The Abelian Manna model on various lattices in one and two dimensions (original) (raw)
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Abelian Manna model in three dimensions and below
Physical Review E, 2012
The Abelian Manna model of self-organized criticality is studied on various three-dimensional and fractal lattices. The exponents for avalanche size, duration, and area distribution of the model are obtained by using a high-accuracy moment analysis. Together with earlier results on lower-dimensional lattices, the present results reinforce the notion of universality below the upper critical dimension and allow us to determine the coefficients of an expansion. By rescaling the critical exponents by the lattice dimension and incorporating the random walker dimension, a remarkable relation is observed, satisfied by both regular and fractal lattices.
Abelian Manna model on two fractal lattices
Physical Review E, 2010
We analyze the avalanche size distribution of the Abelian Manna model on two different fractal lattices with the same dimension dg = ln 3/ ln 2, with the aim to probe for scaling behavior and to study the systematic dependence of the critical exponents on the dimension and structure of the lattices. We show that the scaling law D(2 − τ) = dw generalizes the corresponding scaling law on regular lattices, in particular hypercubes, where dw = 2. Furthermore, we observe that the lattice dimension dg, the fractal dimension of the random walk on the lattice dw, and the critical exponent D, form a plane in 3D parameter space, i.e. they obey the linear relationship D = 0.632(3)dg + 0.98(1)dw − 0.49(3).
2012
The Abelian Manna model of self-organized criticality is studied on various three-dimensional and fractal lattices. The exponents for avalanche size, duration and area distribution of the model are obtained by using a high-accuracy moment analysis. Together with earlier results on lowerdimensional lattices, the present results reinforce the notion of universality below the upper critical dimension and allow us to determine the coefficients of an ǫ-expansion. By rescaling the critical exponents by the lattice dimension and incorporating the random walker dimension , a remarkable relation is observed, satisfied by both regular and fractal lattices.
Physical Review E, 1995
We study the Abelian sandpile model on decorated one-dimensional chains. We show that there are two types of avalanches, and determine the effects of finite, though large, system size I on the asymptotic form of distributions of avalanche sizes, and show that these differ qualitatively from the behavior on a simple linear chain. For large L, we find that the probability distribution of the total number of topplings 8 is not described by a simple finite-size scaling form, but by a linear combination of two simple scaling forms: ProbL, (s) = z fi(z) + b f2(~z), where fi and f2 are nonuniversal scaling functions of one argument.
Avalanche size distribution in a random walk model
arXiv (Cornell University), 1996
We introduce a simple model for the size distribution of avalanches based on the idea that the front of an avalanche can be described by a directed random walk. The model captures some of the qualitative features of earthquakes, avalanches and other self-organized critical phenomena in one dimension. We find scaling laws relating the frequency, size and width of avalanches and an exponent 4/3 in the size distribution law.
Journal of the Physical Society of Japan, 1999
A site-percolation model with two different sizes of particles is newly introduced on a square lattice. To estimate the fractal dimension and critical exponents with high accuracy, a finite-size scaling analysis is performed with a Monte Carlo simulation. The obtained fractal dimension coincides with that of the ordinary model, while the estimated exponents β, γ, and ν are slightly different from those of the ordinary model. However, the normalized exponentsβ = β/ν and γ = γ/ν remain the same as in the ordinary model, which is a manifestation of weak universality.
Scaling behavior in very small percolation lattices
Physical Review C, 1997
We examine the average cluster distribution as a function of lattice probability for a very small (Lϭ6) lattice and determine the scaling function of three-dimensional percolation. The behavior of the second moment, calculated from the average cluster distribution of Lϭ6 and Lϭ63 lattices, is compared to power-law behavior predicted by the scaling function. We also examine the finite-size scaling of the critical point and the size of the largest cluster at the critical point. This analysis leads to estimates of the critical exponent and the ratio of critical exponents /. ͓S0556-2813͑97͒02703-9͔
Rare events and breakdown of simple scaling in the Abelian sandpile model
Physical Review E, 1998
Due to intermittency and conservation, the Abelian sandpile in 2D obeys multifractal, rather than finite size scaling. In the thermodynamic limit, a vanishingly small fraction of large avalanches dominates the statistics and a constant gap scaling is recovered in higher moments of the toppling distribution. Thus, rare events shape most of the scaling pattern and preserve a meaning for effective exponents, which can be determined on the basis of numerical and exact results.
Series study of percolation moments in general dimension
Physical Review B, 1990
Series expansions for general moments of the bond-percolation cluster-size distribution on hypercubic lattices to 15th order in the concentration have been obtained. This is one more than the previously published series for the mean cluster size in three dimensions and four terms more for higher moments and higher dimensions. Critical exponents, amplitude ratios, and thresholds have been calculated from these and other series by a variety of independent analysis techniques. A comprehensive summary of extant estimates for exponents, some universal amplitude ratios, and thresholds for percolation in all dimensions is given, and our results are shown to be in excellent agreement with the ε expansion and some of the most accurate simulation estimates. We obtain threshold values of 0.2488±0.0002 and 0.180 25±0.000 15 for the three-dimensional bond problem on the simple-cubic and body-centered-cubic lattices, respectively, and 0.160 05±0.000 15 and 0.118 19±0.000 04, for the hypercubic bond problem in four and five dimensions, respectively. Our direct exponent estimates are γ=1.805±0.
Ordered Avalanches on the Bethe Lattice
Entropy, 2019
We discuss deterministic sequences of avalanches on a directed Bethe lattice. The approach is motivated by the phenomenon of self-organized criticality. Grains are added only at one node of the network. When the number of grains at any node exceeds a threshold b, each of k out-neighbors gets one grain. The probability of an avalanche of size s is proportional to s - τ . When the avalanche mass is conserved ( k = b ), we get τ = 1 . For an application of the model to social phenomena, the conservation condition can be released. Then, the exponent τ is found to depend on the model parameters; τ ≈ l o g ( b ) / l o g ( k ) . The distribution of the time duration of avalanches is exponential. Multifractal analysis of the avalanche sequences reveals their strongly non-uniform fractal organization. Maximal value of the singularity strength α m a x in the bifractal spectrum is found to be 1 / τ .