Avalanche prediction in Abelian sandpile model

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, 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, 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 Letters, 1999

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).

Physical Review Letters, 1997

We introduce and study a new directed sandpile model with threshold dynamics and stochastic toppling rules. We show that particle conservation law and the directed percolation-like local evolution of avalanches lead to the formation of a spatial structure in the steady state, with the density developing a power law tail away from the top. We determine the scaling exponents characterizing the avalanche distributions in terms of the critical exponents of directed percolation in all dimensions.

2010

In this paper we prove that the avalanche problem for the Kadanoff sandpile model (KSPM) is P-complete for two-dimensions. Our proof is based on a reduction from the monotone circuit value problem by building logic gates and wires which work with configurations in KSPM. The proof is also related to the known prediction problem for sandpile which is in NC for one-dimensional sandpiles and is P-complete for dimension 3 or greater. The computational complexity of the prediction problem remains open for two-dimensional sandpiles.

Sankhyā: The Indian Journal of Statistics, 2005

Since its introduction by Bak, Tang and Wiesenfeld, the abelian sandpile dynamics has been studied extensively in finite volume. There are many problems posed by the existence of a sandpile dynamics in an infinite volume S: its invariant distribution should be the thermodynamic limit (does the latter exist?) of the invariant measure for the finite volume dynamics; the extension of the sand grains addition operator to infinite volume is related to the boundary effects of the dynamics in finite volume; finally, the crucial difficulty of the definition of a Markov process in infinite volume is that, due to sand avalanches, the interaction is long range, so that no use of the Hille-Yosida theorem is possible. In that review paper, we recall the needed results in finite volume, then explain how to deal with infinite volume when S = Z, S = T is an infinite tree, S = Z d with d large, and when the dynamics is dissipative (i.e. sand grains may disappear at each toppling) 1 . The abelian sandpile model in a finite volume V describes the evolution on a lattice of configurations η of discrete height-variables, which can be thought as local slopes of a sandpile. Sand grains are randomly added on the sites x ∈ V , and if at a site the height value for configuration η exceeds some critical value γ, then that 'unstable' site 'topples', i.e., gives an equal portion of its grains to each of its neighboring sites which in turn can become unstable and topple etc., until every site has again a subcritical height-value. An unstable site thus creates an 'avalanche' involving possibly the toppling of many sites around it. The range of this avalanche depends on the configuration, making the dynamics highly non-local. The action of the 'addition operator' a x,V consists in the instantaneous passage from configuration η to which a sand grain has been added on site x to the stable configuration a x,V η reached after the avalanche has ended. This model has a rich mathematical structure, first discovered by Dhar (see for instance ). The main tool in its analysis is the 'abelian group' of addition operators, identified with the set R V of recurrent configurations for the dynamics. The stationary measure for the dynamics is the uniform measure µ V on R V .

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 E, 1993

We model the dynamics of avalanches in granular assemblies in partly filled rotating cylinders using a mean-field approach. We show that, upon varying the cylinder angular velocity ω, the system undergoes a hysteresis cycle between an intermittent and a continuous flow regimes. In the intermittent flow regime, and approaching the transition, the avalanche duration exhibits critical slowing down with a temporal power-law divergence. Upon adding a white noise term, and close to the transition, the distribution of avalanche durations is also a power-law. The hysteresis, as well as the statistics of avalanche durations, are in good qualitative agreement with recent experiments in partly filled rotating cylinders.

We investigate numerically temporal correlations in a one-dimensional critical-slope sandpile model with rules that on average conserve the number of particles. Our work is motivated by the existence of two well-separated time scales in self-organized sandpile models, one related to the spreading of avalanches and the other imposed by the external driving. We assume that avalanches are instantaneous events on the time scale imposed by the external deposition and study the autocorrelation function of the series of successive avalanche amplitudes. We find that the autocorrelation function has a log-normal form and for large system sizes tends to a constant, implying that the temporal correlations become stronger in the limit of large system size. We independently test this result by calculating the power spectrum of the series of successive avalanche lifetimes and sizes. For large system sizes L there is a frequency regime where the power spectrum tends to a 1/f type of noise, in agreement with the tendency of the autocorrelation function to approach a constant in large systems. ͓S1063-651X͑96͒07212-1͔ PACS number͑s͒: 64.60.Lx