Large Scale Structures, Symmetry, and Universality in Sandpiles

Hughes, David; Paczuski, Maya

doi:10.1103/physrevlett.88.054302

Cited by 33 publications

(29 citation statements)

References 14 publications

(29 reference statements)

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“…More recently, a number of works have focused on certain variants of abelian models: they include the random distribution of toppling grains onto a restricted number of neighboring sites [6][7][8], and the complete toppling of all grains from an unstable site [9][10][11]. The first modification does keep the models in the abelian class, and an exact solution for the probability distribution function of events (PDF) has been discussed.…”

Section: Introductionmentioning

confidence: 99%

Exact solution for the self-organized critical rainfall model

Andrade

2003

Braz. J. Phys.

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This work presents an analytical investigation for a Self-Organized Criticality abelian model that describes basic properties of rainfall phenomena. The knowledge of the exact solution for the probability that a site topples when mass is added to any other site of the lattice leads to a large number properties of the model, including the exponent of the power law that describes presence of the events as function of their magnitude. It is shown that the model belongs to the same universality class of a first model proposed by Dhar and Ramaswamy (DR). However, for finite size lattices, it is found that its exponent is larger than that one for the DR model. I IntroductionAbelian sand pile models (ASM) [1] became quite important for the understanding of basic properties of Self-Organized Criticality (SOC) [2,3]. They satisfy the remarkable property that the final state of the model, after subsequent addition of grains to any two sites, is independent of the order in which the grains have been added.The first analyses by Dhar and Ramaswamy [4] considered a model (DR) based on a critical height criterion for toppling, i.e., a site becomes unstable and topples when its amount of mass exceeds a threshold value m th . In this model, a site topples to exactly d distinct neighbors. This is a directed and deterministic model as, i) the toppling rules breaks the isotropy of the lattice and avalanches develop along one direction; and ii) a deterministic rule indicates the fixed number of grains that each site receives when an unstable site topples. These two properties impose the condition that each site topples only once during an avalanche.Abelian models constitute a special set where analytical investigation has lead to exact results, what are still rare subjects in the SOC landscape. More recently, a number of works have focused on certain variants of abelian models: they include the random distribution of toppling grains onto a restricted number of neighboring sites [6][7][8], and the complete toppling of all grains from an unstable site [9][10][11]. The first modification does keep the models in the abelian class, and an exact solution for the probability distribution function of events (PDF) has been discussed. The properties of such models are distinct from those with deterministic toppling rules. On the other hand, the second modification breaks the commutativity of the models, that seem to be nonintegrable.Other SOC models, that were initially proposed based on the so-called gradient condition for toppling, as the Bak-Tang-Wiesenfeld (BTW) model [2], also belong to this class, provided the variables are conveniently re-interpreted. But, as far as they allow for multiple toppling for a single avalanche, they are not exactly integrable.The exact solution for the DR model is based on the evaluation of the probability distribution function P (s 1 ; s 0 ). It measures the probability that a site s 1 topples when one grain is added in a site s 0 , and takes into account all configurations of the lattice which satisfies th...

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Section: Introductionmentioning

confidence: 99%

Exact solution for the self-organized critical rainfall model

Andrade

2003

Braz. J. Phys.

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“…Notice that, although other randomneighbor versions of sandpile models have been proposed during last two decades, e.g. [3][4][5][6][7][8][9], none of them was self-adaptive in the sense of random connection probability. We have realized [11][12][13] that self-adapted P c plays a crucial role to the intermittency in the LRCS model.…”

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confidence: 98%

“…Since the original nearest-neighboring sandpile model was introduced by Bak et al [1,2], various numerical and analytical studies of modified sandpile models have been a considerable subject of researches, e.g. [3][4][5][6][7][8][9][10][11]. Among them, the annealed random-neighbor sandpile models where an avalanche can propagate within the system were first (perhaps) proposed by Christensen and Olami [4] and then extensively studied on a long-range connected (small-world) network by, for example, de Arcangelis and Herrmann [7], Lahtinen et al [9], and Chen et al [10,11].…”

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confidence: 99%

Transition on the relationship between fractal dimension and Hurst exponent in the long-range connective sandpile models

et al. 2011

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“…The first-principle analysis of these models is complicated by the involvement of higher-order statistical moments (Vespignani & Zapperi 1998;Sethna et al 2001) requiring full dynamic renormalization group treatment (Chang 1992). If the behavior of solar corona was fully analogous to such models, the free energy landscape in the corona would be solely formed by preceding flaring events independently of the driver, as exemplified by non-Abelian sandpile models (Hughes & Paczuski 2002;Uritsky et al 2004). …”

Section: Routes To Multiscale Dissipationmentioning

confidence: 99%

Stochastic Coupling of Solar Photosphere and Corona

Uritsky¹,

Davila²,

Ofman³

et al. 2013

ApJ

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The observed solar activity is believed to be driven by the dissipation of nonpotential magnetic energy injected into the corona by dynamic processes in the photosphere. The enormous range of scales involved in the interaction makes it difficult to track down the photospheric origin of each coronal dissipation event, especially in the presence of complex magnetic topologies. In this paper, we propose an ensemble-based approach for testing the photosphere-corona coupling in a quiet solar region as represented by intermittent activity in Solar and Heliospheric Observatory Michelson Doppler Imager and Solar TErrestrial RElations Observatory Extreme Ultraviolet Imager image sets. For properly adjusted detection thresholds corresponding to the same degree of intermittency in the photosphere and corona, the dynamics of the two solar regions is described by the same occurrence probability distributions of energy release events but significantly different geometric properties. We derive a set of scaling relations reconciling the two groups of results and enabling statistical description of coronal dynamics based on photospheric observations. Our analysis suggests that multiscale intermittent dissipation in the corona at spatial scales >3 Mm is controlled by turbulent photospheric convection. Complex topology of the photospheric network makes this coupling essentially nonlocal and non-deterministic. Our results are in an agreement with the Parker's coupling scenario in which random photospheric shuffling generates marginally stable magnetic discontinuities at the coronal level, but they are also consistent with an impulsive wave heating involving multiscale Alfvénic wave packets and/or magnetohydrodynamic turbulent cascade. A back-reaction on the photosphere due to coronal magnetic reconfiguration can be a contributing factor.

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Large Scale Structures, Symmetry, and Universality in Sandpiles

Cited by 33 publications

References 14 publications

Exact solution for the self-organized critical rainfall model

Exact solution for the self-organized critical rainfall model

Transition on the relationship between fractal dimension and Hurst exponent in the long-range connective sandpile models

Stochastic Coupling of Solar Photosphere and Corona

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