Diffusion in stochastic sandpiles

Cunha, S. D. da; Vidigal, Ronaldo; Silva, L. R. da; Dickman, Ronald

doi:10.1140/epjb/e2009-00367-0

Cited by 13 publications

(13 citation statements)

References 28 publications

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“…Some recent studies (e.g., Bonachela, Muñoz [5]) have focussed their attention to sandpile models with stochastic update rules. Although the paradigm of Dickman, Vespignani et al for SOC is widely believed to contain both deterministic and stochastic sandpile models [9,44], adding randomness can lead to a qualitatively different critical behaviour. The best known example in this random setting is the Stochastic Sandpile model (SSM).…”

Section: The Model Description and Main Resultsmentioning

confidence: 99%

Non-fixation for Conservative Stochastic Dynamics on the Line

Basu

Ganguly

Hoffman

2017

Commun. Math. Phys.

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ABSTRACT. We consider Activated Random Walk (ARW), a model which generalizes the Stochastic Sandpile, one of the canonical examples of self organized criticality. Informally ARW is a particle system on Z with mass conservation. One starts with a mass density µ > 0 of initially active particles, each of which performs a symmetric random walk at rate one and falls asleep at rate λ > 0. Sleepy particles become active on coming in contact with other active particles. We investigate the question of fixation/non-fixation of the process and show for small enough λ the critical mass density for fixation is strictly less than one. Moreover, the critical density goes to zero as λ tends to zero. This settles a long standing open question. THE MODEL DESCRIPTION AND MAIN RESULTSelf organized criticality (SOC) is a universal property describing systems that have critical points as an attractor. In most of the examples of SOC, simple local moves give rise to complex global properties. These systems are driven under their natural evolution to the boundary between stable and unstable states without any fine-tuning of parameters. A canonical and widely studied example of such a model is the Abelian Sandpile model proposed by Bak, Tang and Weisenfeld [3]. In this model, in finite volume, particles are added at random, which dissipate across the boundary; the corresponding closed system in the infinite volume setting is a so-called fixed energy sandpile where the total number of particles is conserved.In a sequence of influential papers Dickman, Vespignani and their co-authors [14,15,42,43] developed a theory of SOC, which has since become widely accepted in statistical physics literature. Their theory predicts a specific relationship between driven dissipative systems and the corresponding systems where total number of particles is conserved; and in particular an absorbing state phase transition, exhibited by the latter system as the particle density is varied. In the finite state, even though there is no tuning parameter, they argued that the loss of particles through the sink(s) is balanced with the dynamical addition of new particles, driving the system to the edge of instability. However subsequently, there has been some controversy in the mathematics and physics community surrounding their predictions. We elaborate more on this in § 1.1.In the last fifteen years, there has also been significant progress in studying these systems via a combinatorial approach, in particular exploiting the so-called Abelian property. Roughly, the Abelian property in a distributed network stipulates that the system produces the same output regardless of the order in which a sequence of local moves are performed. A systematic mathematical treatment of such systems can be found in [19]. Even though this might seem a severe restriction, systems with Abelian property are known to produce intricate large-scale patterns starting with relatively simple local rules. In addition to the Abelian sandpile model and a number of cellular automata introduced by...

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Section: The Model Description and Main Resultsmentioning

confidence: 99%

Non-fixation for Conservative Stochastic Dynamics on the Line

Basu

Ganguly

Hoffman

2017

Commun. Math. Phys.

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“…In the present work we impose a height restriction on the conserved Oslo model. Since the symmetries and conserved quantities of the restricted and unrestricted models are the same, one expects, on the basis of experience with critical phenomena both in and out of equilibrium, that the models belong to the same universality class, as is indeed borne out for conserved versions of the Manna model [19][20][21]. The symmetries here are limited to spatial translation and inversion, while the conservation law is that of particle number.…”

Section: Introductionmentioning

confidence: 87%

Conserved Sandpile with A Variable Height Restriction

2013

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We study a restricted-height version of the one-dimensional Oslo sandpile with conserved density, using periodic boundary conditions. Each site has a limiting height which can be either two or three. When a site reaches its limiting height it becomes active and may topple, loosing two particles, which move randomly to nearest-neighbor sites. After a site topples it is randomly assigned a new limiting height. We study the model using meanfield theory and Monte Carlo simulation, focusing on the quasi-stationary state, in which the number of active sites fluctuates about a stationary value. Using finite-size scaling analysis, we determine the critical particle density and associated critical exponents.

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“…It has been recently observed [18] that in some onedimensional stochastic sand-piles the tracer diffusion coefficient D t ( R 2 (t) = 2 D t t, where R(t) is the displacement of a particle at time t) scales in the same manner as the density of active particles, so that it represents an alternative definition of an order parameter. Here we show that the same behavior is found in the DLG, which enables us to obtain independent estimates of the scaling exponents β and σ.…”

Section: Tracer Diffusion Constant As An Order Parametermentioning

confidence: 99%

Connecting the microdynamics to the emergent macrovariables: Self-organized criticality and absorbing phase transitions in the deterministic lattice gas

Giometto

2012

Phys. Rev. E

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We reinvestigate the Deterministic Lattice Gas introduced as a paradigmatic model of the 1/f spectra [Phys. Rev. Lett. 64, 3103 (1990)] arising according to the self-organized criticality scenario. We demonstrate that the density fluctuations exhibit an unexpected dependence on systems size and relate the finding to effective Langevin equations. The low-density behavior is controlled by the critical properties of the gas at the absorbing state phase transition. We also show that the deterministic lattice gas is in the Manna universality class of absorbing state phase transitions. This is in contrast to expectations in the literature that suggested that the entirely deterministic nature of the dynamics would put the model in a different universality class. To our knowledge this is the first fully deterministic member of the Manna universality class.

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Diffusion in stochastic sandpiles

Cited by 13 publications

References 28 publications

Non-fixation for Conservative Stochastic Dynamics on the Line

Non-fixation for Conservative Stochastic Dynamics on the Line

Conserved Sandpile with A Variable Height Restriction

Connecting the microdynamics to the emergent macrovariables: Self-organized criticality and absorbing phase transitions in the deterministic lattice gas

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