Universal versus drive-dependent exponents for sandpile models

Nakanishi, Hiizu; Sneppen, Kim

doi:10.1103/physreve.55.4012

Cited by 33 publications

(47 citation statements)

References 23 publications

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“…(7) 2.73(7) 3.36 (7) 4.00 (7) 2.78 (7) 3.03 (7) The last unexpected conclusion pertains to the relation between the dimensions d g and d w , and the critical exponent D. We found that they obey the general linear relationship: D = αd g + βd w + γ. This was uncovered by plotting the following six points in 3D parameter space: [15], the square [7,17], the cube [18,19], the hypercube [20], the arrowhead and the crab lattices respectively. It is interesting that while the first four points due to the hypercubic lattices are found to lie on a straight line, all six points together make up a plane instead of a tetrahedron.…”

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confidence: 99%

Abelian Manna model on two fractal lattices

2010

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We analyze the avalanche size distribution of the Abelian Manna model on two different fractal lattices with the same dimension dg = ln 3/ ln 2, with the aim to probe for scaling behavior and to study the systematic dependence of the critical exponents on the dimension and structure of the lattices. We show that the scaling law D(2 − τ ) = dw generalizes the corresponding scaling law on regular lattices, in particular hypercubes, where dw = 2. Furthermore, we observe that the lattice dimension dg, the fractal dimension of the random walk on the lattice dw, and the critical exponent D, form a plane in 3D parameter space, i.e. they obey the linear relationship D = 0.632(3)dg + 0.98(1)dw − 0.49(3). Although extensive research has been performed on self-organized criticality [1] for models on hypercubic lattices, far less work has been done on fractal lattices [2,3]. It remains somewhat unclear what to conclude from the latter studies. Fractal lattices are important for the understanding of critical phenomena for a number of reasons. Firstly, results for critical exponents in lattices with non-integer dimensions might provide a means to determine the terms of their = 4 − d expansion. Secondly, fractal lattices are particularly suitable for a real space renormalization group procedures, in particular that by . Thirdly, scaling relations that are derived in a straightforward fashion on hypercubic lattices can be put to test in a more general setting. In this Brief Report, we address the first and the third aspect, by examining both numerically and analytically the scaling behaviour of the Abelian version of the Manna model [7][8][9] on two different fractal lattices.The fractal lattices used in this study are generated from the arc-fractal system [10]. The lattice sites are the invariant set of points of the arc-fractal. We consider nearest neighbor interactions among sites. Here, the nearest neighbors of a given site are all sites which have the (same) shortest Euclidean distance to it. Our fractal lattices have no natural boundary; instead, they have only two end points at which two copies can join to form a bigger lattice. The dimension of the lattices is the same as the arc-fractal that generates them.In this study, we shall consider two fractal lattices: the Sierpinski arrowhead and the crab (see Fig. 1). The former is named "Sierpinski arrowhead" because it is the same as the well-known Sierpinski arrowhead [11], whereas the latter is termed "crab" because the overall shape of the generated lattice looks like a crab. These fractal lattices are generated through the arc-fractal system with number of segments n = 3 and opening angle of the arc α = π. For the Sierpinski arrowhead, the rule for orientating the arc at each iteration is "in-out-in", while the rule is "out-in-out" for the crab. Both lattices have the same dimension d g = ln 3/ ln 2 ≈ 1.58. The total number of sites on the lattice at the i-th iteration is N i = 3 i + 1. The coordination number of sites on these lattices varies between two and three. Asy...

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Abelian Manna model on two fractal lattices

2010

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“…A simple one dimensional "Oslo" model was proposed to mimic these experiments, and numerical simulations showed that the model exhibits SOC with dispersive transport [3]. This model was subsequently shown to represent a large universality class of avalanche phenomena including interface depinning, a slip-stick model for earthquakes [4], and maybe other sandpile models [5].Although it is known that SOC can be reached only if the driving rate is very low, very little is known about the transition out of SOC as the driving rate is increased. In particular, for a given system size, there will be some driving rate at which the motion in the system never stops and the avalanches become infinite, signaling a new type of behavior which may or may not be related to the avalanche regime.…”

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Avalanche Merging and Continuous Flow in a Sandpile Model

Corral

Paczuski

1999

Phys. Rev. Lett.

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A dynamical transition separating intermittent and continuous flow is observed in a sandpile model, with scaling functions relating the transport behaviors between both regimes. The width of the active zone diverges with system size in the avalanche regime but becomes very narrow for continuous flow. The change of the mean slope, ∆z, on increasing the driving rate, r, obeys ∆z ∼ r 1/θ . It has nontrivial scaling behavior in the continuous flow phase with an exponent θ given, paradoxically, only in terms of exponents characterizing the avalanches θ = (1 + z − D)/(3 − D).PACS numbers: 64.60. Ht, 45.70.Ht, 05.65.+b Granular systems can exhibit continuous flow or intermittent avalanches depending on the driving rate. Experiments on rice piles have demonstrated that when grains are slowly added, transport of grains through the pile takes place in terms of avalanches of all sizes [1]; these rice piles exhibit self-organized criticality (SOC) [2]. Additional experiments have established that the transport process is dispersive, in the sense that the transit times of grains through the pile are broadly distributed [3]. A simple one dimensional "Oslo" model was proposed to mimic these experiments, and numerical simulations showed that the model exhibits SOC with dispersive transport [3]. This model was subsequently shown to represent a large universality class of avalanche phenomena including interface depinning, a slip-stick model for earthquakes [4], and maybe other sandpile models [5].Although it is known that SOC can be reached only if the driving rate is very low, very little is known about the transition out of SOC as the driving rate is increased. In particular, for a given system size, there will be some driving rate at which the motion in the system never stops and the avalanches become infinite, signaling a new type of behavior which may or may not be related to the avalanche regime. Previously, Tang and Bak [6] measured the increase in average height in the BTW sandpile model [2] as the driving rate was increased. They found a power law behavior (at high rates) for the average height vs. driving rate, but did not study the system at low rates (see in addition Ref. [7]). Here we show, using the Oslo model, that there is an abrupt change in transport behaviors distinguishing two regimes, an avalanche regime and a continuous flow phase, with a different critical exponent θ for each one. This change is associated with a pronounced contraction in the width of the active zone of transport. We utilize the active zone behavior to relate the scaling coefficients in the continuous flow phase to the exponents characterizing the avalanches in the SOC state.The Oslo model is defined as follows: In a one dimensional system of size L, an integer variable h(x) gives the height of the pile at position x, and z(x) = h(x)−h(x+1) is the local slope. Grains are dropped at x = 1 with the opposite boundary open, i.e., h(L+1) ≡ 0. At each time step, all sites are tested for stability. Each unstable site x with z(x) > z c (x) toppl...

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“…One of the most intriguing questions concerns the classes of universality of these models. There are several attempts to shed light on this problem [2,3]. Recently, Ben-Hur and Biham [2] proposed a classification scheme for the 2d models both stochastic and deterministic.…”

Section: Introductionmentioning

confidence: 99%

“…They found that the original Bak, Tang, Wiesenfeld (BTW) model [1] belongs to the universality class of undirected models, directed models form a separate class, and the two-state Manna model [4] belongs to the universality class of random relaxation models. Later on, Nakanishi and Sneppen [3] examined several 1d sandpile models and suggested that the two-state Manna model and rice pile model [5,6] belong to the same universality class.…”

Section: Introductionmentioning

confidence: 99%

Self-organizing height-arrow model: numerical and analytical results

Shcherbakov

Turcotte

2000

Physica A: Statistical Mechanics and its Applications

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The recently introduced self-organizing height-arrow (HA) model is numerically investigated on the square lattice and analytically on the Bethe lattice. The concentration of occupied sites and critical exponents of distributions of avalanches are evaluated for two slightly different versions of the model. The obtained exponents for distributions of avalanches by mass, area, duration and appropriate fractal dimensions are close to those for the BTW model, which suggests that the HA model belongs to the same universality class. For comparison, the concentration of occupied sites in the HA model is calculated exactly on the Bethe lattice of coordination number q = 4 as well. PACS number(s): 05.70. Ln, 05.40.+j, Typeset using

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Universal versus drive-dependent exponents for sandpile models

Cited by 33 publications

References 23 publications

Abelian Manna model on two fractal lattices

Abelian Manna model on two fractal lattices

Avalanche Merging and Continuous Flow in a Sandpile Model

Self-organizing height-arrow model: numerical and analytical results

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