show intermittent fluctuations. The q-th moments of wave size differences possess local multiscaling and global simple scaling regimes analogous to those holding for velocity structure functions in fluid turbulence. The correspondence involves identity of a basic scaling relation and of the form of relevant probability distributions. The sandpile provides a qualitative analog of many features of turbulent phenomena.
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.
The self-organized criticality in Ehrenfest's historical dog-flea model is analyzed by simulating the underlying stochastic process. The fluctuations around the thermal equilibrium in the model are treated as avalanches. We show that the distributions for the fluctuation length differences at subsequent time steps are in the shape of a $q$-Gaussian (the distribution which is obtained naturally in the context of nonextensive statistical mechanics) if one avoids the finite size effects by increasing the system size. We provide a clear numerical evidence that the relation between the exponent $τ$…
The $1/f$-like decay observed in the power spectrum of electro-physiological signals, along with scale-free statistics of the so-called neuronal avalanches, constitute evidences of criticality in neuronal systems. Recent in vitro studies have shown that avalanche dynamics at criticality corresponds to some specific balance of excitation and inhibition, thus suggesting that this is a basic feature of the critical state of neuronal networks. In particular, a lack of inhibition significantly alters the temporal structure of the spontaneous avalanche activity and leads to an anomalous abundance of…
According to Pruessner and Peters [Phys. Rev. E {\bf 73}, 025106(R) (2006)], the finite size scaling exponents of the order parameter in sandpile models depend on the tuning of driving and dissipation rates with system size. We point out that the same is not true for {\em avalanches} in the slow driving limit.
Self-organized criticality can be translated into the language of absorbing state phase transitions. Most models for which this analogy is established have been investigated for their absorbing state characteristics. In this article, we transform the self-organized critical Oslo model into an absorbing state Oslo model and analyze the avalanche behavior. We find that the resulting gap exponent, D, is consistent with its value in the self-organized critical model. For the avalanche size exponent, τ, an analysis of the effect of the external drive and the boundary conditions is required.
We study the BTW-height model of self-organized criticality on a square lattice with some long range connections giving to the lattice the character of small world network. We find that as function of the fraction $p$ of long ranged bonds the power law of the avalanche size and lifetime distribution changes following a crossover scaling law with crossover exponents 2/3 and 1 for size and lifetime respectively.
Self-organized critical models are used to describe the 1/f-spectra of rather different physical situations like snow avalanches, noise of electric currents, luminosities of stars or topologies of landscapes. The prototype of the SOC-models is the sandpile model of Bak, Tang and Wiesenfeld (Phys. Rev. Lett. 59, (1987) 351). We implement this model on non-periodic graphs where it can become either isotropic or anisotropic and compare its properties with the periodic counterpart on the square lattice.
