Abelian Manna model in three dimensions and below

Huynh, Hoai Nguyen; Pruessner, Gunnar

doi:10.1103/physreve.85.061133

Cited by 17 publications

(33 citation statements)

References 26 publications

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“…Tab. I shows a comparison between this result and the numerical values found by simulations [9,17,18] on lattices in dimensions d ∈ {1, 2, 3, 5}. While our estimate Eq.…”

Section: Critical Densitymentioning

confidence: 58%

“…Numerical simulations have established that a range of other models belong to the same universality class [10, p. 178], in particular the Oslo Model [11,12] and the conserved lattice gas [13,14]. The stationary density of the AMM has been estimated with very high precision on hypercubic lattices of dimensions d = 1 to d = 5 [9,[15][16][17][18]. Yet, theoretical understanding of the Manna Model is far from complete.…”

Section: Introductionmentioning

confidence: 99%

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Critical density of the Abelian Manna model via a multitype branching process

Wei

Pruessner

2019

Phys. Rev. E

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A multi-type branching process is introduced to mimic the evolution of the avalanche activity and determine the critical density of the Abelian Manna model. This branching process incorporates partially the spatio-temporal correlations of the activity, which are essential for the dynamics, in particular in low dimensions. An analytical expression for the critical density in arbitrary dimensions is derived, which significantly improves the results over mean-field theories, as confirmed by comparison to the literature on numerical estimates from simulations. The method can easily be extended to lattices and dynamics other than those studied in the present work.

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“…Tab. I shows a comparison between this result and the numerical values found by simulations [9,17,18] on lattices in dimensions d ∈ {1, 2, 3, 5}. While our estimate Eq.…”

Section: Critical Densitymentioning

confidence: 58%

Section: Introductionmentioning

confidence: 99%

Critical density of the Abelian Manna model via a multitype branching process

Wei

Pruessner

2019

Phys. Rev. E

Self Cite

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“…It is widely accepted that the BTW model has a multiscaling behaviour [16,17] due to its complete toppling balance [18] and positive auto-correlation in avalanche wave series [19,20], whereas the SSM does not show such correlation and consequently follows finite size scaling (FSS) ansatz. Recent numerical studies of the SSM have been carried out not only on various regular lattices of integer dimension but also on various kind of deterministic fractal lattices [21,22,23] and the results confirm the existence of robust FSS behaviour of the SSM across different regular, as well as fractal lattices though the universality class depends on the space or fractal dimension of these lattices. The fractal lattices considered for such studies were deterministic, the properties of SSM as well as BTW on random fractal lattices are yet to be studied.…”

Section: Introductionmentioning

confidence: 85%

“…Note that taking the measured values of the exponents the quantity D s (2 − τ s ) has the value 2.59 which is again within the error bar of the measured value of σ s (1). Recently, based on an extensive numerical study Huynh and Pruessner [22,23] proposed a relation among D s , d w and spatial dimension (d) as,…”

Section: Probability Distribution Functionmentioning

confidence: 99%

Critical properties of deterministic and stochastic sandpile models on two-dimensional percolation backbone

Bhaumik

Santra

2020

Physica A: Statistical Mechanics and its Applications

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Both the deterministic and stochastic sandpile models are studied on the percolation backbone, a random fractal, generated on a square lattice in 2-dimensions. In spite of the underline random structure of the backbone, the deterministic Bak Tang Wiesenfeld (BTW) model preserves its positive time auto-correlation and multifractal behaviour due to its complete toppling balance, whereas the critical properties of the stochastic sandpile model (SSM) still exhibits finite size scaling (FSS) as it exhibits on the regular lattices. Analyzing the topography of the avalanches, various scaling relations are developed. While for the SSM, the extended set of critical exponents obtained is found to obey various the scaling relation in terms of the fractal dimension d B f of the backbone, whereas the deterministic BTW model, on the other hand, does not. As the critical exponents of the SSM defined on the backbone are related to d B f , the backbone fractal dimension, they are found to be entirely different from those of the SSM defined on the regular lattice as well as on other deterministic fractals. The SSM on the percolation backbone is found to obey FSS but belongs to a new stochastic universality class.

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“…Not only SSM shows robust scaling behaviour than the deterministic BTW model, it is also able to explain certain experimentally observed avalanche behaviour [22]. SSM found to be one of the most studied models in various dimensions in SOC literature [23,24,25]. However, there are not many studies that report the scaling behaviour of SSM on SWN.…”

Section: Introductionmentioning

confidence: 93%

Stochastic sandpile model on small-world networks: Scaling and crossover

Bhaumik

Santra

2018

Physica A: Statistical Mechanics and its Applications

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A dissipative stochastic sandpile model is constructed on one and two dimensional smallworld networks with different shortcut densities φ where φ = 0 and 1 represent a regular lattice and a random network respectively. In the small-world regime (2 −12 ≤ φ ≤ 0.1), the critical behaviour of the model is explored studying different geometrical properties of the avalanches as a function of avalanche size s. For both the dimensions, three regions of s, separated by two crossover sizes s 1 and s 2 (s 1 < s 2 ), are identified analyzing the scaling behaviour of average height and area of the toppling surface associated with an avalanche. It is found that avalanches of size s < s 1 are compact and follow Manna scaling on the regular lattice whereas the avalanches with size s > s 1 are sparse as they are on network and follow mean-field scaling. Coexistence of different scaling forms in the small-world regime leads to violation of usual finite-size scaling, in contrary to the fact that the model follows the same on the regular lattice as well as on the random network independently. Simultaneous appearance of multiple scaling forms are characterized by developing a coexistence scaling theory. As SWN evolves from regular lattice to random network, a crossover from diffusive to super-diffusive nature of sand transport is observed and scaling forms of such crossover is developed and verified.

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Abelian Manna model in three dimensions and below

Cited by 17 publications

References 26 publications

Critical density of the Abelian Manna model via a multitype branching process

Critical density of the Abelian Manna model via a multitype branching process

Critical properties of deterministic and stochastic sandpile models on two-dimensional percolation backbone

Stochastic sandpile model on small-world networks: Scaling and crossover

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