Scaling theory of self-organized criticality

Emergence of power-law distributions in self-segregation reaction-diffusion processes

Many natural or human-made systems encompassing local reactions and diffusion processes exhibit spatially distributed patterns of some relevant dynamical variable. These interactions, through self-organization and critical phenomena, give rise to power-law distributions, where emergent patterns and structures become visible across vastly different scales. Recent observations reveal power-law distributions in the spatial organization of, e.g., tree clusters and forest patch sizes. Crucially, these patterns do not follow a spatially periodic order but rather a statistical one. Unlike the spatially periodic patterns elucidated by the Turing mechanism, the statistical order of these particular vegetation patterns suggests an incomplete understanding of the underlying mechanisms. Here, we present a self-segregation mechanism, driving the emergence of power-law scalings in pattern-forming systems. The model incorporates an Allee-logistic reaction term, responsible for the local growth, and a nonlinear diffusion process accounting for positive interactions and limited resources. According to a self-organized criticality (SOC) principle, after an initial decrease, the system mass reaches an analytically predictable threshold, beyond which it self-segregates into distinct clusters, due to local positive interactions that promote cooperation. Numerical investigations show that the distribution of cluster sizes obeys a power law with an exponential cutoff.

Eigen microstates in self-organized criticality

We employ the eigen microstates approach to explore the self-organized criticality (SOC) in two celebrated sandpile models, namely the BTW model and the Manna model. In both models, phase transitions from the absorbing state to the critical state can be understood by the emergence of dominant eigen microstates with significantly increased weights. Spatial eigen microstates of avalanches can be uniformly characterized by a linear system size rescaling. The first temporal eigen microstates reveal scaling relations in both models. Furthermore, by finite-size scaling analysis of the first eigen microstates, we numerically estimate critical exponents, i.e., sqrt[σ_{0}w_{1}]/v[over ̃]_{1}∝L^{D} and v[over ̃]_{1}∝L^{D(1-τ_{s})/2}. Our findings could provide profound insights into eigen microstates of the universality and phase transition in nonequilibrium complex systems governed by self-organized criticality.

Universal predictability of large avalanches in the Manna sandpile model

Substantiated explanations of the unpredictability regarding sandpile models of self-organized criticality (SOC) gave way to efficient forecasts of extremes in a few models. The appearance of extremes requires a preparation phase that ends with general overloading of the system and spatial clustering of the local stress. Here, we relate the predictability of large events to the system volume in the Manna and Bak-Tang-Wiesenfeld sandpiles, which are basic models of SOC. We establish that in the Manna model, the events located to the right of the power-law segment of the size-frequency relationship are predictable and the prediction efficiency is described by the universal linear dependence on the event size scaled by a power-law function of the lattice volume. Our scaling-based approach to predictability contributes to the theory of SOC and may elucidate the forecast of extremes in the dynamics of such systems with SOC as neuronal networks and earthquakes.

Finite-size scaling of critical avalanches

We examine probability distribution for avalanche sizes observed in self-organized critical systems. While a power-law distribution with a cutoff because of finite system size is typical behavior, a systematic investigation reveals that it may also decrease with increasing the system size at a fixed avalanche size. We implement the scaling method and identify scaling functions. The data collapse ensures a correct estimation of the critical exponents and distinguishes two exponents related to avalanche size and system size. Our simple analysis provides striking implications. While the exact value for avalanches size exponent remains elusive for the prototype sandpile on a square lattice, we suggest the exponent should be 1. The simulation results represent that the distribution shows a logarithmic system size dependence, consistent with the normalization condition. We also argue that for the train or Oslo sandpile model with bulk drive, the avalanche size exponent is slightly less than 1, which differs significantly from the previous estimate of 1.11.

Self-organized quantization and oscillations on continuous fixed-energy sandpiles

Atmospheric self-organization and activator-inhibitor dynamics in biology provide examples of checkerboardlike spatiotemporal organization. We study a simple model for local activation-inhibition processes. Our model, first introduced in the context of atmospheric moisture dynamics, is a continuous-energy and non-Abelian version of the fixed-energy sandpile model. Each lattice site is populated by a nonnegative real number, its energy. Upon each timestep all sites with energy exceeding a unit threshold redistribute their energy at equal parts to their nearest neighbors. The limit cycle dynamics gives rise to a complex phase diagram in dependence on the mean energy μ: For low μ, all dynamics ceases after few redistribution events. For large μ, the dynamics is well-described as a diffusion process, where the order parameter, spatial variance σ, is removed. States at intermediate μ are dominated by checkerboardlike period-two phases which are however interspersed by much more complex phases of far longer periods. Phases are separated by discontinuous jumps in σ or ∂_{μ}σ-akin to first- and higher-order phase transitions. Overall, the energy landscape is dominated by few energy levels which occur as sharp spikes in the single-site density of states and are robust to noise.

Self-consistent model of the plasma staircase and nonlinear Schrödinger equation with subquadratic power nonlinearity

A new basis has been found for the theory of self-organization of transport avalanches and jet zonal flows in L-mode tokamak plasma, the so-called "plasma staircase" [Dif-Pradalier et al., Phys. Rev. E 82, 025401(R) (2010)PLEEE81539-375510.1103/PhysRevE.82.025401]. The jet zonal flows are considered as a wave packet of coupled nonlinear oscillators characterized by a complex time- and wave-number-dependent wave function; in a mean-field approximation this function is argued to obey a discrete nonlinear Schrödinger equation with subquadratic power nonlinearity. It is shown that the subquadratic power leads directly to a white Lévy noise, and to a Lévy fractional Fokker-Planck equation for radial transport of test particles (via wave-particle interactions). In a self-consistent description the avalanches, which are driven by the white Lévy noise, interact with the jet zonal flows, which form a system of semipermeable barriers to radial transport. We argue that the plasma staircase saturates at a state of marginal stability, in whose vicinity the avalanches undergo an ever-pursuing localization-delocalization transition. At the transition point, the event-size distribution of the avalanches is found to be a power law w_{τ}(Δn)∼Δn^{-τ}, with the drop-off exponent τ=(sqrt[17]+1)/2≃2.56. This value is an exact result of the self-consistent model. The edge behavior bears signatures enabling to associate it with the dynamics of a self-organized critical (SOC) state. At the same time the critical exponents, pertaining to this state, are found to be inconsistent with classic models of avalanche transport based on sand piles and their generalizations, suggesting that the coupled avalanche-jet zonal flow system operates on different organizing principles. The results obtained have been validated in a numerical simulation of the plasma staircase using flux-driven gyrokinetic code for L-mode Tore-Supra plasma.

Coexistence of scale-invariant and rhythmic behavior in self-organized criticality

Scale-free behavior as well as oscillations are frequently observed in the activity of many natural systems. One important example is the cortical tissues of mammalian brain where both phenomena are simultaneously observed. Rhythmic oscillations as well as critical (scale-free) dynamics are thought to be important, but theoretically incompatible, features of a healthy brain. Motivated by the above, we study the possibility of the coexistence of scale-free avalanches along with rhythmic behavior within the framework of self-organized criticality. In particular, we add an oscillatory perturbation to local threshold condition of the continuous Zhang model and characterize the subsequent activity of the system. We observe regular oscillations embedded in well-defined avalanches which exhibit scale-free size and duration in line with observed neuronal avalanches. The average amplitude of such oscillations are shown to decrease with increasing frequency consistent with real brain oscillations. Furthermore, it is shown that optimal amplification of oscillations occur at the critical point, further providing evidence for functional advantages of criticality.

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.

Oslo model, hyperuniformity, and the quenched Edwards-Wilkinson model

We present simulations of the one-dimensional Oslo rice pile model in which the critical height at each site is randomly reset after each toppling. We use the fact that the stationary state of this sand-pile model is hyperuniform to reach system of sizes >10^{7}. Most previous simulations were seriously flawed by important finite-size corrections. We find that all critical exponents have values consistent with simple rationals: ν=4/3 for the correlation length exponent, D=9/4 for the fractal dimension of avalanche clusters, and z=10/7 for the dynamical exponent. In addition, we relate the hyperuniformity exponent to the correlation length exponent ν. Finally, we discuss the relationship with the quenched Edwards-Wilkinson model, where we find in particular that the local roughness exponent is α_{loc}=1.

Synaptic plasticity and neuronal refractory time cause scaling behaviour of neuronal avalanches

Neuronal avalanches measured in vitro and in vivo in different cortical networks consistently exhibit power law behaviour for the size and duration distributions with exponents typical for a mean field self-organized branching process. These exponents are also recovered in neuronal network simulations implementing various neuronal dynamics on different network topologies. They can therefore be considered a very robust feature of spontaneous neuronal activity. Interestingly, this scaling behaviour is also observed on regular lattices in finite dimensions, which raises the question about the origin of the mean field behavior observed experimentally. In this study we provide an answer to this open question by investigating the effect of activity dependent plasticity in combination with the neuronal refractory time in a neuronal network. Results show that the refractory time hinders backward avalanches forcing a directed propagation. Hebbian plastic adaptation plays the role of sculpting these directed avalanche patterns into the topology of the network slowly changing it into a branched structure where loops are marginal.

Dynamics of social contagions with memory of nonredundant information

A key ingredient in social contagion dynamics is reinforcement, as adopting a certain social behavior requires verification of its credibility and legitimacy. Memory of nonredundant information plays an important role in reinforcement, which so far has eluded theoretical analysis. We first propose a general social contagion model with reinforcement derived from nonredundant information memory. Then, we develop a unified edge-based compartmental theory to analyze this model, and a remarkable agreement with numerics is obtained on some specific models. We use a spreading threshold model as a specific example to understand the memory effect, in which each individual adopts a social behavior only when the cumulative pieces of information that the individual received from his or her neighbors exceeds an adoption threshold. Through analysis and numerical simulations, we find that the memory characteristic markedly affects the dynamics as quantified by the final adoption size. Strikingly, we uncover a transition phenomenon in which the dependence of the final adoption size on some key parameters, such as the transmission probability, can change from being discontinuous to being continuous. The transition can be triggered by proper parameters and structural perturbations to the system, such as decreasing individuals' adoption threshold, increasing initial seed size, or enhancing the network heterogeneity.

Crossover from rotational to stochastic sandpile universality in the random rotational sandpile model

In the rotational sandpile model, either the clockwise or the anticlockwise toppling rule is assigned to all the lattice sites. It has all the features of a stochastic sandpile model but belongs to a different universality class than the Manna class. A crossover from rotational to Manna universality class is studied by constructing a random rotational sandpile model and assigning randomly clockwise and anticlockwise rotational toppling rules to the lattice sites. The steady state and the respective critical behavior of the present model are found to have a strong and continuous dependence on the fraction of the lattice sites having the anticlockwise (or clockwise) rotational toppling rule. As the anticlockwise and clockwise toppling rules exist in equal proportions, it is found that the model reproduces critical behavior of the Manna model. It is then further evidence of the existence of the Manna class, in contradiction with some recent observations of the nonexistence of the Manna class.

Mean-field behavior as a result of noisy local dynamics in self-organized criticality: neuroscience implications

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 (nonconservative) 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 exhibits true criticality for a wide range of noise in various dimensions, given that conservation is respected on the average. Although the system remains critical, exhibiting finite-size scaling, the value of critical exponents change depending on the intensity of local noise. Interestingly, for a sufficiently strong noise level, the critical exponents approach and saturate at their mean-field values, consistent with empirical measurements of neuronal avalanches. This is confirmed for both two and three dimensional models. However, the addition of noise does not affect the exponents at the upper critical dimension (D = 4). In addition to an extensive finite-size scaling analysis of our systems, we also employ a useful time-series analysis method to establish true criticality of noisy systems. Finally, we discuss the implications of our work in neuroscience as well as some implications for the general phenomena of criticality in nonequilibrium systems.

Relation between self-organized criticality and grain aspect ratio in granular piles

We investigate experimentally whether self-organized criticality (SOC) occurs in granular piles composed of different grains, namely, rice, lentils, quinoa, and mung beans. These four grains were selected to have different aspect ratios, from oblong to oblate. As a function of aspect ratio, we determined the growth (β) and roughness (α) exponents, the avalanche fractal dimension (D), the avalanche size distribution exponent (τ), the critical angle (γ), and its fluctuation. At superficial inspection, three types of grains seem to have power-law-distributed avalanches with a well-defined τ. However, only rice is truly SOC if we take three criteria into account: a power-law-shaped avalanche size distribution, finite size scaling, and a universal scaling relation relating characteristic exponents. We study SOC as a spatiotemporal fractal; in particular, we study the spatial structure of criticality from local observation of the slope angle. From the fluctuation of the slope angle we conclude that greater fluctuation (and thus bigger avalanches) happen in piles consisting of grains with larger aspect ratio.

Simulation of a Casimir-like effect in a granular pile with avalanches

Using a modified Bak-Tang-Wiesenfeld model for sand piles, we simulate a Casimir-like effect in a granular pile with avalanches. Results obtained in the simulation are in good agreement with results previously acquired experimentally: two parallel walls are attracted to each other at small separation distances, with a force decreasing with increasing distance. In the simulation only, at medium distances a weak repulsion exists. Additionally, with the aim of avalanche prevention, the possibility of suppressing self-organized criticality with an array of walls placed on the slope of the pile is investigated, but the prevention effect is found to be negligible.

Universality classes and crossover behaviors in non-Abelian directed sandpiles

We study universality classes and crossover behaviors in non-Abelian directed sandpile models in terms of the metastable pattern analysis. The non-Abelian property induces spatially correlated metastable patterns, characterized by the algebraic decay of the grain density along the propagation direction of an avalanche. Crossover scaling behaviors are observed in the grain density due to the interplay between the toppling randomness and the parity of the threshold value. In the presence of such crossovers, we show that the broadness of the grain distribution plays a crucial role in resolving the ambiguity of the universality class. Finally, we claim that the metastable pattern analysis is important as much as the conventional analysis of avalanche dynamics.

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.

RESULTS

Neither the LFP nor the subsampled SOC models showed a power law for f(s). Both, f(s) and sigma, depended sensitively on the subsampling geometry and the dynamics of the model. Only one of the SOC models, the Abelian Sandpile Model, exhibited f(s) and sigma similar to those calculated from LFP activity.

CONCLUSION

Since subsampling can prevent the observation of the characteristic power law and sigma in SOC systems, misclassifications of critical systems as sub- or supercritical are possible. Nevertheless, the system specific scaling of f(s) and sigma under subsampling conditions may prove useful to select physiologically motivated models of brain function. Models that better reproduce f(s) and sigma calculated from the physiological recordings may be selected over alternatives.

Direct evidence for conformal invariance of avalanche frontiers in sandpile models

Appreciation of stochastic Loewner evolution (SLE_{kappa}) , as a powerful tool to check for conformal invariant properties of geometrical features of critical systems has been rising. In this paper we use this method to check conformal invariance in sandpile models. Avalanche frontiers in Abelian sandpile model are numerically shown to be conformally invariant and can be described by SLE with diffusivity kappa=2 . This value is the same as value obtained for loop-erased random walks. The fractal dimension and Schramm's formula for left passage probability also suggest the same result. We also check the same properties for Zhang's sandpile model.

Relevance of Abelian symmetry and stochasticity in directed sandpiles

We provide a comprehensive view of the role of Abelian symmetry and stochasticity in the universality class of directed sandpile models, in the context of the underlying spatial correlations of metastable patterns and scars. It is argued that the relevance of Abelian symmetry may depend on whether the dynamic rule is stochastic or deterministic, by means of the interaction of metastable patterns and avalanche flow. Based on the new scaling relations, we conjecture critical exponents for an avalanche, which is confirmed reasonably well in large-scale numerical simulations.

Scaling properties of a rice-pile model: inertia and friction effects

We present a rice-pile cellular automaton model that includes inertial and friction effects. This model is studied in one dimension, where the updating of metastable sites is done according to a stochastic dynamics governed by a probabilistic toppling parameter p that depends on the accumulated energy of moving grains. We investigate the scaling properties of the model using finite-size scaling analysis. The avalanche size, the lifetime, and the residence time distributions exhibit a power-law behavior. Their corresponding critical exponents, respectively, tau, y, and yr, are not universal. They present continuous variation versus the parameters of the system. The maximal value of the critical exponent tau that our model gives is very close to the experimental one, tau=2.02 [Frette, Nature (London) 379, 49 (1996)], and the probability distribution of the residence time is in good agreement with the experimental results. We note that the critical behavior is observed only in a certain range of parameter values of the system which correspond to low inertia and high friction.

Modeling temporal fluctuations in avalanching systems

We demonstrate how to model the toppling activity in avalanching systems by stochastic differential equations (SDEs). The theory is developed as a generalization of the classical mean-field approach to sandpile dynamics by formulating it as a generalization of Itô's SDE. This equation contains a fractional Gaussian noise term representing the branching of an avalanche into small active clusters and a drift term reflecting the tendency for small avalanches to grow and large avalanches to be constricted by the finite system size. If one defines avalanching to take place when the toppling activity exceeds a certain threshold, the stochastic model allows us to compute the avalanche exponents in the continuum limit as functions of the Hurst exponent of the noise. The results are found to agree well with numerical simulations in the Bak-Tang-Wiesenfeld and Zhang sandpile models. The stochastic model also provides a method for computing the probability density functions of the fluctuations in the toppling activity itself. We show that the sandpiles do not belong to the class of phenomena giving rise to universal non-Gaussian probability density functions for the global activity. Moreover, we demonstrate essential differences between the fluctuations of total kinetic energy in a two-dimensional turbulence simulation and the toppling activity in sandpiles.

Confirming and extending the hypothesis of universality in sandpiles

Stochastic sandpiles self-organize to an absorbing-state critical point with scaling behavior different from directed percolation (DP) and characterized by the presence of an additional conservation law. This is usually called the C-DP or Manna universality class. There remains, however, an exception to this universality principle: a sandpile automaton introduced by Maslov and Zhang, which was claimed to be in the DP class despite the existence of a conservation law. We show, by means of careful numerical simulations as well as by constructing and analyzing a field theory, that (contrarily to what was previously thought) this sandpile is also in the C-DP or Manna class. This confirms the hypothesis of universality for stochastic sandpiles and gives rise to a fully coherent picture of self-organized criticality in systems with conservation. In passing, we obtain a number of results for the C-DP class and introduce a strategy to easily discriminate between DP and C-DP scaling.

Emergence of quasiunits in the one-dimensional Zhang model

We study the Zhang model of sandpile on a one-dimensional chain of length L , where a random amount of energy is added at a randomly chosen site at each time step. We show that in spite of this randomness in the input energy, the probability distribution function of energy at a site in the steady state is sharply peaked, and the width of the peak decreases as L(-1/2) for large L . We discuss how the energy added at one time is distributed among different sites by topplings with time. We relate this distribution to the time-dependent probability distribution of the position of a marked grain in the one-dimensional Abelian model with discrete heights. We argue that in the large L limit, the variance of energy at site x has a scaling form L(-1)g(x/L) , where g(xi) varies as ln(1/xi) for small xi , which agrees very well with the results from numerical simulations.

Asymptotic behavior and synchronizability characteristics of a class of recurrent neural networks

We propose an approach to the analysis of the influence of the topology of a neural network on its synchronizability in the sense of equal output activity rates given by a particular neural network model. The model we introduce is a variation of the Zhang model. We investigate the time-asymptotic behavior of the corresponding dynamical system (in particular, the conditions for the existence of an invariant compact asymptotic set) and apply the results of the synchronizability analysis on a class of random scale free networks and to the classical random networks with Poisson connectivity distribution.

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.

Sticky grains do not change the universality class of isotropic sandpiles

We revisit the sandpile model with "sticky" grains introduced by Mohanty and Dhar [Phys. Rev. Lett. 89, 104303 (2002)] whose scaling properties were claimed to be generically in the universality class of directed percolation for both isotropic and directed models. While for directed models this conclusion is unquestionable, for isotropic models we present strong evidence that the asymptotic scaling in the self-organized regime (in which a stationary critical state exists in the limit of slow driving and vanishing dissipation rate) is, like other stochastic sandpiles, generically in the Manna universality class. This conclusion is drawn from extensive Monte Carlo simulations, and is strengthened by the analysis of the Langevin equations (proposed by the same authors to account for this problem), argued to converge upon coarse-graining to the well-established set of Langevin equations for the Manna class.

Behavior of coupled automata

We study the nature of statistical correlations that develop between systems of interacting self-organized critical automata (sandpiles). Numerical and analytical findings are presented describing the emergence of "synchronization" between sandpiles and the dependency of this synchronization on factors such as variations in coupling strength, toppling rule probabilities, symmetric versus asymmetric coupling rules, and numbers of sandpiles.

Numerical study of the Langevin theory for fixed-energy sandpiles

The recently proposed Langevin equation, aimed to capture the relevant critical features of stochastic sandpiles and other self-organizing systems, is studied numerically. The equation is similar to the Reggeon field theory, describing generic systems with absorbing states, but it is coupled linearly to a second conserved and static (nondiffusive) field. It has been claimed to represent a different universality class, including different discrete models: the Manna as well as other sandpiles, reaction-diffusion systems, etc. In order to integrate the equation, and surpass the difficulties associated with its singular noise, we follow a numerical technique introduced by Dickman. Our results coincide remarkably well with those of discrete models claimed to belong to this universality class, in one, two, and three dimensions. This provides a strong backing for the Langevin theory of stochastic sandpiles, and to the very existence of this meagerly understood universality class.

Self-organized criticality in a bead pile

Self-organized criticality has been proposed to explain complex dynamical systems near their critical points. This experiment examined a monodisperse conical bead pile and how the distribution of avalanches is affected by the pattern of beads glued on a base, by the size or shape of the base, and by the height at which each bead was dropped onto the pile. By measuring the number of avalanches for a given size that occurred during the experiment, the resulting distribution could be compared to a power law description. When the beads were dropped from a small height, all data were consistent with a simple power law of exponent -1.5, which is the mean-field model value. The data showed that neither the bead pattern on the base nor the base size or shape significantly affected the power law behavior. However, when the bead is dropped from different heights, then the power law description breaks down and a power law times an exponential is more appropriate. We found a scaling relationship in the distribution of avalanches for different heights and relate the data to an energy dissipation model. We both confirm self-organized criticality and observe deviations from it.

Oslo rice pile model is a quenched Edwards-Wilkinson equation

The Oslo rice pile model is a sandpile-like paradigmatic model of self-organized criticality (SOC). In this paper it is shown that the Oslo model is in fact exactly a discrete realization of the much studied quenched Edwards-Wilkinson equation (qEW) [Nattermann et al., J. Phys. II France 2, 1483 (1992)]. This is possible by choosing the correct dynamical variable and identifying its equation of motion. It establishes for the first time an exact link between SOC models and the field of interface growth with quenched disorder. This connection is obviously very encouraging as it suggests that established theoretical techniques can be brought to bear with full strength on some of the hitherto elusive problems of SOC.

Local rigidity in sandpile models

We address the problem of the role of the concept of local rigidity in the family of sandpile systems. We define rigidity as the ratio between the critical energy and the amplitude of the external perturbation and we show, in the framework of the dynamically driven renormalization group, that any finite value of the rigidity in a generalized sandpile model renormalizes to an infinite value at the fixed point, i.e., on a large scale. The fixed-point value of the rigidity allows then for a nonambiguous distinction between sandpilelike systems and diffusive systems. Numerical simulations support our analytical results.

Self-organized criticality: robustness of scaling exponents

We investigate a deterministic, conservative, undirected, critical height sandpile model with dissipation of an energy at boundaries that can simulate avalanche dynamics. It was derived from the Bak-Tang-Wiesenfeld model [P. Bak, C. Tang, and K. Wiesenfeld, Phys. Rev. Lett. 59, 381 (1987)] introducing an additional second-higher threshold so the model has two distinct thresholds. Our computer simulations for a two-dimensional lattice show that scaling properties of the model depend on the higher-threshold values and site concentrations. These results are not therefore consistent with the present self-organized criticality hypothesis where the scaling properties are independent of the model parameters.

Anomalous scaling and Lee-Yang zeros in self-organized criticality

We show that the generating functions of probability distributions in self-organized criticality (SOC) models exhibit a Lee-Yang phenomenon [Phys. Rev. 87, 404 (1952)]. Namely, their zeros pinch the real axis at z=1, as the system size goes to infinity. This establishes a new link between the classical theory of critical phenomena and SOC. A scaling theory of the Lee-Yang zeros is proposed in this setting.

Lyapunov exponents and transport in the Zhang model of self-organized criticality

We discuss the role played by Lyapunov exponents in the dynamics of Zhang's model of self-organized criticality. We show that a large part of the spectrum (the slowest modes) is associated with energy transport in the lattice. In particular, we give bounds on the first negative Lyapunov exponent in terms of the energy flux dissipated at the boundaries per unit of time. We then establish an explicit formula for the transport modes that appear as diffusion modes in a landscape where the metric is given by the density of active sites. We use a finite size scaling ansatz for the Lyapunov spectrum, and relate the scaling exponent to the scaling of quantities such as avalanche size, duration, density of active sites, etc.

Evidence for universality within the classes of deterministic and stochastic sandpile models

Recent numerical studies have provided evidence that within the family of conservative, undirected sandpile models with short range dynamic rules, deterministic models such as the Bak-Tang-Wiesenfeld model [P. Bak, C. Tang, and K. Wiesenfeld, Phys. Rev. Lett. 59, 381 (1987)] and stochastic models such as the Manna model [S. S. Manna, J. Phys. A 24, L363 (1991)] belong to different universality classes. In this paper we examine the universality within each of the two classes in two dimensions by numerical simulations. To this end we consider additional deterministic and stochastic models and use an extended set of critical exponents, scaling functions, and geometrical features. Universal behavior is found within the class of deterministic Abelian models, as well as within the class of stochastic models (which includes both Abelian and non-Abelian models). In addition, it is observed that deterministic but non-Abelian models exhibit critical exponents that depend on a parameter, namely they are nonuniversal.

Large-scale synchrony in weakly interacting automata

We study the behavior of two spatially distributed (sandpile) models which are weakly linked with one another. Using a Monte Carlo implementation of the renormalization-group and algebraic methods, we describe how large-scale correlations emerge between the two systems, leading to synchronized behavior.

Distribution of sizes of erased loops of loop-erased random walks in two and three dimensions

We show that in the loop-erased random-walk problem, the exponent characterizing the probability distribution of areas of erased loops is superuniversal. In d dimensions, the probability that the erased loop has an area A varies as A(-2) for large A, independent of d, for 2< or =d< or =4. We estimate the exponents characterizing the distribution of perimeters and areas of erased loops in d=2 and 3 by large-scale Monte Carlo simulations. Our estimate of the fractal dimension z in two dimensions is consistent with the known exact value 5/4. In three dimensions, we get z=1.6183+/-0.0004. The exponent for the distribution of the durations of avalanches in the three-dimensional Abelian sandpile model is determined from this by using scaling relations.

Discretized diffusion processes

We study the properties of the "rigid Laplacian" operator; that is we consider solutions of the Laplacian equation in the presence of fixed truncation errors. The dynamics of convergence to the correct analytical solution displays the presence of a metastable set of numerical solutions, whose presence can be related to granularity. We provide some scaling analysis in order to determine the value of the exponents characterizing the process. We believe that this prototype model is also suitable to provide an explanation of the widespread presence of power law in a social and economic system where information and decision diffuse, with errors and delay from agent to agent.

Crossover phenomenon in self-organized critical sandpile models

We consider a stochastic sandpile where the sand grains of unstable sites are randomly distributed to the nearest neighbors. Increasing the value of the threshold condition the stochastic character of the distribution is lost and a crossover to the scaling behavior of a different sandpile model takes place where the sand grains are equally transferred to the nearest neighbors. The crossover behavior is analyzed numerically in detail; especially we consider the exponents which determine the scaling behavior.

Absorbing-state phase transitions in fixed-energy sandpiles

We study sandpile models as closed systems, with the conserved energy density zeta playing the role of an external parameter. The critical energy density zeta(c) marks a nonequilibrium phase transition between active and absorbing states. Several fixed-energy sandpiles are studied in extensive simulations of stationary and transient properties, as well as the dynamics of roughening in an interface-height representation. Our primary goal is to identify the universality classes of such models, in hopes of assessing the validity of two recently proposed approaches to sandpiles: a phenomenological continuum Langevin description with absorbing states, and a mapping to driven interface dynamics in random media.

Proof of breaking of self-organized criticality in a nonconservative abelian sandpile model

By computer simulations, it was reported that the Bak-Tang-Wiesenfeld (BTW) model loses self-organized criticality (SOC) when some particles are annihilated in a toppling process in the bulk of system. We give a rigorous proof that the BTW model loses SOC as soon as the annihilation rate becomes positive. To prove this, a nonconservative Abelian sandpile model is defined on a square lattice, which has a parameter alpha (>/=1) representing the degree of breaking of the conservation law. This model is reduced to be the BTW model when alpha=1. By calculating the average number of topplings in an avalanche <T> exactly, it is shown that for any alpha>1, <T><infinity even in the infinite-volume limit. The power-law divergence of <T> with an exponent 1 as alpha-->1 gives a scaling relation 2nu(2-a)=1 for the critical exponents nu and a of the distribution function of T. The 1-1 height correlation C11(r) is also calculated analytically and we show that C11(r) is bounded by an exponential function when alpha>1, although C11(r) approximately r(-2d) was proved by Majumdar and Dhar for the d-dimensional BTW model. A critical exponent nu(11) characterizing the divergence of the correlation length xi as alpha-->1 is defined as xi approximately |alpha-1|(-nu(11)) and our result gives an upper bound nu(11)</=1/2.

Moment analysis of the probability distribution of different sandpile models

We reconsider the moment analysis of the Bak-Tang-Wiesenfeld and the stochastic sandpile model introduced by Manna [J. Phys. A 24, L363 (1991)] in two and three dimensions. In contrast to recently performed investigations our analysis reveals that the models are characterized by different scaling behavior, i.e., they belong to different universality classes.

Scaling of waves in the bak-tang-wiesenfeld sandpile model

We study probability distributions of waves of topplings in the Bak-Tang-Wiesenfeld model on hypercubic lattices for dimensions D>/=2. Waves represent relaxation processes which do not contain multiple toppling events. We investigate bulk and boundary waves by means of their correspondence to spanning trees, and by extensive numerical simulations. While the scaling behavior of avalanches is complex and usually not governed by simple scaling laws, we show that the probability distributions for waves display clear power-law asymptotic behavior in perfect agreement with the analytical predictions. Critical exponents are obtained for the distributions of radius, area, and duration of bulk and boundary waves. Relations between them and fractal dimensions of waves are derived. We confirm that the upper critical dimension D(u) of the model is 4, and calculate logarithmic corrections to the scaling behavior of waves in D=4. In addition, we present analytical estimates for bulk avalanches in dimensions D>/=4 and simulation data for avalanches in D</=3. For D=2 they seem not easy to interpret.

Dynamical real space renormalization group applied to sandpile models

A general framework for the renormalization group analysis of self-organized critical sandpile models is formulated. The usual real space renormalization scheme for lattice models when applied to nonequilibrium dynamical models must be supplemented by feedback relations coming from the stationarity conditions. On the basis of these ideas the dynamically driven renormalization group is applied to describe the boundary and bulk critical behavior of sandpile models. A detailed description of the branching nature of sandpile avalanches is given in terms of the generating functions of the underlying branching process.

Universality and self-similarity of an energy-constrained sandpile model with random neighbors

We study an energy-constrained sandpile model with random neighbors. The critical behavior of the model is in the same universality class as the mean-field self-organized criticality sandpile. The critical energy E(c) depends on the number of neighbors n for each site, but the various exponents are independent of n. A self-similar structure with n-1 major peaks is developed for the energy distribution p(E) when the system approaches its stationary state. The avalanche dynamics contributes to the major peaks appearing at E(p(k))=2k/(2n-1) with k=1,2,...,n-1, while the fine self-similar structure is a natural result of the way the system is disturbed.