Boundary effects in a two-dimensional Abelian sandpile

This article is based on a talk given by one of us (EVI) at the conference "StatPhys-Taipei-1997". It overviews the exact results in the theory of the sandpile model and discusses shortly yet unsolved problem of calculation of avalanche distribution exponents. The key ingredients include the analogy with the critical reaction-diffusion system, the spanning tree representation of height configurations and the decomposition of the avalanche process into waves of topplings. PACS number(s): 05.40.+j

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“…Moreover, on the boundary of the lattice only this operator determines the correlations of all other heights [22,23].…”

Section: Correlation Functions and The Continuous Limitmentioning

confidence: 99%

Introduction to the sandpile model

Ivashkevich

Priezzhev

1998

Physica A: Statistical Mechanics and its Applications

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“…The way these conflicting requirements of particle conservation and locally critical DP-like evolution are satisfied in our model is that the critical steady state develops a spatial structure. The density r, and hence P 1 , are not uniform throughout the system, but vary from layer to layer [11]. Let r͑ᐉ͒ be the average density of sites with nonzero height in the ᐉth layer.…”

Section: (Received 28 April 1997)mentioning

confidence: 99%

Emergent Spatial Structures in Critical Sandpiles

Tadić

Dhar

1997

Phys. Rev. Lett.

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We introduce and study a new directed sandpile model with threshold dynamics and stochastic toppling rules. We show that the particle conservation law and the directed percolationlike local evolution of avalanches lead to the formation of a spatial structure in the steady state, with the density developing a power-law tail away from the top. We determine the scaling exponents characterizing the avalanche distributions in terms of the critical exponents of directed percolation in all dimensions. [S0031-9007(97)03872-6] PACS numbers: 64.60. Lx, 05.40.+ j, 46.10.+ z, 64.60.Ak Many extended slowly driven dissipative systems in nature evolve into self-organized critical (SOC) steady states which show long-range spatial and temporal correlations. Since the pioneering work of Bak et al. [1], sandpile models have served as paradigms of SOC systems. A great deal of understanding of SOC has been achieved by numerically studying sandpile models with different evolution rules [2]. For a special subclass of these models, i.e., the Abelian sandpile models (ASM), several analytical results are available [3]. Recent studies of models with stochastic toppling rules have shown that these models usually belong to a universality class different from deterministic automata; they have robust critical states with respect to changes of a control parameter, and may also exhibit a dynamical phase transition between qualitatively different steady states [4]. However, the spatial structures in the steady states of these models are much less investigated [5]. Because of long-range correlations in the critical states, influence of the boundary can be felt deep inside, and this can give rise to large-scale spatial structures. Indeed, emergent spatial structures are sine qua non for a SOC theory of fractals occurring in nature, e.g., mountain landscapes, river networks, and earthquake fault zones.In this Letter, we propose a new stochastic sandpile model which shows emergent spatial structures in the steady states. The model is a stochastic generalization of the directed ASM [6] and contains a probabilistic control parameter p. In the case p 1 the exponents are known exactly in all dimensions [6]. We show that, for p fi 1, the model is in a new universality class and can be related to a directed percolation (DP) [7] problem with a nonuniform concentration profile. A steady state exists only for p . p , where p equals the critical threshold for directed site percolation. Above p the system evolves towards a steady state which is arbitrarily close to the generalized DP critical line. We also show that in the critical state of our model a spatial structure results from an interplay of DP-like local evolution rules on the one side, and the dynamic conservation law on the other. We find a power-law density profile, which further enables us to determine the exact expressions for the scaling exponents of avalanches in terms of the DP critical exponents in all dimensions d. Our numerical simulations in two dimensions support these conclusions.For co...

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“…+l6 prob l23ll +leprob 22 (8.4) %e note that a similar series is encountered in calculating the probability that on deleting a bond at random in a spanning tree, exactly s, sites get disconnected [11] (10) Tj F + 1 j for 1~j~n~ (12) Experimentation with small systems shows that this is usually true. (Its general validity was erroneously claimed in [2]. )…”

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confidence: 97%

“…Only recently has it been possible to calculate the fractional number of sites having different heights in the steady state in two dimensions [8]. It is also known that correlations between sites with minimum allowed height vary as r in the critical state, where r is the separation between sites, in the bulk, and on the surface [9,10]. However, so far it has not been possible to relate any of the exponents referring to distribution of sizes of avalanches to known exponents.…”

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confidence: 99%

Inverse avalanches in the Abelian sandpile model

Dhar

Manna

1994

Phys. Rev. E

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We define and study the inverse of particle addition process in the Abelian sandpile model. We show how to obtain the unique recurrent configuration corresponding to a single particle deletion by a sequence of operations called inverse avalanches. We study the probability distribution of sl, the number of "untopplings" in the first inverse avalanche. For a square lattice, we determine Prob (sl ) [7]. If from a spanning tree we delete a bond at random, the probability distribution of the number of sites disconnected is known to have power law tails. In two dimensions, this probability varies as s "~, where s is the number of sites disconnected [11].In this paper we show that the exponent in the ASM case that corresponds to deleting an edge in the spanning tree problem is related to the process of removing

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Boundary effects in a two-dimensional Abelian sandpile

Cited by 28 publications

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Introduction to the sandpile model

Introduction to the sandpile model

Emergent Spatial Structures in Critical Sandpiles

Inverse avalanches in the Abelian sandpile model

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