Oslo Ricepile Model Research Articles

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.

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.

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.

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.