Abelian deterministic self-organized-criticality model: complex dynamics of avalanche waves.

The aim of this study is to investigate a wave dynamics and a size scaling of avalanches which were created by the mathematical model [J. Cernák, Phys. Rev. E 65, 046141 (2002)]. Numerical simulations were carried out on a two-dimensional lattice L×L in which two constant thresholds E(c)(I) = 4 and E(c)(II) > E(c)(I) were randomly distributed. The density of sites c of the thresholds E(c)(II) and threshold E(c)(II) are parameters of the model. Autocorrelations of avalanche size waves, Hurst exponents, avalanche structures, and avalanche size moments were determined for several densities c and thresholds E(c)(II). The results show correlated avalanche size waves and multifractal scaling of avalanche sizes not only for specific conditions, densities c = 0.0,1.0 and thresholds 8 ≤ E(c)(II) ≤ 32, in which relaxation rules were precisely balanced, but also for more general conditions, densities 0.0 < c < 1.0 and thresholds 8 ≤ E(c)(II) ≤ 32, in which relaxation rules were unbalanced. The results suggest that the hypothesis of a precise relaxation balance could be a specific case of a more general rule.

Aggregation of magnetic holes in a rotating magnetic field.

We experimentally investigated field-induced aggregation of nonmagnetic particles confined in a magnetic fluid layer when rotating magnetic fields were applied. After application of a magnetic field rotating in the plane of the fluid layer, the single particles start to form two-dimensional clusters, like dimers, trimers, and more complex structures. These clusters aggregated again and again to form bigger clusters. During this nonequilibrium process, a broad range of cluster sizes was formed, and the scaling exponents z and z;{'} of the number of clusters N(t) approximately t;{-z;{'}} and average cluster size S(t) approximately t;{z} were calculated. The process could be characterized as diffusion-limited cluster-cluster aggregation. We found that all sizes of clusters that occurred during an experiment fall on a single curve, as the dynamic scaling theory predicts. However, the characteristic scaling exponents z;{'},z and crossover exponents Delta were not universal. A particle tracking method was used to find the dependence of the diffusion coefficients D_{s} on cluster size s . The cluster motions show features of Brownian motion. The average diffusion coefficients D_{s} depend on the cluster size s as a power law D_{s} proportional, variants;{gamma} where values of gamma as different as gamma=-0.62+/-0.19 and gamma=-2.08+/-0.51 were found in two of the experiments.

Inhomogeneous sandpile model: Crossover from multifractal scaling to finite-size scaling.

We study an inhomogeneous sandpile model in which two different toppling rules are defined. For any site only one rule is applied corresponding to either the Bak, Tang, and Wiesenfeld model [P. Bak, C. Tang, and K. Wiesenfeld, Phys. Rev. Lett. 59, 381 (1987)] or the Manna two-state sandpile model [S. S. Manna, J. Phys. A 24, L363 (1991)]. A parameter c is introduced which describes a density of sites which are randomly deployed and where the stochastic Manna rules are applied. The results show that the avalanche area exponent tau a, avalanche size exponent tau s, and capacity fractal dimension Ds depend on the density c. A crossover from multifractal scaling of the Bak, Tang, and Wiesenfeld model (c = 0) to finite-size scaling was found. The critical density c is found to be in the interval 0 < c < 0.01. These results demonstrate that local dynamical rules are important and can change the global properties of the model.

Aggregation dynamics of nonmagnetic particles in a ferrofluid.

Nonmagnetic microspheres confined in a ferrofluid layer are denoted by magnetic holes. They form aggregates due to dipolar interactions when an external magnetic field is exerted. Their cluster-cluster aggregation was studied for various magnetic fields using optical microscopy, both for small spheres of diameters, d=1.9 and 4 microm, for which Brownian motion was important and for large spheres of diameter, d=14 microm, for which Brownian motion was not important. The results for the two smaller sizes were in agreement with standard dynamic scaling theory and the dynamic scaling exponent z for the average cluster length S(t) approximately t(z) was found to be slightly smaller than 0.5, while for the largest spheres the z exponent showed a strong dependence on the magnetic-field strength.

Self-organized criticality: robustness of scaling exponents.

We investigate a deterministic, conservative, undirected, critical height sandpile model with dissipation of an energy at boundaries that can simulate avalanche dynamics. It was derived from the Bak-Tang-Wiesenfeld model [P. Bak, C. Tang, and K. Wiesenfeld, Phys. Rev. Lett. 59, 381 (1987)] introducing an additional second-higher threshold so the model has two distinct thresholds. Our computer simulations for a two-dimensional lattice show that scaling properties of the model depend on the higher-threshold values and site concentrations. These results are not therefore consistent with the present self-organized criticality hypothesis where the scaling properties are independent of the model parameters.