Scaling behavior of the first arrival time of a random-walking magnetic domain

We report a universal scaling behavior of the first arrival time of a traveling magnetic domain wall into a finite space-time observation window of a magneto-optical microscope enabling direct visualization of a Barkhausen avalanche in real time. The first arrival time of the traveling magnetic domain wall exhibits a nontrivial fluctuation and its statistical distribution is described by universal power-law scaling with scaling exponents of 1.34+/-0.07 for CoCr and CoCrPt films, despite their quite different domain evolution patterns. Numerical simulation of the first arrival time with an assumption that the magnetic domain wall traveled as a random walker well matches our experimentally observed scaling behavior, providing an experimental support for the random-walking model of traveling magnetic domain walls.     

Direct observation of Barkhausen avalanche in Co thin films

We report direct full-field magneto-optical observations of Barkhausen avalanches in Co polycrystalline thin films at criticality. We provide experimental evidence for the validity of a phenomenological model of the Barkhausen avalanche originally proposed by Cizeau, Zapperi, Durin, and Stanley [Phys. Rev. Lett. 79, 4669 (1997)]], where the model describes a 180 degrees -type flexible domain wall deformed by a localized defect with consideration of long-range dipolar interaction. The Barkhausen jump areas show a power-law scaling distribution with critical exponent tau approximately 1.33 for all the samples having different thickness from 5 to 50 nm, which is in accord with the two-dimensional prediction of the model.     

Anomalous Scaling Behaviors in a Rice-Pile Model with Two Different Driving Mechanisms

The moment analysis is applied to perform large scale simulations of the rice-pile model. We find that this model shows different scaling behavior depending on the driving mechanism used. With the noisy driving, the rice-pile model violates the finite-size scaling hypothesis, whereas, with fixed driving, it shows well defined avalanche exponents and displays good finite size scaling behavior for the avalanche size and time duration distributions.     

Anomalous Scaling Behaviors in a Rice-Pile Model with Two Different Driving Mechanisms

The moment analysis is applied to perform large scale simulations of the rice-pile model. We find that this model shows different scaling behavior depending on the driving mechanism used. With the noisy driving, the rice-pile model violates the finite-size scaling hypothesis, whereas, with fixed driving, it shows well defined avalanche exponents and displays good finite size scaling behavior for the avalanche size and time duration distributions.     

Percolation of diffusional creep: a new universality class

We study the percolation aspects of diffusional "Coble" creep on heterogeneous grain boundary networks, assuming free grain boundary sliding. A novel percolation threshold is obtained for the honeycomb lattice when two representative types of grain boundaries are randomly distributed, p(cc)=0.5416+/-0.0036. The creep viscosity diverges near the percolation threshold with power-law exponents t=1.69+/-0.09 and s=1.88+/-0.12, different from the standard conduction and rigidity percolation exponents. The moments of both the force and flux distributions all conform to finite-size scaling at p(cc), but with new exponents. These new scaling behaviors seen in the creeping system are proposed to arise from the unique coupling of both force and flux balances in the network.     

Sandpile on scale-free networks

We investigate the avalanche dynamics of the Bak-Tang-Wiesenfeld sandpile model on scale-free (SF) networks, where the threshold height of each node is distributed heterogeneously, given as its own degree. We find that the avalanche size distribution follows a power law with an exponent tau. Applying the theory of the multiplicative branching process, we obtain the exponent tau and the dynamic exponent z as a function of the degree exponent gamma of SF networks as tau=gamma divided by (gamma-1) and z=(gamma-1) divided by (gamma-2) in the range 23, with a logarithmic correction at gamma=3. The analytic solution supports our numerical simulation results. We also consider the case of a uniform threshold, finding that the two exponents reduce to the mean-field ones.     

Driving rate effects on crackling noise

Many systems respond to slowly changing external conditions with crackling noise, created by avalanches or pulses with a broad range of sizes. Examples range from Barkhausen noise (BN) in magnets to earthquakes. In this Letter, we discuss the effects of increasing driving rate Omega on the scaling behavior of the avalanche size and duration distributions as well as qualitative effects of Omega on the power spectra. We derive an exponent inequality as a criteria for the relevance of Omega. To illustrate these general results, we use recent experiments on BN as a successful example.     

Critical Behaviors in a Stochastic One-Dimensional Sand-Pile Model

A one-dimensional sand-pile model (Manna model), which has a stochastic redistribution process, is studied both in discrete and continuous manners. The system evolves into a critical state after a transient period. A detailed analysis of the probability distribution of the avalanche size and duration is numerically investigated. Interestingly,contrary to the deterministic one-dimensional sand-pile model, where multifractal analysis works well, the analysis based on simple finite-size scaling is suited to fitting the data on the distribution of the avalanche size and duration. The exponents characterizing these probability distributions are measured. Scaling relations of these scaling exponents and their universality class are discussed.     

Critical Behaviors in a Stochastic One-Dimensional Sand-Pile Model

A one-dimensional sand-pile model （Manna model）, which has a stochastic redistribution process, is studied both in discrete and continuous manners. The system evolves into a critical state after a transient period. A detailed analysis of the probability distribution of the avalanche size and duration is numerically investigated. Interestingly,contrary to the deterministic one-dimensional sand-pile model, where multifractal analysis works well, the analysis based on simple finite-size scaling is suited to fitting the data on the distribution of the avalanche size and duration. The exponents characterizing these probability distributions are measured. Scaling relations of these scaling exponents and their universality class are discussed.     

Time distribution and loss of scaling in granular flow

Two cellular automata models with directed mass flow and internal time scales are studied by numerical simulations. Relaxation rules are a combination of probabilistic critical height (probability of toppling p) and deterministic critical slope processes with internal correlation time t_c equal to the avalanche lifetime, in model A, and ,in model B. In both cases nonuniversal scaling properties of avalanche distributions are found for , where is related to directed percolation threshold in d=3. Distributions of avalanche durations for are studied in detail, exhibiting multifractal scaling behavior in model A, and finite size scaling behavior in model B, and scaling exponents are determined as a function of p. At a phase transition to noncritical steady state occurs. Due to difference in the relaxation mechanisms, avalanche statistics at approaches the parity conserving universality class in model A, and the mean-field universality class in model B. We also estimate roughness exponent at the transition. Received: 29 May 1998 / Revised: 8 September 1998 / Accepted: 10 September 1998     

On self-organised criticality in one dimension

In critical phenomena, many of the characteristic features encountered in higher dimensions such as scaling, data collapse and associated critical exponents are also present in one dimension. Likewise for systems displaying self-organised criticality. We show that the one-dimensional Bak–Tang–Wiesenfeld sandpile model, although trivial, does indeed fall into the general framework of self-organised criticality. We also investigate the Oslo ricepile model, driven by adding slope units at the boundary or in the bulk. We determine the critical exponents by measuring the scaling of the kth moment of the avalanche size probability with system size. The avalanche size exponent depends on the type of drive but the avalanche dimension remains constant.     

Avalanche dynamics of idealized neuron function in the brain on an uncorrelated random scale-free network

We study a simple model for a neuron function in a collective brain system. The neural network is composed of an uncorrelated configuration model (UCM) for eliminating the degree correlation of dynamical processes. The interaction of neurons is assumed to be isotropic and idealized. These neuron dynamics are similar to biological evolution in extremal dynamics with locally isotropic interaction but has a different time scale. The functioning of neurons takes place as punctuated patterns based on avalanche dynamics. In our model, the avalanche dynamics of neurons exhibit self-organized criticality which shows power-law behavior of the avalanche sizes. For a given network, the avalanche dynamic behavior is not changed with different degree exponents of networks, γ≥2.4 and various refractory periods referred to the memory effect, T_r. Furthermore, the avalanche size distributions exhibit power-law behavior in a single scaling region in contrast to other networks. However, return time distributions displaying spatiotemporal complexity have three characteristic time scaling regimes Thus, we find that UCM may be inefficient for holding a memory.     

Critical Behaviors in a Stochastic Local Limited One-Dimensional Rice-Pile Model

A stochastic local limited one-dimensional rice-pile model is numerically investigated. The distributions for avalanche sizes have a clear power-law behavior and it displays a simple finite size scaling. We obtain the avalanche exponents τ_s=1.54±0.10, β_s=2.17±0.10 and τ_T=1.80±0.10, β_T=1.46±0.10. This self-organized critical model belongs to the same universality class with the Oslo rice-pile model studied by K. Christensen et al. [Phys. Rev. Lett. 77 (1996) 107], a rice-pile model studied by L.A.N. Amaral et al. [Phys. Rev. E 54 (1996) 4512], and a simple deterministic self-organized critical model studied by M.S. Vieira [Phys. Rev. E 61 (2000) 6056].     

Rapid roughening in thin film growth of an organic semiconductor (diindenoperylene)

The scaling exponents alpha, beta, and 1/z in thin films of the organic molecule diindenoperylene deposited on SiO2 under UHV conditions are determined. Atomic-force microscopy, x-ray reflectivity, and diffuse x-ray scattering were employed. The surface width displays power law scaling over more than 2 orders of magnitude in film thickness. We obtained alpha = 0.684+/-0.06, beta = 0.748+/-0.05, and 1/zeta = 0.92+/-0.20. The derived exponents point to an unusually rapid growth of vertical roughness and lateral correlations. We suggest that they could be related to lateral inhomogeneities arising from the formation of grain boundaries between tilt domains in the early stages of growth.     

Anomalous roughening of viscous fluid fronts in spontaneous imbibition

We report experiments on spontaneous imbibition of a viscous fluid by a model porous medium in the absence of gravity. The average position of the interface satisfies Washburn's law. Scaling of the interface fluctuations suggests a dynamic exponent z approximately 3, indicative of global dynamics driven by capillary forces. The complete set of exponents clearly shows that interfaces are not self-affine, exhibiting distinct local and global scaling, both for time (beta = 0.64 +/- 0.02, beta(*) = 0.33 +/- 0.03) and space (alpha = 1.94 +/- 0.20, alpha(loc) = 0.94 +/- 0.10). These values are compatible with an intrinsic anomalous scaling scenario.     

Scaling exponents and probability distributions of DNA end-to-end distance

The scaling of the average gyration radius of polymers as a function of their length can be experimentally determined from ensemble measurements, such as light scattering, and agrees with analytical estimates. Ensemble techniques, yet, do not give access to the full probability distributions. Single molecule techniques, instead, can deliver information on both average quantities and distribution functions. Here we exploit the high resolution of atomic force microscopy over long DNA molecules adsorbed on a surface to measure the average end-to-end distance as a function of the DNA length, and its full distribution function. We find that all the scaling exponents are close to the predicted 3D values (upsilon=0.589+/-0.006 and delta=2.58+/-0.77). These results suggest that the adsorption process is akin to a geometric projection from 3D to 2D, known to preserve the scaling properties of fractal objects of dimension df<2.     

Exact results for quench dynamics and defect production in a two-dimensional model

We show that for a d-dimensional model in which a quench with a rate tau(-1) takes the system across a (d-m)-dimensional critical surface, the defect density scales as n approximately 1/tau(mnu/(znu+1)), where nu and z are the correlation length and dynamical critical exponents characterizing the critical surface. We explicitly demonstrate that the Kitaev model provides an example of such a scaling with d = 2 and m = nu = z = 1. We also provide the first example of an exact calculation of some multispin correlation functions for a two-dimensional model that can be used to determine the correlation between the defects. We suggest possible experiments to test our theory.     

Scaling of a Slope: The Erosion of Tilted Landscapes

We formulate a stochastic equation to model the erosion of a surface with fixed inclination. Because the inclination imposes a preferred direction for material transport, the problem is intrinsically anisotropic. At zeroth order, the anisotropy manifests itself in a linear equation that predicts that the prefactor of the surface height–height correlations depends on direction. The first higher order nonlinear contribution from the anisotropy is studied by applying the dynamic renormalization group. Assuming an inhomogeneous distribution of soil substrate that is modeled by a source of static noise, we estimate the scaling exponents at first order in an ε-expansion. These exponents also depend on direction. We compare these predictions with empirical measurements made from real landscapes and find good agreement. We propose that our anisotropic theory applies principally to small scales and that a previously proposed isotropic theory applies principally to larger scales. Lastly, by considering our model as a transport equation for a driven diffusive system, we construct scaling arguments for the size distribution of erosion “events” or “avalanches.” We derive a relationship between the exponents characterizing the surface anisotropy and the avalanche size distribution, and indicate how this result may be used to interpret previous findings of power-law size distributions in real submarine avalanches.     

Defect production in nonlinear quench across a quantum critical point

We show that the defect density n, for a slow nonlinear power-law quench with a rate tau(-1) and an exponent alpha>0, which takes the system through a critical point characterized by correlation length and dynamical critical exponents nu and z, scales as n approximately tau(-alphanud/(alphaznu+1)) [n approximately (alphag((alpha-1)/alpha)/tau)(nud/(znu+1))] if the quench takes the system across the critical point at time t=0 [t=t(0) not = 0], where g is a nonuniversal constant and d is the system dimension. These scaling laws constitute the first theoretical results for defect production in nonlinear quenches across quantum critical points and reproduce their well-known counterpart for a linear quench (alpha=1) as a special case. We supplement our results with numerical studies of well-known models and suggest experiments to test our theory.     

Corrections to scaling and probability distribution of avalanches for the stochastic Zhang sandpile model

We study the distributions of dissipative and nondissipative avalanches separately in the stochastic Zhang (SP-Z) sandpile in two dimension. We find that dissipative and nondissipative avalanches obey simple power laws and do not have the logarithmic correction, while the avalanche distributions in the Abelian Manna model should include a logarithmic correction. We use the moment analysis to determine the numerical critical exponents of dissipative and nondissipative avalanches, respectively, and find that they are different from the corresponding values in the Abelian Manna model. All these indicate that the stochastic Zhang model and the Abelian Manna model belong to distinct universality classes, which imply that the Abelian symmetry breaking changes the scaling behavior of the avalanches in the case of the stochastic sandpile model.