Cascades and self-organized criticality

Manna, Sourav; Kiss, L.B.; Kertész, János

doi:10.1007/bf01027312

Cited by 100 publications

(98 citation statements)

References 16 publications

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“…The result is that the introduction of dissipation turns the fixed point into a trivial one, and this means that the system is no longer in a critical state. This is in agreement with the simulation on dissipative sandpile models [12], while other dissipative models [13] appear to belong to different universality classes.…”

supporting

confidence: 80%

“…In the past few years extensive numerical simulations have been devoted to the study of discrete and continuous sandpile models, and critical exponents have been measured for various dimensionality in several models [1,14,[16][17][18][19][20]. The BTW, Zhang, and two state models are supposed to belong to the same universality class, and the exponents measured are very close, although some discrepancies are observed, probably due to finite size effects.…”

Section: Sandpile Modelsmentioning

confidence: 99%

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Renormalization approach to the self-organized critical behavior of sandpile models

1995

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We introduce a renormalization scheme of a type that is able to describe the self-organized critical state (SOC) of sandpile models. We have defined a characterization of the phase space that allows us to study the evolution of the dynamics under change of scale. In addition, a stationarity condition provides a feedback mechanism that drives the system to its critical state. We obtain an attractive fixed point in the phase space of the parameters that clarifies the self-organized critical nature of these systems. The universality class of several models is identified by studying the properties of the basin of attraction of this fixed point. We compute analytically the avalanche exponent 'T and the dynamical exponent z for sandpile models in d = 2. The values obtained are in very good agreement with computer simulations.The renormalization scheme can also be applied to study nonconservative sandpile models. The result is that the introduction of a dissipation parameter destroys the critical properties as suggested from simulations. The concept of self-organized criticality (SOC) has been invoked by Bak, Tang, and Wiesenfeld [1] as a possible unifying framework to describe a vast class of dynamically driven systems which evolve spontaneously toward a critical stationary state with no characteristic time or length scale. The name self-organized criticality comes from the fact that, unlike phase transitions in eqUilibrium statistical physics, the critical state is reached without the need to fine tune any control parameter; i.e., the critical state is an attractor of the dynamics. To illustrate the basic ideas of SOC, Bak, Tang, and Wiesenfeld used a cellular automaton inspired by the flow of avalanches in a pile of sand. In these models the sand is added grain by grain on a d-dimensional lattice until unstable sand (too large local slope of the pile) slides off. In this way the pile reaches a stationary critical state, characterized by a critical slope, in which additional grains of sand will fall off the pile via avalances that can be small or cover the entire size of the system. The critical state is characterized by a power law for the size and lifetime distribution of avalanches which therefore do not have a characteristic length scale. This class of automaton can be used to model many avalanche phenomena, interpreting the sand as energy, mechanical stress, heat memory, etc. One can add more examples from other fields like economy, social sciences and biology, as well [2].Other examples of SOC behavior can be found in fractal growth phenomena, such as diffusion limited aggregation (DLA) [3,4]. These models lead spontaneously to a statistically stationary state where a self-similar pattern is generated. Also in this case the critical state is an attractor for the dynamics, and it does not require any tuning parameter.1063-651X/95/51(3l1171.1(14)/S06.00 51The general conditions under which a physical system shows SOC properties have recently been discussed [5,6]. For a sandpile these conditions are identified in ...

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supporting

confidence: 80%

Section: Sandpile Modelsmentioning

confidence: 99%

Renormalization approach to the self-organized critical behavior of sandpile models

1995

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“…Actually, dissipation associated with SOC was already considered in [14,15] and more recently in [16]. There non-trivial power-laws in the power spectrum where found, strongly dependent on the dissipation coefficient.…”

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confidence: 99%

Universal1/fNoise from Dissipative Self-Organized Criticality Models

Rios

Zhang

1999

Phys. Rev. Lett.

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“…We provide a general scheme in which the details of different models are included via effective parameters and constraints. We mainly discuss sandpile models [2] with and without dissipation [7], but the formalism can be directly applied to other stochastic SOC models, such as the forest-fire model [8]. We find two independent critical parameters, i.e., relevant scaling fields, both with critical value equal to zero, and, just in this double limit, criticality is reached.…”

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confidence: 99%

Order Parameter and Scaling Fields in Self-Organized Criticality

1997

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We present a unified dynamical mean-field theory for stochastic self-organized critical models. We use a single site approximation, and we include the details of different models by using effective parameters and constraints. We identify the order parameter and the relevant scaling fields in order to describe the critical behavior in terms of the usual concepts of nonequilibrium lattice models with steady states. We point out the inconsistencies of previous mean-field approaches, which lead to different predictions. Numerical simulations confirm the validity of our results beyond mean-field theory.[ S0031-9007(97) The origin of scaling in Nature [1] has become in recent years a challenging problem in physics. Bak, Tang, and Wiesenfeld (BTW) [2] have proposed self-organized criticality (SOC) as a unifying theoretical framework to describe a vast class of driven systems that evolve "spontaneously" to a stationary state, characterized by power law distributions of dissipation events. Despite the insights SOC concepts have brought to a number of problems, an agreement on the precise definition of SOC has still not been reached, and the exact meaning of the word "spontaneous" is quite unclear. Originally, SOC was associated with the absence of tuning parameters, but it has been noted [3] that the driving rate acts as a tuning parameter in most, if not all, SOC models. This ambiguity has hindered the formulation of precise relations between SOC and other nonequilibrium critical phenomena [4,5].In this Letter we reformulate SOC in terms of typical concepts of nonequilibrium critical phenomena [5] by using the dynamical single site mean-field (MF) theory [6]. We provide a general scheme in which the details of different models are included via effective parameters and constraints. We mainly discuss sandpile models [2] with and without dissipation [7], but the formalism can be directly applied to other stochastic SOC models, such as the forest-fire model [8]. We find two independent critical parameters, i.e., relevant scaling fields, both with critical value equal to zero, and, just in this double limit, criticality is reached. We study the behavior of the order parameter and evaluate critical exponents. The results we obtain are in contrast with previous MF approaches [9,10]. This is due to a subtle inconsistency in the way critical parameters have been chosen in previous works. We show that some MF exponents are exact also in lowdimensional systems because of conservation laws. Our predictions are confirmed by numerical simulations of two-dimensional sandpile models.Sandpile models are cellular automata with an integer (or continuous) variable z i (energy) defined in a ddimensional lattice. At each time step an energy grain is added to a randomly chosen site, until the energy of a site reaches a threshold z c . When this happens the site relaxes ͑z i ! z i 2 z c ͒, and energy is transferred to the nearest neighbors ͑z j ! z j 1 y j ͒. For conservative models the transferred energy equals the energy lost by the relaxing...

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Cascades and self-organized criticality

Cited by 100 publications

References 16 publications

Renormalization approach to the self-organized critical behavior of sandpile models

Renormalization approach to the self-organized critical behavior of sandpile models

Universal1/fNoise from Dissipative Self-Organized Criticality Models

Order Parameter and Scaling Fields in Self-Organized Criticality

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