A Stochastic Variant of the Abelian Sandpile Model

Journal of Statistical Physics - We introduce a natural stochastic extension, called SSP, of the abelian sandpile model (ASM), which shares many mathematical properties with ASM, yet radically...     

Sandpile Dynamics Driven by Degree on Scale-Free Networks

We introduce a sandpile model driven by degree on scale-free networks, where the perturbation is triggered at nodes with the same degree. We numerically investigate the avalanche behaviour of sandpile driven by different degrees on scale-free networks. It is observed that the avalanche area has the same behaviour with avalanche size. When the sandpile is driven at nodes with the minimal degree, the avalanches of our model behave similarly to those of the original Bak-Tang-Wiesenfeld （BTW） model on scale-free networks. As the degree of driven nodes increases from the minimal value to the maximal value, the avalanche distribution gradually changes from a clean power law, then a mixture of Poissonian and power laws, finally to a Poisson-like distribution. The average avalanche area is found to increase with the degree of driven nodes so that perturbation triggered on higher-degree nodes will result in broader spreading of avalanche propagation.     

Landslides, forest fires, and earthquakes: examples of self-organized critical behavior

Per Bak conceived self-organized criticality as an explanation for the behavior of the sandpile model. Subsequently, many cellular automata models were found to exhibit similar behavior. Two examples are the forest-fire and slider-block models. Each of these models can be associated with a serious natural hazard: the sandpile model with landslides, the forest-fire model with actual forest fires, and the slider-block model with earthquakes. We examine the noncumulative frequency–area statistics for each natural hazard, and show that each has a robust power-law (fractal) distribution. We propose an inverse-cascade model as a general explanation for the power-law frequency–area statistics of the three cellular-automata models and their ‘associated’ natural hazards.     

Avalanche Dynamics in Quenched Random Directed Sandpile Models

We numerically investigate the quenched random directed sandpile models which are local, conservative and Abelian. A local flow balance between the outflow of grains during a single toppling at a site and the total number of grains flowing into the same site plays an important role when all the nearest-neighbouring sites of the above-mentioned site topple for once. The quenched model has the same critical exponents with the Abelian deterministic directed sandpile model when the local flow balance exists, otherwise the critical exponents of this quenched model and the annealed Abelian random directed sandpile model are the same. These results indicate that the presence or absence of this local flow balance determines the universality class of the Abelian directed sandpile model.     

颗粒堆内微观力学结构的离散元模拟研究

将离散单元法应用到三维堆积过程的模拟计算,探讨了滑动摩擦及滚动摩擦对堆积形成的影响,得到了颗粒堆内部的应力分布规律,发现颗粒堆的形态是由滑动摩擦和滚动摩擦共同决定的,在堆内颗粒间的作用力基本呈树状结构.在模拟得到的颗粒堆中出现了应力分布奇异现象,在堆积角较大的情况下,颗粒堆与地面间作用力的最大值常发生在距堆底中心不远的环状区域,而并非发生在堆底的中心;在堆积角相对较小时颗粒堆与地面间作用力的最大值较容易发生在堆底的中心.对于一个颗粒堆,具体会发生哪种受力情况具有一定的偶然性. 关键词： 堆积 离散单元法 计算颗粒力学     

Corrections to scaling and probability distribution of avalanches for the stochastic Zhang sandpile model

We study the distributions of dissipative and nondissipative avalanches separately in the stochastic Zhang (SP-Z) sandpile in two dimension. We find that dissipative and nondissipative avalanches obey simple power laws and do not have the logarithmic correction, while the avalanche distributions in the Abelian Manna model should include a logarithmic correction. We use the moment analysis to determine the numerical critical exponents of dissipative and nondissipative avalanches, respectively, and find that they are different from the corresponding values in the Abelian Manna model. All these indicate that the stochastic Zhang model and the Abelian Manna model belong to distinct universality classes, which imply that the Abelian symmetry breaking changes the scaling behavior of the avalanches in the case of the stochastic sandpile model.     

ONE-DIMENSIONAL SANDPILE MODEL WITH STOCHASTIC SLIDE

A new kind of theoretical one-dimensional sandpile model is proposed. In contrast to the models studied previously, the sliding process in this model is assumed to be of stochastic nature. Numerical simulations show that the behavior of this model is apparently closer to the reality of true sandpile than the models considered previously. The universality and sealing of this model is also discussed.     

Stabilization of avalanche processes on dynamical networks

The stabilization of avalanches on dynamical networks has been studied. Dynamical networks are networks where the structure of links varies in time owing to the presence of the individual “activity” of each site, which determines the probability of establishing links with other sites per unit time. An interesting case where the times of existence of links in a network are equal to the avalanche development times has been examined. A new mathematical model of a system with the avalanche dynamics has been constructed including changes in the network on which avalanches are developed. A square lattice with a variable structure of links has been considered as a dynamical network within this model. Avalanche processes on it have been simulated using the modified Abelian sandpile model and fixed-energy sandpile model. It has been shown that avalanche processes on the dynamical lattice under study are more stable than a static lattice with respect to the appearance of catastrophic events. In particular, this is manifested in a decrease in the maximum size of an avalanche in the Abelian sandpile model on the dynamical lattice as compared to that on the static lattice. For the fixed-energy sandpile model, it has been shown that, in contrast to the static lattice, where an avalanche process becomes infinite in time, the existence of avalanches finite in time is always possible.     

Universality Class in Abelian Sandpile Models with Stochastic Toppling Rules

We present a stochastic critical slope sandpile model, where the amount of grains that fall in an overturning event is stochastic variable. The model is local, conservative, and Abelian. We apply the moment analysis to evaluate critical exponents and finite size scaling method to consistently test the obtained results. Numerical results show that this model, Oslo model, and one-dimensional Abelian Manna model have the same critical behavior although the three models have different stochastic toppling rules, which provides evidences suggesting that Abelian sandpile models with different stochastic toppling rules are in the same universality class.     

Minimal Sandpiles on Hexagonal Lattice

We study the minimal recurrent configurations of the Abelian sandpile model on the hexagonal lattice referred to the dynamics of a nonconservative sandpile model. The one-to-one correspondence between these configurations and the set of maximally oriented spanning trees on the triangular sublattice is constructed. We derive the correlation functions in minimal recurrent configurations on a quasi-one-dimensional 2 × N lattice, compare them with correlations for ordinary recurrent configurations, and argue for asymptotic equivalence between them.     

Universality Class in Abelian Sandpile Models with Stochastic Toppling Rules

We present a stochastic critical slope sandpile model, where the amount of grains that fall in an overturning event is stochastic variable. The model is local, conservative, and Abelian. We apply the moment analysis to evaluate critical exponents and finite size scaling method to consistently test the obtained results. Numerical results show that this model, Oslo model, and one-dimensional Abelian Manna model have the same critical behavior although the three models have different stochastic toppling rules, which provides evidences suggesting that Abelian sandpile models with different stochastic toppling rules are in the same universality class.     

The Computational Complexity of Sandpiles

Journal of Statistical Physics - Given an initial distribution of sand in an Abelian sandpile, what final state does it relax to after all possible avalanches have taken place? In d≥3, we...     

Exactly solvable sandpile with fractal avalanching

A simple one-dimensional sandpile model is constructed which possesses exact analytical solvability while displaying both scale-free behavior and fractal properties. The sandpile grows by avalanching on all scales, yet its shape and energy content are described by a simple, continuous (but nowhere differentiable) analytical formula. The avalanche energy distribution and the avalanche time series are both power laws with index -1 ("1/f spectra").     

Structure of two-dimensional sandpile. I. Height probabilities

The height probabilities of the two-dimensional Abelian sandpile model are the fractionial numbers of lattice sites having heights 1, 2, 3, 4. A combinatorial method for evaluation of these quantities is proposed. The method is based on mapping the set of allowed sandpile configurations onto the set of spanning trees covering a given lattice. Exact analytical expressions for all probabilities are obtained.     

Precise toppling balance, quenched disorder, and universality for sandpiles

A single sandpile model with quenched random toppling matrices captures the crucial features of different models of self-organized criticality. With symmetric matrices avalanche statistics falls in the multiscaling Bak-Tang-Wiesenfeld universality class. In the asymmetric case the simple scaling of the Manna model is observed. The presence or absence of a precise toppling balance between the amount of sand released by a toppling site and the total quantity the same site receives when all its neighbors topple once determines the appropriate universality class.     

Intermittent criticality in the long-range connective sandpile (LRCS) model

We here propose a long-range connective sandpile model with variable connection probability Pc which has an important impact on the slope of the power-law frequency-size distribution of avalanches. The long-range connection probability Pc is changed according to an explicit function of the size of the latest event, although the evolution rule of Pc may be different in various physical systems. Such version of the sandpile model demonstrates large fluctuations in the dynamical variable 〈Z〉(t) (the spatially averaged amount of grains retained within the grid at each time step), indicating the state of intermittent criticality in the system. Many researches have suggested that the earthquake fault system is an intermittent criticality system, which would imply some level of statistical predictability of great events. Our modified sandpile model thus provides a testing ground for many proposed precursory measures related to great earthquakes.     

Freezing Transitions in Non-Fellerian Particle Systems

Non-Fellerian particle systems are characterized by nonlocal interactions, somewhat analogous to non-Gibbsian distributions. They exhibit new phenomena that are unseen in standard interacting particle systems. We consider freezing transitions in one-dimensional non-Fellerian processes which are built from the abelian sandpile additions to which in one case, spin flips are added, and in another case, so called anti-sandpile subtractions. In the first case and as a function of the sandpile addition rate, there is a sharp transition from a non-trivial invariant measure to the trivial invariant measure of the sandpile process. For the combination sandpile plus anti-sandpile, there is a sharp transition from one frozen state to the other anti-state.     

Abelian Sandpiles and the Harmonic Model

We present a construction of an entropy-preserving equivariant surjective map from the d-dimensional critical sandpile model to a certain closed, shift-invariant subgroup of \mathbbT^\mathbbZd{\mathbb{T}^{\mathbb{Z}^d}} (the ‘harmonic model’). A similar map is constructed for the dissipative abelian sandpile model and is used to prove uniqueness and the Bernoulli property of the measure of maximal entropy for that model.     

Asymmetric abeiian sandpile models

In the Abelian sandpile models introduced by Dhar, long-time behavior is determined by an invariant measure supported uniformly on a set of implicitly defined recurrent configurations of the system. Dhar proposed a simple procedure, theburning algorithm, as a possible test of whether a configuration is recurrent, and later with Majumdar verified the correctness of this test when the toppling rules of the sandpile are symmetric. We observe that the test is not valid in general and give a new algorithm which yields a test correct for all sandpiles; we also obtain necessary and sufficient conditions for the validity of the original test. The results are applied to a family of deterministic one-dimensional sandpile models originally studied by Lee, Liang, and Tzeng.     

Precursory phenomena associated with large avalanches in the long-range connective sandpile (LRCS) model

Reduction in b-values before a large earthquake is a very popular topic for discussion. This study proposes an alternative sandpile model being able to demonstrate reduction in scaling exponents before large events through adaptable long-range connections. The distant connection between two separated cells was introduced in the sandpile model. We found that our modified long-range connective sandpile (LRCS) system repeatedly approaches and retreats from a critical state. When a large avalanche occurs in the LRCS model, accumulated energy dramatically dissipates and the system simultaneously retreats from criticality. The system quickly approaches the critical state accompanied by the increase in the slopes of the power-law frequency-size distributions of events. Afterwards, and most interestingly, the power-law slope declines before the next large event. The precursory b-value reduction before large earthquakes observed from earthquake catalogues closely mimics the evolution in power-law slopes for the frequency-size distributions of events derived in the LRCS models. Our paper, thus, provides a new explanation for declined b-values before large earthquakes.