Bethe lattice representation for sandpiles

Physica A: Statistical Mechanics and its Applications, 2002

We study the simple two-dimensional height sandpile model. The action of adding one grain of sand to a randomly chosen node as the ÿrst, we calculate the step-by-step probability of occurrence of avalanches of a given size. An avalanche determination method is devised that allows to determine exactly the di erent avalanches that can take place and the corresponding probabilities of occurrence in each step of the evolution of the sandpile.

Physical review. E, Statistical, nonlinear, and soft matter physics, 2002

The probability distribution function of the avalanche size in the sandpile model does not verify strict self-similarity under changes of the sandpile size. Here we show the existence of avalanches with different space-time structure, and each type of avalanche has a different scaling with the sandpile size. This is the main cause of the lack of self-similarity of the probability distribution function of the avalanche sizes, although the boundary effects can also play a role.

Physical Review E, 2008

Stochastic sandpiles self-organize to a critical point with scaling behavior different from directed percolation (DP) and characterized by the presence of an additional conservation law. This is usually called C-DP or Manna universality class. There remains, however, an exception to this universality principle: a sandpile automaton introduced by Maslov and Zhang, which was claimed to be in the directed percolation class despite of the existence of a conservation law. In this paper we show, by means of careful numerical simulations as well as by constructing and analyzing a field theory, that (contrarily to what previously thought) this sandpile is also in the C-DP or Manna class. This confirms the hypothesis of universality for stochastic sandpiles, and gives rise to a fully coherent picture of self-organized criticality in systems with a conservation law. In passing, we obtain a number of results for the C-DP class and introduce a new strategy to easily discriminate between DP and C-DP scaling.

Lecture Notes in Computer Science, 2011

Sand pile models are dynamical systems emphasizing the phenomenon of Self Organized Criticality (SOC). From N stacked grains, iterating evolution rules leads to some critical configuration where a small disturbance has deep consequences on the system, involving numerous steps of grain fall. Physicists L. Kadanoff et al inspire KSPM, a model presenting a sharp SOC behavior, extending the well known Sand Pile Model. In KSPM with parameter D we start from a pile of N stacked grains and apply the rule: D −1 grains can fall from column i onto the D − 1 adjacent columns to the right if the difference of height between columns i and i+1 is greater or equal to D. We propose an iterative study of KSPM evolution where one single grain addition is repeated on a heap of sand. The sequence of grain falls following a single grain addition is called an avalanche. From a certain column precisely studied for D = 3, we provide a plain process describing avalanches.

Physica A: Statistical Mechanics and its Applications, 2004

Avalanche dynamics is an indispensable feature of complex systems. Here we study the self-organized critical dynamics of avalanches on scale-free networks with degree exponent γ through the Bak-Tang-Wiesenfeld (BTW) sandpile model. The threshold height of a node i is set as k 1−η i with 0 ≤ η < 1, where k i is the degree of node i. Using the branching process approach, we obtain the avalanche size and the duration distribution of sand toppling, which follow power-laws with exponents τ and δ, respectively. They are given as τ = (γ−2η)/(γ− 1 − η) and δ = (γ − 1 − η)/(γ − 2) for γ < 3 − η, 3/2 and 2 for γ > 3 − η, respectively. The power-law distributions are modified by a logarithmic correction at γ = 3 − η.

Physical Review E, 2000

We study a directed stochastic sandpile model of Self-Organized Criticality, which exhibits recurrent, multiple topplings, putting it in a separate universality class from the exactly solved model of Dhar and Ramaswamy. We show that in the steady-state all stable states are equally likely. Then we explicitly derive a discrete dynamical equation for avalanches on the lattice. By coarse-graining we arrive at a continuous Langevin equation for the propagation of avalanches and calculate all the critical exponents characterizing them. The avalanche equation is similar to the Edwards-Wilkinson equation, but with a noise amplitude that is a threshold function of the local avalanche activity, or interface height, leading to a stable absorbing state when the avalanche dies. It represents a new type of absorbing state phase transition.

Physical Review E, 2004

The Oslo sandpile model, or if one wants to be precise, ricepile model, is a cellular automaton designed to model experiments on granular piles displaying self-organized criticality. We present an analytic treatment that allows the calculation of the transition probabilities between the different configurations of the system; from here, using the theory of Markov chains, we can obtain the stationary occupation distribution, which tell us that the phase space is occupied with probabilities that vary in many orders of magnitude from one state to another. Our results show how the complexity of this simple model is built as the number of elements increases, and allows, for a given system size, the exact calculation of the avalanche size distribution and other properties related to the profile of the pile.

Physical Review Letters, 1998

We use a phenomenological field theory, reflecting the symmetries and conservation laws of sandpiles, to compare the driven dissipative sandpile, widely studied in the context of self-organized criticality, with the corresponding fixed-energy model. The latter displays an absorbing-state phase transition with upper critical dimension d c = 4. We show that the driven model exhibits a fundamentally different approach to the critical point, and compute a subset of critical exponents. We present numerical simulations in support of our theoretical predictions. PACS numbers: 64.60.Lx, 05.40.+j, 05.70.Ln Typeset using REVT E X 1 A wide variety of nonequilibrium systems display transitions between "active" and "absorbing" states: examples are epidemic processes [1] catalysis [2], directed percolation (DP) [3], and the depinning of interfaces in quenched disorder [4]. When driven continuously, such systems may exhibit stick-slip instabilities, or broadly distributed avalanches, commonly associated with self-organized criticality (SOC) [5,6]. SOC sandpiles [5] possess an infinite number of absorbing configurations (i.e., from which the system cannot escape), and are placed, by definition, at the critical point in a twodimensional parameter space [7,8] resembling that of directed percolation (DP) [3] or contact processes [9-11].

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics, 1996

We introduce and study numerically a directed two-dimensional sandpile automaton with probabilistic toppling (probability parameter p) which provides a good laboratory to study both self-organized criticality and the far-from-equilibrium phase transition. In the limit p = 1 our model reduces to the critical height model in which the self-organized critical behavior was found by exact solution [D. Dhar and R. Ramaswamy, Phys. Rev. Lett. 63, 1659 (1989)]. For 0 < p < 1 metastable columns of sand may be formed, which are relaxed when one of the local slopes exceeds a critical value σc. By varying the probability of toppling p we find that a continuous phase transition occurs at the critical probability pc, at which the steady states with zero average slope (above pc) are replaced by states characterized by a finite average slope (below pc). We study this phase transition in detail by introducing an appropriate order parameter and the order-parameter susceptibility χ. In a certain range of p < 1 we find the self-organized critical behavior which is characterized by nonuniversal p−dependent scaling exponents for the probability distributions of size and length of avalanches. We also calculate the anisotropy exponent ζ and the fractal dimension d f of relaxation clusters in the entire range of values of the toppling parameter p. We show that the relaxation clusters in our model are anisotropic and can be described as fractals for values of p above the transition point. Below the transition they are isotropic and compact.

The IMA Volumes in Mathematics and its Applications, 2000

To study the self{organization of systems, their approach t o w ards a critical state, and the statistical properties at criticality, so{called mathematical sandpiles have been suggested. In this paper we analyze elementary properties of a slope{based one{dimensional model, for which one boundary is an abyss, the other is a wall. Our analysis is based on properties of the Markov matrix. Some numerical results for sandpiles with small lattice sizes are also included.

Physical Review E, 1995

We study the Abelian sandpile model on decorated one-dimensional chains. We show that there are two types of avalanches, and determine the effects of finite, though large, system size I on the asymptotic form of distributions of avalanche sizes, and show that these differ qualitatively from the behavior on a simple linear chain. For large L, we find that the probability distribution of the total number of topplings 8 is not described by a simple finite-size scaling form, but by a linear combination of two simple scaling forms: ProbL, (s) = z fi(z) + b f2(~z), where fi and f2 are nonuniversal scaling functions of one argument.

Physical Review E, 1999

We study numerically scaling properties of the distribution of cumulative energy dissipated in an avalanche and the dynamic phase transition in a stochastic directed cellular automaton [B. Tadić and D. Dhar, Phys. Rev. Lett. 79, 1519 (1997)] in d = 1 + 1 dimensions. In the critical steady state occurring for the probability of toppling p ≥ p ⋆ =0.70548, the dissipated energy distribution exhibits scaling behavior with new scaling exponents τE and DE for slope and cutoff energy, respectively, indicating that the sandpile surface is a fractal. In contrast to avalanche exponents, the energy exponents appear to be p-dependent in the region p ⋆ ≤ p < 1, however the product (τE − 1)DE remains universal. We estimate the roughness exponent of the transverse section of the pile as χ = 0.44 ± 0.04. Critical exponents characterizing the dynamic phase transition at p ⋆ are obtained by direct simulation and scaling analysis of the survival probability distribution and the average outflow current. The transition belongs to a new universality class with the critical exponents ν = γ = 1.22±0.02, β = 0.56±0.02 and ν ⊥ = 0.761±0.029, with apparent violation of hyperscaling. Generalized hyperscaling relation leads to β+β ′ = (d−1)ν ⊥ , where β ′ = 0.195±0.012 is the exponent governed by the ultimate survival probability.

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics, 2000

Time series resulting from wave decomposition show the existence of different correlation patterns for avalanche dynamics. For the d=2 Bak-Tang-Wisenfeld model, long range correlations determine a modification of the wave size distribution under coarse graining in time, and multifractal scaling for avalanches. In the Manna model, the distribution of avalanche coincides with that of waves, which are uncorrelated and obey finite size scaling, a result expected also for the d=3 Bak-Tang-Wiesenfeld sandpile.

Physica A: Statistical Mechanics and its Applications, 2000

We have described a lattice model of a sandpile that includes a coupling between evolving granular structures and dynamic responses. The coupling manifests as a granular memory. We have illustrated the role of memory by observing avalanches in a (three)-dimensional model sandpile. There are two distinct classes of dynamic events that depend on the process history and mimic, closely, two categories of avalanches recently observed in piles of glass beads. The origins of the di erent dynamic regimes are explained in terms of the statistical distribution of local stability criteria.

Physical review letters, 2002

We introduce a sandpile model where, at each unstable site, all grains are transferred randomly to downstream neighbors. The model is local and conservative, but not Abelian. This does not appear to change the universality class for the avalanches in the self-organized critical state. It does, however, introduce long-range spatial correlations within the metastable states. For the transverse direction d(perpendicular)&gt;0, we find a fractal network of occupied sites, whose density vanishes as a power law with distance into the sandpile.