Sandpile models and random walkers on finite lattices

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 .

arXiv (Cornell University), 1996

We introduce a simple model for the size distribution of avalanches based on the idea that the front of an avalanche can be described by a directed random walk. The model captures some of the qualitative features of earthquakes, avalanches and other self-organized critical phenomena in one dimension. We find scaling laws relating the frequency, size and width of avalanches and an exponent 4/3 in the size distribution law.

2002

We study sandpile models with stochastic toppling rules and having sticky grains so that with a non-zero probability no toppling occurs, even if the local height of pile exceeds the threshold value. Dissipation is introduced by adding a small probability of particle loss at each toppling. Generically, for models with a preferred direction, the avalanche exponents are those of critical directed percolation clusters. For undirected models, avalanche exponents are those of directed percolation clusters in one higher dimension.

Physical review, 2017

The self-organized criticality on the random fractal networks has many motivations, like the movement pattern of fluid in the porous media. In addition to the randomness, introducing correlation between the neighboring portions of the porous media has some nontrivial effects. In this paper, we consider the Ising-like interactions between the active sites as the simplest method to bring correlations in the porous media, and we investigate the statistics of the BTW model in it. These correlations are controlled by the artificial "temperature" T and the sign of the Ising coupling. Based on our numerical results, we propose that at the Ising critical temperature T c the model is compatible with the universality class of two-dimensional (2D) self-avoiding walk (SAW). Especially the fractal dimension of the loops, which are defined as the external frontier of the avalanches, is very close to D SAW f = 4 3. Also, the corresponding open curves has conformal invariance with the root-mean-square distance R rms ∼ t 3/4 (t being the parametrization of the curve) in accordance with the 2D SAW. In the finite-size study, we observe that at T = T c the model has some aspects compatible with the 2D BTW model (e.g., the 1/ log(L)-dependence of the exponents of the distribution functions) and some in accordance with the Ising model (e.g., the 1/L-dependence of the fractal dimensions). The finite-size scaling theory is tested and shown to be fulfilled for all statistical observables in T = T c. In the off-critical temperatures in the close vicinity of T c the exponents show some additional power-law behaviors in terms of T − T c with some exponents that are reported in the text. The spanning cluster probability at the critical temperature also scales with L 1 2 , which is different from the regular 2D BTW model.

We provide a comprehensive view of the role of Abelian symmetry and stochasticity in the universality class of directed sandpile models, in the context of the underlying spatial correlations of metastable patterns and scars. It is argued that the relevance of Abelian symmetry may depend on whether the dynamic rule is stochastic or deterministic, by means of the interaction of metastable patterns and avalanche flow. Based on the new scaling relations, we conjecture critical exponents for an avalanche, which is confirmed reasonably well in large-scale numerical simulations.

Physical Review E, 1999

Avalanches in sandpiles are represented throughout a process of percolation in a Bethe lattice with a feedback mechanism. The results indicate that the frequency spectrum and probability distribution of avalanches resemble more to experimental results than other models using cellular automata simulations. Apparent discrepancies between experiments are reconciled. Critical behavior is here expressed throughout the critical properties of percolation phenomena.

Physical Review E, 1998

Due to intermittency and conservation, the Abelian sandpile in 2D obeys multifractal, rather than finite size scaling. In the thermodynamic limit, a vanishingly small fraction of large avalanches dominates the statistics and a constant gap scaling is recovered in higher moments of the toppling distribution. Thus, rare events shape most of the scaling pattern and preserve a meaning for effective exponents, which can be determined on the basis of numerical and exact results.

Physical Review E, 2000

The avalanche statistics in a stochastic sandpile model where toppling takes place with a probability p is investigated. The limiting case p = 1 corresponds to the Bak-Tang-Wiesenfeld (BTW) model with deterministic toppling rule. Based on the moment analysis of the distribution of avalanche sizes we conclude that for 0 < p < pc the model belongs to the DP universality class while for pc < p < 1 it belongs to the BTW universality class, where pc is identified with the critical probability for directed percolation in the corresponding lattice. 64.60.Lx, 05.70.Ln Sandpile automata were proposed as a paradigm of self-organized critical (SOC) phenomena [1]. These simple models capture its essential dynamics, which takes place in the form of avalanches of all sizes. At the early state of SOC theory it was believed that the critical state of sandpile automata is insensitive to changes in model parameters, however some recent works contradict this statement. For instance, Vespignani and Zapperi [2] have shown that driving and dissipation rates actually act as control parameters, criticality is obtained after fine tuning of these fields. On the other hand, we have recently shown that a class of models with stochastic rules display a transition from SOC to directed percolation (DP) with increasing the degree of stochasticity . Nevertheless, before we make our final conclusion, we have to investigate if the original Bak-Tang-Wiesenfeld (BTW) sandpile automaton and these modified models belong to the same universality class, otherwise they would just be different models. One may thus ask: do deterministic and stochastic sandpile models belong to the same universality class?

Springer Theses, 2014

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.

Arxiv preprint cond-mat/0403769, 2004

We study the probability distribution of residence time, T , of the sand grains in the one dimensional abelian sandpile model on a lattice of L sites, for T << L 2 and T >> L 2. The distribution function decays as exp(− K L T L 2). We numerically calculate the coefficient K L for the value of L upto 150. Interestingly the distribution function has a scaling form 1 L a f (T L b) with a = b for large L.

Physical Review E

We give some examples to illustrate that scale invariance may not be a manifestation of complex behavior in one-dimensional sandpile models. The multiscaling statistical properties and the existence of intrinsic length scales observed in the local limited one-dimensional model reflects a certain level of complexity. The local, limited, and limited to no traps model presents scale invariance due to the inhomogeneous way of perturbing the lattice. It behaves, however, as the trivial one-dimensional version of the Bak, Tang, and Wisenfeld ͓Phys. Rev. Lett. 59, 381 ͑1987͒; Phys. Rev. A 38, 364 ͑1988͔͒ model. A nonlocal limited model presents scaling statistical properties and displays the same level of complexity as the nontrivial two-dimensional models.

Physical Review Letters, 2012

Fixed-energy sandpiles with stochastic update rules are known to exhibit a nonequilibrium phase transition from an active phase into infinitely many absorbing states. Examples include the conserved Manna model, the conserved lattice gas, and the conserved threshold transfer process. It is believed that the transitions in these models belong to an autonomous universality class of nonequilibrium phase transitions, the so-called Manna class. Contrarily, the present numerical study of selected (1+1)-dimensional models in this class suggests that their critical behavior converges to directed percolation after very long time, questioning the existence of an independent Manna class.

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.

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.