Self-organized Critical Model Research Articles

Multifractal Aspects for a Self Organized Critical Model

We investigate multifractal properties for an Abelian directed model of self-organized criticality that describes the growth of droplets inside a cloud and the subsequent rainfall. The probability distribution of events of the model satisfies finite-size scaling. We obtain the singularity spectra f(α) associated with temporal records for avalanche size and for the potential energy, defined by the total sum of the product between mass and height of each site. The measure defined by avalanche size has a clear cut multifractal character, while the obtained f(α) for potential energy may include a spurious branch.

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.

Entropy in the Bak-Sneppen Model for Self-Organized Criticality

The distributions of fitness on the sites of one- andtwo-dimensional lattices are studied for the nearest-neighbourBak-Sneppen model on self-organized criticality. The distributions showcomplicated behaviour showing that the system is far from equilibrium.By introducing the ``energy'' of a site, the entropy flow from thesystem to its environment is investigated.

Random walk through fractal environments.

We analyze random walk through fractal environments, embedded in three-dimensional, permeable space. Particles travel freely and are scattered off into random directions when they hit the fractal. The statistical distribution of the flight increments (i.e., of the displacements between two consecutive hittings) is analytically derived from a common, practical definition of fractal dimension, and it turns out to approximate quite well a power-law in the case where the dimension D(F) of the fractal is less than 2, there is though, always a finite rate of unaffected escape. Random walks through fractal sets with D(F)< or =2 can thus be considered as defective Levy walks. The distribution of jump increments for D(F)>2 is decaying exponentially. The diffusive behavior of the random walk is analyzed in the frame of continuous time random walk, which we generalize to include the case of defective distributions of walk increments. It is shown that the particles undergo anomalous, enhanced diffusion for D(F)<2, the diffusion is dominated by the finite escape rate. Diffusion for D(F)>2 is normal for large times, enhanced though for small and intermediate times. In particular, it follows that fractals generated by a particular class of self-organized criticality models give rise to enhanced diffusion. The analytical results are illustrated by Monte Carlo simulations.

Instabilities in slowly driven granular packing.

In this work, a digital imaging technique is used to study the superficial fluctuations observed when a granular packing is slowly driven to the threshold of instability. The experimental results show the presence of three types of events. Small superficial rearrangements of grains are observed during all the experiments. They present a power-law behavior although the system is not in a critical state as predicted by self-organized criticality models. In thick granular piles, large rearrangements are detected at regular angular intervals. They are related to the threshold of instability of the contact network that relaxes to stable configurations producing internal rearrangements of the grains. Finally, an avalanche is triggered when the superficial beads that are set in motion acquire enough momentum to destabilize grains from layers below.

Bursty events and incremental diffusion in a local diffusion and multi-scale convection system

A one-dimensional cellular automaton is defined without the critical gradient rule (Δh>Δhc) which is essential to the existence of avalanches in self-organized criticality (SOC) models. Instead, only the local diffusion rule is used, however, the characteristics of SOC, such as the bursty behaviour, power-law decay in fluctuation spectra, self-similarity over a broad range of scales and long-time correlations, are still observed in these numerical experiments. This numerical model is established to suggest that the bursty events and the incremental diffusion observed universally in fusion experiments do not necessarily imply the submarginal dynamics.

A minimalist model of characteristic earthquakes

Abstract. In a spirit akin to the sandpile model of self-organized criticality, we present a simple statistical model of the cellular-automaton type which simulates the role of an asperity in the dynamics of a one-dimensional fault. This model produces an earthquake spectrum similar to the characteristic-earthquake behaviour of some seismic faults. This model, that has no parameter, is amenable to an algebraic description as a Markov Chain. This possibility illuminates some important results, obtained by Monte Carlo simulations, such as the earthquake size-frequency relation and the recurrence time of the characteristic earthquake.

Self-Organized Criticality Processes in HL-1M Tokamak Plasma

We study the dynamics of laminar time between successive bursts inanomalous particle flux measured at the HL-1M tokamak plasma edge. Theresults reveal that the flux fluctuations are self-similar in a narrowrange of time scales and that their probability distribution functionis not Gaussian. These properties are not consistent with thosepredicted by self-organized criticality (SOC) models as well as therunning sand-pile SOC model developed by Hwa and Kardar.

Self-organized criticality - a model for recrystallization?

Abstract A novel X-ray diffraction method, allowing the position-resolved imaging of a polycrystalline specimen using the diffracted radiation, was applied for in situ investigation of recrystallization of cold-rolled copper. A large area of the specimen could be observed simultaneously, yielding information about nucleation and growth of many individual crystallites. The recrystallization process showed a stochastic behavior which can be described by the model of self-organized criticality.

Dealing with transients in models with self-organized criticality

The problems of identifying and eliminating long transients are common to various numerical models in statistical mechanics. These problems are particularly relevant for models of self-organized criticality, as the Olami–Feder–Christensen (OFC) model, for which most of the results were, and still are, obtained through numerical simulations. In order to obtain reliable numerical results, it is usually necessary to simulate models on lattices as large as possible. However, in general, this is not an easy task, because transients increase fast with lattice size. So it is often necessary to wait long computer runs to obtain good statistics. In this paper we present an efficient algorithm to reduce transient times and to identify with a certain degree of precision if the statistical stationary state is reached, avoiding long runs to obtain good statistics. The efficiency of the algorithm is exemplified in the OFC model for the dynamics of earthquakes, but it can be useful as well in many other situations. Our analysis also shows that the OFC model approaches stationarity in qualitatively different ways in the conservative and non-conservative cases.

Continuum analysis of an avalanche model for solar flares.

We investigate the continuum limit of a class of self-organized critical lattice models for solar flares. Such models differ from the classical numerical sandpile model in their formulation of stability criteria in terms of the curvature of the nodal field, and are known to belong to a different universality class. A fourth-order nonlinear hyperdiffusion equation is reverse engineered from the discrete model's redistribution rule. A dynamical renormalization-group analysis of the equation yields scaling exponents that compare favorably with those measured in the discrete lattice model within the relevant spectral range dictated by the sizes of the domain and the lattice grid. We argue that the fourth-order nonlinear diffusion equation that models the behavior of the discrete model in the continuum limit is, in fact, compatible with magnetohydrodynamics (MHD) of the flaring phenomenon in the regime of strong magnetic field and the effective magnetic diffusivity characteristic of strong MHD turbulence.

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.

Universality in critical exponents for toppling waves of the BTW sandpile model on two-dimensional lattices

Universality and scaling for systems driven to criticality by a tuning parameter has been well studied. However, there are very few corresponding studies for the models of self-organized criticality, e.g., the Bak, Tang, and Wiesenfeld (BTW) sandpile model. It is well known that every avalanche of the BTW sandpile model may be represented as a sequence of waves and the asymptotic probability distributions of all waves and last waves have critical exponents, 1 and 11/8, respectively. By an inversion symmetry, Hu, Ivashkevich, Lin, and Priezzhev showed that in the BTW sandpile model the probability distribution of dissipating waves of topplings that touch the boundary of the system shows a power-law relationship with critical exponent 5/8 and the probability distribution of those dissipating waves that are also last in an avalanche has an exponent of 1 (Phys. Rev. Lett. 85 (2000) 4048). Such predictions have been confirmed by extensive numerical simulations of the BTW sandpile model on square lattices. Very recently, we used Monte Carlo simulations to find that the waves of the BTW model on square, honeycomb, triangular, and random lattices have the same set of critical exponents.

Self-Organized Criticality and Thermodynamic Formalism

We introduce a dissipative version of the Zhang's model of Self-Organized Criticality, where a parameter allows to tune the local energy dissipation. We analyze the main dynamical features of the model and relate in particular the Lyapunov spectrum with the transport properties in the stationary regime. We develop a thermodynamic formalism where we define formal Gibbs measure, partition function and pressure characterizing the avalanche distributions. We discuss the infinite size limit in this setting. We show in particular that a Lee-Yang phenomenon occurs in this model, for the only conservative case. This suggests new connexions to classical critical phenomena.

New toppling dynamics for the self-organized critical rainfall model

In this paper we introduce two new dynamical features of rainfall phenomena into the self-organized critical rainfall model. The new dynamics lead to the constrained random and the complete toppling versions, both of which lose the abelian character of the original model. The constrained random rainfall model is also a self-organized criticality model, but it belongs to another universal class. The complete toppling rainfall model presents two different regimes related to small and large avalanches, respectively.

Nonconservative earthquake model of self-organized criticality on a random graph.

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.

Self-organized criticality on small world networks

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.

Geometrical properties of avalanches in self-organized critical models of solar flares.

We investigate the geometrical properties of avalanches in self-organized critical models of solar flares. Traditionally, such models differ from the classical sandpile model in their formulation of stability criteria in terms of the curvature of the nodal field, and belong to a distinct universality class. With a view toward comparing these properties to those inferred from spatially and temporally resolved flare observations, we consider the properties of avalanche peak snapshots, time-integrated avalanches in two and three dimensions, and the two-dimensional projections of the latter. The nature of the relationship between the avalanching volume and its projected area is an issue of particular interest in the solar flare context. Using our simulation results we investigate this relationship, and demonstrate that proper accounting of the fractal nature of avalanches can bring into agreement hitherto discrepant results of observational analyses based on simple, nonfractal geometries for the flaring volume.

Scale invariance of incident size distributions in response to sizes of their causes.

Incidents can be defined as low-probability, high-consequence events and lesser events of the same type. Lack of data on extremely large incidents makes it difficult to determine distributions of incident size that reflect such disasters, even though they represent the great majority of total losses. If the form of the incident size distribution can be determined, then predictive Bayesian methods can be used to assess incident risks from limited available information. Moreover, incident size distributions have generally been observed to have scale invariant, or power law, distributions over broad ranges. Scale invariance in the distributions of sizes of outcomes of complex dynamical systems has been explained based on mechanistic models of natural and built systems, such as models of self-organized criticality. In this article, scale invariance is shown to result also as the maximum Shannon entropy distribution of incident sizes arising as the product of arbitrary functions of cause sizes. Entropy is shown by simulation and derivation to be maximized as a result of dependence, diversity, abundance, and entropy of multiplicative cause sizes. The result represents an information-theoretic explanation of invariance, parallel to those of mechanistic models. For example, distributions of incident size resulting from 30 partially dependent causes are shown to be scale invariant over several orders of magnitude. Empirical validation of power law distributions of incident size is reviewed, and the Pareto (power law) distribution is validated against oil spill, hurricane, and insurance data. The applicability of the Pareto distribution, in particular, for assessment of total losses over a planning period is discussed. Results justify the use of an analytical, predictive Bayesian version of the Pareto distribution, derived previously, to assess incident risk from available data.

QUANTUM FIELD THEORY RENORMALIZATION GROUP APPROACH TO SELF-ORGANIZED CRITICAL MODELS: THE CASE OF RANDOM BOUNDARIES

The long time and large scale asymptotic behavior for a stochastic problem related to self-organized critical (SOC) models is studied in the framework of advanced quantum field theory renormalization group (RG) method. The threshold condition and time scale separation between the slow dynamics of energy injection and the fast dynamics of avalanches (relaxation) are taken into account in the model. Herewith, the reciprocal correlation time at wavenumber k scales as tc(k) ∝ k-2+2η with some phenomenological parameter η &gt; 0 corresponding to the anomalous diffusion coefficient z = 2(1 - η). The quantum field theory corresponding to the nonlinear stochastic problem is multiplicatively renormalizable and has an infinite number of coupling constants. The RG equations have a two-dimensional manifold of fixed points. Some of them relate to the stable asymptotic solutions and stipulate a general scaling with the critical dimensions of time Δ[t] = -2 + 2η and the energy field Δ[E] = d/2 - 3(1 - η). Possible corrections to the leading asymptotic behavior are discussed.