Number Of Topplings Research Articles

Geometry-induced nonequilibrium phase transition in sandpiles.

We study the sandpile model on three-dimensional spanning Ising clusters with the temperature T treated as the control parameter. By analyzing the three-dimensional avalanches and their two-dimensional projections (which show scale-invariant behavior for all temperatures), we uncover two universality classes with different exponents (an ordinary BTW class, and SOC_{T=∞}), along with a tricritical point (at T_{c}, the critical temperature of the host) between them. The transition between these two criticalities is induced by the transition in the support. The SOC_{T=∞} universality class is characterized by the exponent of the avalanche size distribution τ^{T=∞}=1.27±0.03, consistent with the exponent of the size distribution of the Barkhausen avalanches in amorphous ferromagnets Durin and Zapperi [Phys. Rev. Lett. 84, 4705 (2000)PRLTAO0031-900710.1103/PhysRevLett.84.4705]. The tricritical point is characterized by its own critical exponents. In addition to the avalanche exponents, some other quantities like the average height, the spanning avalanche probability (SAP), and the average coordination number of the Ising clusters change significantly the behavior at this point, and also exhibit power-law behavior in terms of ε≡T-T_{c}/T_{c}, defining further critical exponents. Importantly, the finite-size analysis for the activity (number of topplings) per site shows the scaling behavior with exponents β=0.19±0.02 and ν=0.75±0.05. A similar behavior is also seen for the SAP and the average avalanche height. The fractal dimension of the external perimeter of the two-dimensional projections of avalanches is shown to be robust against T with the numerical value D_{f}=1.25±0.01.

Universality of high-dimensional spanning forests and sandpiles

We prove that the wired uniform spanning forest exhibits mean-field behaviour on a very large class of graphs, including every transitive graph of at least quintic volume growth and every bounded degree nonamenable graph. Several of our results are new even in the case of mathbb {Z}^d, dge 5. In particular, we prove that every tree in the forest has spectral dimension 4/3 and walk dimension 3 almost surely, and that the critical exponents governing the intrinsic diameter and volume of the past of a vertex in the forest are 1 and 1/2 respectively. (The past of a vertex in the uniform spanning forest is the union of the vertex and the finite components that are disconnected from infinity when that vertex is deleted from the forest.) We obtain as a corollary that the critical exponent governing the extrinsic diameter of the past is 2 on any transitive graph of at least five dimensional polynomial growth, and is 1 on any bounded degree nonamenable graph. We deduce that the critical exponents describing the diameter and total number of topplings in an avalanche in the Abelian sandpile model are 2 and 1/2 respectively for any transitive graph with polynomial growth of dimension at least five, and are 1 and 1/2 respectively for any bounded degree nonamenable graph. In the case of mathbb {Z}^d, dge 5, some of our results regarding critical exponents recover earlier results of Bhupatiraju et al. (Electron J Probab 22(85):51, 2017). In this case, we improve upon their results by showing that the tail probabilities in question are described by the appropriate power laws to within constant-order multiplicative errors, rather than the polylogarithmic-order multiplicative errors present in that work.

Spanning trees and recurrent configurations of a graph

A new explicit relationship between spanning trees and recurrent configurations of a graph is given by constructing subtree. Applying this relationship, we determine the total number of topplings in an avalanche. As applications, we calculate the sizes of avalanches of the Abelian sandpile model on the complete graph, the wheel and the complete bipartite graph.

The Divisible Sandpile at Critical Density

The divisible sandpile starts with i.i.d. random variables ("masses") at the vertices of an infinite, vertex-transitive graph, and redistributes mass by a local toppling rule in an attempt to make all masses at most 1. The process stabilizes almost surely if m<1 and it almost surely does not stabilize if m>1, where $m$ is the mean mass per vertex. The main result of this paper is that in the critical case m=1, if the initial masses have finite variance, then the process almost surely does not stabilize. To give quantitative estimates on a finite graph, we relate the number of topplings to a discrete biLaplacian Gaussian field.

Precursors of catastrophe in the Bak-Tang-Wiesenfeld, Manna, and random-fiber-bundle models of failure.

We have studied precursors of the global failure in some self-organized critical models of sandpile [in Bak-Tang-Wiesenfeld (BTW) and Manna models] and in the random-fiber-bundle model (RFB). In both BTW and Manna model, as one adds a small but fixed number of sand grains (heights) to any central site of the stable pile, the local dynamics starts and continues for an average relaxation time tau and an average number of topplings Delta spread over a radial distance xi. We find that these quantities all depend on the average height h(av) of the pile and they all diverge as h(av) approaches the critical height h(c) from below: Delta approximately (h(c)-h(av))(-delta), tau approximately (h(c)-h(av))(-gamma), and xi approximately (h(c)-h(av))(-nu). Numerically, we find delta approximately 2.0, gamma approximately 1.2, and nu approximately 1.0 for both BTW and Manna model in two dimensions. In the strained RFB model, we find that the breakdown susceptibility chi (giving the differential increment of the number of broken fibers due to increase in external load) and the relaxation time tau, both diverge as the applied load or stress sigma approaches the network failure threshold sigma(c) from below: chi approximately (sigma(c)-sigma)(-1/2) and tau approximately (sigma(c)-sigma)(-1/2). These self-organized dynamical models of failure, therefore, show some definite precursors with robust power laws long before the failure point. Such well-characterized precursors should help predicting the global failure point of the systems in advance.

Proof of breaking of self-organized criticality in a nonconservative abelian sandpile model

By computer simulations, it was reported that the Bak-Tang-Wiesenfeld (BTW) model loses self-organized criticality (SOC) when some particles are annihilated in a toppling process in the bulk of system. We give a rigorous proof that the BTW model loses SOC as soon as the annihilation rate becomes positive. To prove this, a nonconservative Abelian sandpile model is defined on a square lattice, which has a parameter alpha (>/=1) representing the degree of breaking of the conservation law. This model is reduced to be the BTW model when alpha=1. By calculating the average number of topplings in an avalanche <T> exactly, it is shown that for any alpha>1, <T><infinity even in the infinite-volume limit. The power-law divergence of <T> with an exponent 1 as alpha-->1 gives a scaling relation 2nu(2-a)=1 for the critical exponents nu and a of the distribution function of T. The 1-1 height correlation C11(r) is also calculated analytically and we show that C11(r) is bounded by an exponential function when alpha>1, although C11(r) approximately r(-2d) was proved by Majumdar and Dhar for the d-dimensional BTW model. A critical exponent nu(11) characterizing the divergence of the correlation length xi as alpha-->1 is defined as xi approximately |alpha-1|(-nu(11)) and our result gives an upper bound nu(11)</=1/2.

Exact results for the directed Abelian sandpile models

We define the directed Abelian sandpile models by introducing a parameter, c, representing the degree of anisotropy in the avalanche processes, where c = 1 is for the isotropic case. We calculate some quantities characterizing the self-organized critical states on the one- and two-dimensional lattices. In particular, we obtain the expected number of topplings per added particle, (T), which shows the dependence on the lattice size L as Lx for large L. We show that the critical exponent x does not depend on the dimensionality d, at least for d = 1 and 2, and that when any anisotropy is included in the system x = 1, while x = 2 in the isotropic system. This result gives a rigorous proof of the conjecture by Kadanoff et al (1989 A 39 6524-37) that the anisotropy will distinguish different universality classes. We introduce a new critical exponent, θ, defined by X≡limL→∞(T)/L with as for . Both in d = 1 and 2, we obtain θ=1.

Extended operator algebra for abelian sandpile models

We generalize the definition of the operators corresponding to particle addition in the abelian sandpile models to include a phase factor e iϵn , where n is the number of topplings in the avalanche, and ϵ is a real parameter. The new operators so defined are still abelian, and their derivatives with respect to ϵ satisfy an equation similar to the Heisenberg equation for the time-derivative of operators in standard quantum mechanics. The role of the Hamiltonian is played by the toppling function, an operator linear in the height variables of the sandpile, which does not commute with the particle addition operators. We show that moments of the distribution of the number of topplings in avalanches in the steady state are expressed in a simple way in terms of these operators.

On the avalanche-finiteness of Abelian Sandpiles

We prove a necessary and sufficient condition for an Abelian Sandpile Model (ASM) to be avalanche-finite, namely: all unstable states of the system can be brought back to stability in finite number of topplings. The method is also computationally feasible since it involves no greater than O( N 3) arithmetic computations where N is the total number of sites of the system.

Breakdown of simple scaling in Abelian sandpile models in one dimension

We study the abelian sandpile model on decorated one dimensional chains. We determine the structure and the asymptotic form of distribution of avalanche-sizes in these models, and show that these differ qualitatively from the behavior on a simple linear chain. We find that the probability distribution of the total number of topplings $s$ on a finite system of size $L$ is not described by a simple finite size scaling form, but by a linear combination of two simple scaling forms $Prob_L(s) = 1/L f_1(s/L) + 1/L^2 f_2(s/L^2)$, for large $L$, where $f_1$ and $f_2$ are some scaling functions of one argument.

Abelian sandpiles

A class of models for self-organized critical phenomena possesses an isomorphism between the recursive states under addition and the Abelian operator algebra on them. Several exact results follow, including the existence of a unique identity state, which when added to any configuration C in the recursive set relaxes back to that configuration. In this relaxation process, the number of topplings at any lattice site is independent of the configuration C.