Asymptotic Behavior of the Order Parameter in a Stochastic Sandpile

Vidigal, Ronaldo; Dickman, Ronald

doi:10.1007/s10955-004-8775-7

Cited by 7 publications

(15 citation statements)

References 45 publications

(63 reference statements)

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“…In this section we apply the operator formalism and perturbation theory derived in [23] to the evaluation of the collective diffusion coefficient D c of model III on a ring of N sites.…”

Section: Collective Diffusion Coefficient: Theory and Simulationmentioning

confidence: 99%

“…Subsequent terms in the expansion may be evaluated using the diagrammatic perturbation approach developed in [23]. …”

Section: Collective Diffusion Coefficient: Theory and Simulationmentioning

confidence: 99%

“…D c is related to the relaxation time, τ k , of this mode via τ k = 1/(D c k 2 ), in the small-k limit. Using the path-integral based perturbation theory developed in [23], we calculate the first three terms in the expansion of D c in inverse powers of density p, for the stochastic sandpile in which the toppling rate is…”

Section: Introductionmentioning

confidence: 99%

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Diffusion in stochastic sandpiles

et al. 2009

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We study diffusion of particles in large-scale simulations of one-dimensional stochastic sandpiles, in both the restricted and unrestricted versions. The results indicate that the diffusion constant scales in the same manner as the activity density, so that it represents an alternative definition of an order parameter. The critical behavior of the unrestricted sandpile is very similar to that of its restricted counterpart, including the fact that a data collapse of the order parameter as a function of the particle density is only possible over a very narrow interval near the critical point. We also develop a series expansion, in inverse powers of the density. for the collective diffusion coefficient in a variant of the stochastic sandpile in which the toppling rate at a site with n particles is n(n − 1), and compare the theoretical prediction with simulation results. §

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“…In this section we apply the operator formalism and perturbation theory derived in [23] to the evaluation of the collective diffusion coefficient D c of model III on a ring of N sites.…”

Section: Collective Diffusion Coefficient: Theory and Simulationmentioning

confidence: 99%

“…Subsequent terms in the expansion may be evaluated using the diagrammatic perturbation approach developed in [23]. …”

Section: Collective Diffusion Coefficient: Theory and Simulationmentioning

confidence: 99%

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Diffusion in stochastic sandpiles

et al. 2009

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“…N3 N4 N5 N6 N7 N8 N9 N10 3 4 0 4 7 40 0 0 8 24 11 136 0 0 0 8 0 120 15 1304 0 0 0 4 32 48 288 560 19 3024 0 0 0 0 0 8 0 288 0 2432 Table 1 Degeneracies arise if one of the ai is zero in a solution of Eq. (14). In the table, g denotes the total number of solutions with one of the ai = 0 i.e.…”

Section: The Modelmentioning

confidence: 99%

“…the 1-dimensional Oslo rice pile model, but a straightforward direct depth-first calculation of the exact probabilities of different configurations in the steady state takes O(exp(L 3 )) steps where L is the system length [9]. While the exact values of the critical exponents have been conjectured for (1 + 1) dimensional directed Manna model [10,11], the prototypical undirected Manna model in one dimension has resisted an exact solution so far [12,13,14]. In higher dimensions, most of the studies are only numerical, and deal with the estimation of the critical exponents of avalanche distribution, and the universality class of the model [4,15,16,17,18,19,20,21].…”

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Steady State of Stochastic Sandpile Models

Sadhu

Dhar

2009

J Stat Phys

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We study the steady state of the abelian sandpile models with stochastic toppling rules. The particle addition operators commute with each other, but in general these operators need not be diagonalizable. We use their abelian algebra to determine their eigenvalues, and the Jordan block structure. These are then used to determine the probability of different configurations in the steady state. We illustrate this procedure by explicitly determining the numerically exact steady state for a one dimensional example, for systems of size ≤ 12, and also study the density profile in the steady state.

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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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Asymptotic Behavior of the Order Parameter in a Stochastic Sandpile

Cited by 7 publications

References 45 publications

Diffusion in stochastic sandpiles

Diffusion in stochastic sandpiles

Steady State of Stochastic Sandpile Models

Non-fixation for Conservative Stochastic Dynamics on the Line

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