Rice-pile Model Research Articles

Many universality classes in an interface model restricted to non-negative heights.

We present a simple one-dimensional stochastic model with three control parameters and a surprisingly rich zoo of phase transitions. At each (discrete) site x and time t, an integer n(x,t) satisfies a linear interface equation with added random noise. Depending on the control parameters, this noise may or may not satisfy the detailed balance condition, so that the growing interfaces are in the Edwards-Wilkinson or in the Kardar-Parisi-Zhang universality class. In addition, there is also a constraint n(x,t)≥0. Points x where n>0 on one side and n=0 on the other are called "fronts." These fronts can be "pushed" or "pulled," depending on the control parameters. For pulled fronts, the lateral spreading is in the directed percolation (DP) universality class, while it is in a different universality class for pushed fronts, and another universality class in between. In the DP case, the activity at each active site can in general be arbitrarily large, in contrast to previous realizations of DP. Finally, we find two different types of transitions when the interface detaches from the line n=0 (with 〈n(x,t)〉→const on one side, and →∞ on the other), again with new universality classes. We also discuss a mapping of this model to the avalanche propagation in a directed Oslo rice pile model in specially prepared backgrounds.

Oslo model, hyperuniformity, and the quenched Edwards-Wilkinson model.

We present simulations of the one-dimensional Oslo rice pile model in which the critical height at each site is randomly reset after each toppling. We use the fact that the stationary state of this sand-pile model is hyperuniform to reach system of sizes >10^{7}. Most previous simulations were seriously flawed by important finite-size corrections. We find that all critical exponents have values consistent with simple rationals: ν=4/3 for the correlation length exponent, D=9/4 for the fractal dimension of avalanche clusters, and z=10/7 for the dynamical exponent. In addition, we relate the hyperuniformity exponent to the correlation length exponent ν. Finally, we discuss the relationship with the quenched Edwards-Wilkinson model, where we find in particular that the local roughness exponent is α_{loc}=1.

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.

Critical Behaviors in a Stochastic Local Limited One-Dimensional Rice-Pile Model

A stochastic local limited one-dimensional rice-pile model is numerically investigated. The distributions for avalanche sizes have a clear power-law behavior and it displays a simple finite size scaling. We obtain the avalanche exponents τs = 1.54 ± 0.10, βs = 2.17 ± 0.10 and τT = 1.80 ± 0.10, βT = 1.46 ± 0.10. This self-organized critical model belongs to the same universality class with the Oslo rice-pile model studied by K. Christensen et al. [Phys. Rev. Lett. 77 (1996) 107], a rice-pile model studied by L.A.N. Amaral et al. [Phys. Rev. E 54 (1996) 4512], and a simple deterministic self-organized critical model studied by M.S. Vieira [Phys. Rev. E 61 (2000) 6056].

Influence of the driving rate in a two-dimensional rice pile model

We study the influence of the driving rate in the two-dimensional Oslo rice pile model. We find that the usual power-law behavior of the avalanche size distribution still holds for small avalanches, independent of the driving rate. The signature of fast driving is, however, the increase of the incidence rate of large avalanches. For larger driving rates, this increase is more prominent and spreads to smaller avalanche sizes. As a result, the mass flow due to large avalanches is increased much more than would be expected from an increase in driving rate alone. Fast driving leads to a dramatic increase in devastating avalanches, just before the continuous flow regime is reached.

Sampling rare fluctuations of height in the Oslo ricepile model

We describe a new Monte Carlo algorithm for studying large deviations of the height of the pile from its mean value in the Oslo ricepile model. We have used it to generate events which have probabilities of order 10−1000. These simulations check our qualitative argument (Pradhan P and Dhar D 2006 Phys. Rev. E 73 021303) that in the steady state of the Oslo ricepile model the probability of large negative height fluctuations Δh = −αL about the mean varies as exp(−κα4L3) as L → ∞ with α held fixed and κ > 0.

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.

Anomalous Scaling Behaviors in a Rice-Pile Model with Two DifferentDriving Mechanisms**The project supported by National Natural ScienceFoundation of China under Grant No. 50272022

The moment analysis is applied to perform large scalesimulations of the rice-pile model. We find that this model showsdifferent scaling behavior depending on the driving mechanismused. With the noisy driving, the rice-pile model violates thefinite-size scaling hypothesis, whereas, with fixed driving,it shows well defined avalanche exponents and displays good finitesize scaling behavior for the avalanche size and time durationdistributions.

Moment Analysis of a Rice-Pile Model

Large scale simulations of a rice-pile model are performed. We use momentanalysis techniques to evaluate critical exponents and data collapse methodto verify the obtained results. The moment analysis yields well-definedavalanche exponents, which show that the rice-pile model can be coherentlydescribed within a finite size scaling framework. The general pictureresulting from our analysis allows us to characterize the large scalebehavior of the present model with great accuracy.

On self-organised criticality in one dimension

In critical phenomena, many of the characteristic features encountered in higher dimensions such as scaling, data collapse and associated critical exponents are also present in one dimension. Likewise for systems displaying self-organised criticality. We show that the one-dimensional Bak–Tang–Wiesenfeld sandpile model, although trivial, does indeed fall into the general framework of self-organised criticality. We also investigate the Oslo ricepile model, driven by adding slope units at the boundary or in the bulk. We determine the critical exponents by measuring the scaling of the kth moment of the avalanche size probability with system size. The avalanche size exponent depends on the type of drive but the avalanche dimension remains constant.

Steady state and relaxation spectrum of the Oslo rice-pile model

We show that the one-dimensional Oslo rice-pile model is a special case of the abelian distributed processors model. The exact steady state of the model is determined. We show that the time evolution operator W for the system satisfies the equation W n+1= W n where n= L( L+1)/2 for a pile with L sites. This implies that W has only one eigenvalue 1 corresponding to the steady state, and all other eigenvalues are exactly zero. Also, all connected time-dependent correlation functions in the steady state of the pile are exactly zero for time difference greater than n. Generalization to other abelian critical height models where the critical thresholds are randomly reset after each toppling is briefly discussed.

Calculation of the transition matrix and of the occupation probabilities for the states of the Oslo sandpile model

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 tells 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 allow, for small system sizes, the exact calculation of the avalanche-size distribution and other properties related to the profile of the pile.

Oslo rice pile model is a quenched Edwards-Wilkinson equation.

The Oslo rice pile model is a sandpile-like paradigmatic model of self-organized criticality (SOC). In this paper it is shown that the Oslo model is in fact exactly a discrete realization of the much studied quenched Edwards-Wilkinson equation (qEW) [Nattermann et al., J. Phys. II France 2, 1483 (1992)]. This is possible by choosing the correct dynamical variable and identifying its equation of motion. It establishes for the first time an exact link between SOC models and the field of interface growth with quenched disorder. This connection is obviously very encouraging as it suggests that established theoretical techniques can be brought to bear with full strength on some of the hitherto elusive problems of SOC.

INTERTEMPORAL DISTRIBUTIONS OF A RICE PILE MODEL

In this work we study the distribution of time intervals between avalanches in a rice pile model. This model has shown that the crossover from power law to stretched exponential behaviors observed experimentally in the granular dynamics of rice piles can be well described as a long-range effect resulting from a change in the transport properties of individual grains. In this work we show that the change in the transport properties is also reflected in the behavior of the distribution of time intervals between avalanches.

Self-organized branching process for a one-dimensional rice-pile model

A self-organized branching process is introduced to describe one-dimensional ricepile model with stochastic topplings. Although the branching processes are generally supposed to describe well the systems in high dimension, our modification grasps some of the peculiarities present in one dimension. We find analytically the crossover behavior from the trivial one-dimensional BTW behavior to self-organized criticality characterized by power-law distribution of avalanches. The finite-size effects, which are crucial in the crossover, are calculated.

Damage spreading in a rice pile model

In a recent experiment with rice piles [V. Frette et al., Nature 379 (1996) 49] it was observed that the distribution of avalanches presents a power-law behavior for elongated rice, while avalanches for round rice present a characteristic length. These experiments showed that granular piles may display Self-Organized Criticality (SOC), but the behavior is not universal. We have recently presented a model for granular flow in a one-dimensional rice pile [P.M. Gleiser et al., Phys. Rev. E, in press] which exhibits the crossover from power law to stretched exponential behavior observed experimentally in the granular dynamics of rice piles. In this work, we use the damage spreading technique to study how a small perturbation (or damage) propagates through the system.

1/f alpha fluctuations in a ricepile model

The temporal fluctuation of the average slope of a ricepile model is investigated. It is found that the power spectrum S(f) scales as 1/f alpha with alpha approximately 1.3 when grains of rice are added only to one end of the pile. If grains are randomly added to the pile, the power spectrum exhibits 1/f2 behavior. The profile fluctuations of the pile under different driving mechanisms are also discussed.

Universality classes for the ricepile model with absorbing properties

The absorbing "ricepile" model with stochastic toppling rules has been numerically studied. Local limited, local unlimited, nonlocal limited, and nonlocal unlimited versions of the absorbing model have been investigated. Transport properties and different dynamical regimes of all of the models have been analyzed, from the point of view of self-organized criticality. Phase transitions between different dynamical regimes were studied in detail. It was shown that the absorbing models belong to two different universality classes.

Disorder-induced phase transition in a one-dimensional model of rice pile

We propose a one-dimensional rice-pile model which connects the 1D BTW sandpile model (Phys. Rev. A 38 (1988) 364) and the Oslo rice-pile model (Phys. Rev. Lett. 77 (1997) 107) in a continuous manner. We found that for a sufficiently large system, there is a sharp transition between the trivial critical behaviour of the 1D BTW model and the self-organized critical (SOC) behaviour. When there is SOC, the model belongs to a known universality class with the avalanche exponent τ=1.53.

A sharp transition between a trivial 1D BTW model and self-organized critical rice-pile model

A one-dimensional model of a rice-pile is numerically studied for different driving mechanisms. We found that for a sufficiently large system, there is a sharp transition between the trivial behaviour of a 1D BTW model and self-organized critical (SOC) behaviour. Depending on the driving mechanism, the self-organized critical rice-pile model belongs to two different universality classes.