Persistent self-organization of sandpiles

Complexity and self-organized criticality in liquid foams. A short review

This short review deals with the work done on liquid foams within the framework of the physics of complexity. It aims to stimulate new theoretical and experimental work on foam dynamics as complex dynamical systems. In particular, it examines these systems in relation to Self-Organized Criticality (SOC), for which foams could be used as an accessible experimental model system.

Subsampling effects in neuronal avalanche distributions recorded in vivo

BACKGROUND

Many systems in nature are characterized by complex behaviour where large cascades of events, or avalanches, unpredictably alternate with periods of little activity. Snow avalanches are an example. Often the size distribution f(s) of a system's avalanches follows a power law, and the branching parameter sigma, the average number of events triggered by a single preceding event, is unity. A power law for f(s), and sigma = 1, are hallmark features of self-organized critical (SOC) systems, and both have been found for neuronal activity in vitro. Therefore, and since SOC systems and neuronal activity both show large variability, long-term stability and memory capabilities, SOC has been proposed to govern neuronal dynamics in vivo. Testing this hypothesis is difficult because neuronal activity is spatially or temporally subsampled, while theories of SOC systems assume full sampling. To close this gap, we investigated how subsampling affects f(s) and sigma by imposing subsampling on three different SOC models. We then compared f(s) and sigma of the subsampled models with those of multielectrode local field potential (LFP) activity recorded in three macaque monkeys performing a short term memory task.

RESULTS

Neither the LFP nor the subsampled SOC models showed a power law for f(s). Both, f(s) and sigma, depended sensitively on the subsampling geometry and the dynamics of the model. Only one of the SOC models, the Abelian Sandpile Model, exhibited f(s) and sigma similar to those calculated from LFP activity.

CONCLUSION

Since subsampling can prevent the observation of the characteristic power law and sigma in SOC systems, misclassifications of critical systems as sub- or supercritical are possible. Nevertheless, the system specific scaling of f(s) and sigma under subsampling conditions may prove useful to select physiologically motivated models of brain function. Models that better reproduce f(s) and sigma calculated from the physiological recordings may be selected over alternatives.

Scaling properties of a rice-pile model: inertia and friction effects

We present a rice-pile cellular automaton model that includes inertial and friction effects. This model is studied in one dimension, where the updating of metastable sites is done according to a stochastic dynamics governed by a probabilistic toppling parameter p that depends on the accumulated energy of moving grains. We investigate the scaling properties of the model using finite-size scaling analysis. The avalanche size, the lifetime, and the residence time distributions exhibit a power-law behavior. Their corresponding critical exponents, respectively, tau, y, and yr, are not universal. They present continuous variation versus the parameters of the system. The maximal value of the critical exponent tau that our model gives is very close to the experimental one, tau=2.02 [Frette, Nature (London) 379, 49 (1996)], and the probability distribution of the residence time is in good agreement with the experimental results. We note that the critical behavior is observed only in a certain range of parameter values of the system which correspond to low inertia and high friction.

Edge effect on the power law distribution of granular avalanches

Many punctuated phenomena in nature are claimed [e.g., by the theory of self-organized criticality (SOC)] to be power-law distributed. In our experiments on a three-dimensional pile of long-grained rice, we find that by only changing the boundary condition of the system, we switch from such power-law-distributed avalanche sizes to quasiperiodic system-spanning avalanches. Conversely, by removing ledges the incidence of system-spanning avalanches is significantly reduced. This may offer a perspective on new avalanche prevention schemes. In addition, our findings may help to explain why the archetype of SOC, the sandpile, was found to have power-law-distributed avalanches in some experiments, while in other experiments quasiperiodic system-spanning avalanches were found.

Probability distribution of residence times of grains in models of rice piles

We study the probability distribution of residence time of a grain at a site, and its total residence time inside a pile, in different rice pile models. The tails of these distributions are dominated by the grains that get deeply buried in the pile. We show that, for a pile of size L, the probabilities that the residence time at a site or the total residence time is greater than t, both decay as 1/t(ln t)x for L(omega) << t << exp(L(gamma)) where gamma is an exponent > or = 1, and values of x and omega in the two cases are different. In the Oslo rice pile model we find that the probability of the residence time T(i) at a site i being greater than or equal to t is a nonmonotonic function of L for a fixed t and does not obey simple scaling. For model in d dimensions, we show that the probability of minimum slope configuration in the steady state, for large L, varies as exp(-kappaL(d+2)) where kappa is a constant, and hence gamma=d+2.

Scaling properties of a one-dimensional sandpile model with grain dissipation

We have studied a stochastic sandpile model with grain dissipation as a generalization of the Oslo sandpile model. During a toppling event, grains are removed from the pile with a probability p. Scaling arguments and simulations suggest that an arbitrarily small dissipation rate p yields a noncritical behavior, in contrast to the robust critical behavior of the Oslo sandpile model.

Sandpile formation by revolving rivers

Experimental observation of a new mechanism of sandpile formation is reported. As a steady stream of dry sand is poured onto a horizontal surface, a pile forms which has a thin river of sand on one side flowing from the apex of the pile to the edge of its base. The river rotates about the pile, depositing a new layer of sand with each revolution, thereby causing the pile to grow. For small piles the river is steady and the pile formed is smooth. For larger piles, the river becomes intermittent and the surface of the pile becomes undulating. The essential features of the system that produce the phenomenon are discussed, and the robustness of the phenomena is demonstrated with experiments using different boundary conditions and sands.

Self-organized criticality in a bead pile

Self-organized criticality has been proposed to explain complex dynamical systems near their critical points. This experiment examined a monodisperse conical bead pile and how the distribution of avalanches is affected by the pattern of beads glued on a base, by the size or shape of the base, and by the height at which each bead was dropped onto the pile. By measuring the number of avalanches for a given size that occurred during the experiment, the resulting distribution could be compared to a power law description. When the beads were dropped from a small height, all data were consistent with a simple power law of exponent -1.5, which is the mean-field model value. The data showed that neither the bead pattern on the base nor the base size or shape significantly affected the power law behavior. However, when the bead is dropped from different heights, then the power law description breaks down and a power law times an exponential is more appropriate. We found a scaling relationship in the distribution of avalanches for different heights and relate the data to an energy dissipation model. We both confirm self-organized criticality and observe deviations from it.

Instabilities in slowly driven granular packing

In this work, a digital imaging technique is used to study the superficial fluctuations observed when a granular packing is slowly driven to the threshold of instability. The experimental results show the presence of three types of events. Small superficial rearrangements of grains are observed during all the experiments. They present a power-law behavior although the system is not in a critical state as predicted by self-organized criticality models. In thick granular piles, large rearrangements are detected at regular angular intervals. They are related to the threshold of instability of the contact network that relaxes to stable configurations producing internal rearrangements of the grains. Finally, an avalanche is triggered when the superficial beads that are set in motion acquire enough momentum to destabilize grains from layers below.

Breakdown of self-organized criticality in sandpiles

We introduce two sandpile models which show the same behavior of real sandpiles, that is, an almost self-organized critical behavior for small systems and a dominance of large avalanches as the system size increases. The systems become fully self-organized critical, with the critical exponents of the Bak, Tank, and Wiesenfeld model [P. Bak, C. Tang, and K. Wiesenfeld, Phys. Rev. Lett. 59, 381 (1987)] as the system parameters are changed, showing that they can make a bridge between the well known theoretical and numerical results and what is observed in real experiments.

Anomalous transport in conical granular piles

Experiments on 2+1-dimensional piles of elongated particles are performed. Comparison with previous experiments in 1+1 dimensions shows that the addition of one extra dimension to the dynamics changes completely the avalanche properties, with the appearance of a characteristic avalanche size. Nevertheless, the time which single grains need to cross the whole pile varies smoothly between several orders of magnitude, from a few seconds to more than 100 hours. This behavior is described by a power-law distribution, signaling the existence of scale invariance in the transport process.

Basin of attraction of a bounded self-organized critical state

The robustness of the self-organized critical (SOC) state observed in the motion of an annular plate rotating over a granular medium is studied in this paper. In particular, we investigate the effect of parameters to which the emergent SOC state may be sensitive, including the initialization scheme, driving velocity, and confining pressure. The results indicate that the critical state is not a universal attractor, but has a finite basin of attraction. Furthermore, this state is only one of the three observed, which compare well with subcritical, critical, and supercritical states. The results call into question the precise definition of the term "self-organized criticality," an issue we address.

Modeling relaxation and jamming in granular media

We introduce a stochastic microscopic model to investigate the jamming and reorganization of grains induced by an object moving through a granular medium. The model reproduces the experimentally observed periodic sawtooth fluctuations in the jamming force and predicts the period and the power spectrum in terms of the controllable physical parameters. It also predicts that the avalanche sizes, defined as the number of displaced grains during a single advance of the object, follow a power law P(s) approximately s(-tau), where the exponent is independent of the physical parameters.

Avalanches in one-dimensional piles with different types of bases

We perform a systematic experimental study of the influence of the type of base on the avalanche dynamics of slowly driven 1D ball piles. The control of base details allows us to explore a wide spectrum of pile structures and dynamics. The scaling properties of the observed avalanche distributions suggest that self-organized critical behavior is approached as the "base-induced" disorder at the pile profile increases.

Self-organized criticality in a sheared granular stick-slip system

We present an analysis of results obtained from a mechanical apparatus consisting of an annular plate shearing over a granular bed. The size, energy dissipation, and duration of slips in the system exhibit power-law distributions and a 1/f(2) power spectrum, in accordance with self-organized criticality. We draw similarities with earthquakes.

Hillslope Evolution by Bedrock Landslides

Bedrock landsliding is a dominant geomorphic process in a number of high-relief landscapes, yet is neglected in landscape evolution models. A physical model of sliding in beans is presented, in which incremental lowering of one wall simulates baselevel fall and generates slides. Frequent small slides produce irregular hillslopes, on which steep toes and head scarps persist until being cleared by infrequent large slides. These steep segments are observed on hillslopes in high-relief landscapes and have been interpreted as evidence for increases in tectonic or climatic process rates. In certain cases, they may instead reflect normal hillslope evolution by landsliding.