Dynamic transitions in Domany-Kinzel cellular automata on small-world network

We study Domany-Kinzel cellular automata on small-world network. Every link on a one dimensional chain is rewired and coupled with any node with probability p. We observe that, the introduction of long-range interactions does not remove the critical character of the model and the system still exhibits a well-defined phase transition to absorbing state. In case of directed percolation (DP), we observe a very anomalous behavior as a function of size. The system shows long lived metastable states and a jump in order parameter. This jump vanishes in thermodynamic limit and we recover second-order transition. The critical exponents are not equal to the mean-field values even for large p. However, for compact directed percolation(CDP), the critical exponents reach their mean-field values even for small p.     

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.     

Nonconservative earthquake model of self-organized criticality on a random graph

We numerically investigate the Olami-Feder-Christensen model on a quenched random graph. Contrary to the case of annealed random neighbors, we find that the quenched model exhibits self-organized criticality deep within the nonconservative regime. The probability distribution for avalanche size obeys finite size scaling, with universal critical exponents. In addition, a power law relation between the size and the duration of an avalanche exists. We propose that this may represent the correct mean-field limit of the model rather than the annealed random neighbor version.     

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     

Statistical properties of one-dimensional binary sequences with power-law power spectrum

By the Fourier filtering method, we generate one-dimensional binary sequences from coarse-grained continuous sequences with preset exponents α₀. Using the spectrum analysis, we find that the corresponding binary sequences have pure 1/f^α power spectrum and spectrum exponents α∈[0.0,2.0], where f is the frequency. We evaluate numerically the relation between α and α₀. Using the autocorrelation function analysis, the detrended fluctuation analysis, the duration time analysis and the entropy analysis, we investigate extensively the statistical properties of such binary sequences. We find that the statistical properties are basically different for α<1 and α>1, and binary sequences become more and more ordered as α increases.     

High precision mean-field results for lattice gauge theories: with special attention paid to the gauge dependence of mean-field perturbation theory to finite order

We do mean-field perturbation theory for U(1) lattice gauge theory in the axial gauge, and evaluate corrections from fluctuations up to fourth order for the free energy and plaquette energy. Comparing with similar results previously obtained in the Feynman gauge we find, to those orders studied, a gauge dependence of the size of the first correction term neglected with one exception. This gauge dependence decreases rapidly as the order of the approximation is increased. To any finite order, results in axial gauge are better approximations than results in the Feynman gauge. We speculate why. Assuming it to be generally true, we evaluate the first correction beyond the one-loop mean-field approximation to the free energy of SU(2) gauge theory with Wilson action in the axial gauge. This correction brings the mean-field result very close to Monte Carlo results for β > 1.6. It also makes the mean-field result identical, within a narrow margin, to ressumed strong coupling results in the interval 1.6 < β < 2.4, thus showing the absence of a phase transition.For both groups studied, we find that the asymptotic series of mean-field perturbation theory give much better approximations than do ordinary weak coupling series.     

Dynamic compensation temperature in the kinetic spin-1 Ising model in an oscillating external magnetic field on alternate layers of a hexagonal lattice

The dynamic behavior of a two-sublattice spin-1 Ising model with a crystal-field interaction (D) in the presence of a time-varying magnetic field on a hexagonal lattice is studied by using the Glauber-type stochastic dynamics. The lattice is formed by alternate layers of spins σ=1 and S=1. For this spin arrangement, any spin at one lattice site has two nearest-neighbor spins on the same sublattice, and four on the other sublattice. The intersublattice interaction is antiferromagnetic. We employ the Glauber transition rates to construct the mean-field dynamical equations. Firstly, we study time variations of the average magnetizations in order to find the phases in the system, and the temperature dependence of the average magnetizations in a period, which is also called the dynamic magnetizations, to obtain the dynamic phase transition (DPT) points as well as to characterize the nature (continuous and discontinuous) of transitions. Then, the behavior of the total dynamic magnetization as a function of the temperature is investigated to find the types of the compensation behavior. Dynamic phase diagrams are calculated for both DPT points and dynamic compensation effect. Phase diagrams contain the paramagnetic (p) and antiferromagnetic (af) phases, the p+af and nm+p mixed phases, nm is the non-magnetic phase, and the compensation temperature or the L-type behavior that strongly depend on the interaction parameters. For D<2.835 and H₀>3.8275, H₀ is the magnetic field amplitude, the compensation effect does not appear in the system.     

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 behavior of a three-state Potts model on a Voronoi lattice

We use the single-histogram technique to study the critical behavior of the three-state Potts model on a (random) Voronoi-Delaunay lattice with size ranging from 250 to 8 000 sites. We consider the effect of an exponential decay of the interactions with the distance, , with a>0, and observe that this system seems to have critical exponents and which are different from the respective exponents of the three-state Potts model on a regular square lattice. However, the ratio remains essentially the same. We find numerical evidences (although not conclusive, due to the small range of system size) that the specific heat on this random system behaves as a power-law for a=0 and as a logarithmic divergence for a=0.5 and a=1.0 Received 5 April 2000     

Strong-Disorder Paramagnetic-Ferromagnetic Fixed Point in the Square-Lattice ±J Ising Model

We consider the random-bond ±J Ising model on a square lattice as a function of the temperature T and of the disorder parameter p (p=1 corresponds to the pure Ising model). We investigate the critical behavior along the paramagnetic-ferromagnetic transition line at low temperatures, below the temperature of the multicritical Nishimori point at T ^*=0.9527(1), p ^*=0.89083(3). We present finite-size scaling analyses of Monte Carlo results at two temperature values, T≈0.645 and T=0.5. The results show that the paramagnetic-ferromagnetic transition line is reentrant for T<T ^*, that the transitions are continuous and controlled by a strong-disorder fixed point with critical exponents ν=1.50(4), η=0.128(8), and β=0.095(5). This fixed point is definitely different from the Ising fixed point controlling the paramagnetic-ferromagnetic transitions for T>T ^*. Our results for the critical exponents are consistent with the hyperscaling relation 2β/ν?η=d?2=0.     

Statistical properties of turbulence in a toroidal magnetized ECR plasma

The statistical analyses of fluctuation data measured by electrostatic-probe arrays clearly show that the self-organized criticality (SOC) avalanches are not the dominant behaviors in a toroidal ECR plasma in the SMT (Simple Magnetic Torus) mode of KT-5D device. The f⁻¹ index region in the auto-correlation spectra of the floating potential Vf and the ion saturation current Is, which is a fingerprint of a SOC system, ranges only in a narrow frequency band. By investigating the Hurst exponents at increasingly coarse grained time series, we find that at a time scale of τ>100 μs, there exists no or a very weak long-range correlation over two decades in τ. The difference between the PDFs of Is and Vf clearly shows a more global nature of the latter. The transport flux induced by the turbulence suggests that the natural intermittency of turbulent transport maybe independent of the avalanche induced by near criticality. The drift instability is dominant in a SMT plasma generated by means of ECR discharges.     

Dynamic hysteresis of magnetic aggregates with non-integer dimension

We investigate the dynamic hysteresis of nanoscale magnetic aggregates by employing Monte Carlo simulation, based on Ising model in non-integer dimensional space. The diffusion-limited aggregation (DLA) model with adjustable sticking probability is used to generate magnetic aggregates with different fractal dimension D. It is revealed that the exponential scaling law A(H₀, ω)∼H₀^α·ω^β, where A is the hysteresis area, H₀ and ω the amplitude and frequency of external magnetic field, applies to both the low-ω and high-ω regimes, while exponents α and β decrease with increasing D in the low-ω regime and keep invariant in the high-ω regime. A mean-field approach is developed to explain the simulated results.     

Self-consistent diagrammatics for a correlated superconductor: contrasting two and three dimensions

We use self-consistent diagrammatics to contrast pairing fluctuations in the Attractive Hubbard Model in two and three dimensions. Specifically, we use the self-consistent T-matrix approximation and show that, quite remarkably, this method yields results that are consistent with the exact 2D-XY and 3D-XY critical scaling. We find that dimensionality alone increases the ratio of the mean-field to actual transition temperatures more than 4-fold in 2D, and that the scaling regimes coincide with the normal state pseudogap regime in both 2D and 3D. Together this suggests that the decreasing effective dimensionality when cuprates are underdoped may account for the increasing size of the pseudogap regime.     

Coevolutionary extremal dynamics on gasket fractal

We considered a Bak-Sneppen model on a Sierpinski gasket fractal. We calculated the avalanche size distribution and the distribution of distances between subsequent minimal sites. To observe the temporal correlations of the avalanche, we estimated the return time distribution, the first-return time, and the all-return time distribution. The avalanche size distribution follows the power law, P(s)∼s−τ, with the exponent τ=1.004(7). The distribution of jumping sites also follows the power law, P(r)∼r−π, with the critical exponent π=4.12(4). We observe the periodic oscillation of the distribution of the jumping distances which originated from the jumps of the level when the minimal site crosses the stage of the fractal. The first-return time distribution shows the power law, Pf(t)∼t−τf, with the critical exponent τf=1.418(7). The all-return time distribution is also characterized by the power law, Pa(t)∼t−τa, with the exponent τa=0.522(4). The exponents of the return time satisfy the scaling relation τf+τa=2 for τf?2.     

Cluster growth poised on the edge of break-up: Size distributions with small exponent power-laws

In many naturally occurring growth processes, cluster size distributions of power-law form n(s)∝s^−τ with small exponents 0<τ<1 are observed. We suggest here that such distributions emerge naturally from cluster growth, where size dependent aggregation is counterbalanced by size dependent break-up. The model used in the study is a simple reaction kinetic model including only monomer-cluster processes. It is shown that under such conditions power-law size distributions with small exponents are obtained. Therefore, the results suggest that the ubiquity of small exponent power-law distributions is related to the growth process, where aggregation driven cluster growth is poised on the edge of cluster break-up.     

Critical behaviour in MSA for a symmetric Yukawa mixture

We study a symmetric binary mixture of equidiameter hard-spheres with repulsive Yukawa interaction for unlike atoms and attractive Yukawa interaction for like atoms. Due to a fully analytical approach we find that, for all values of the Yukawa exponentz0, the critical exponents are of the mean-spherical type. The critical region gets narrower with the increase of the range of the potential, causing a failure of the ordinary numerical analysis. Therefore, the previous analysis, based on numerically accessible region near the critical point, that predicted the mean-field structure of the critical exponents for small values ofz, was not adequate.     

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.     

Effect of calcium deficiency on the critical behavior near the paramagnetic to ferromagnetic phase transition temperature in La0.8Ca0.2MnO3 oxides

The critical behavior associated with the magnetic phase transition has been investigated by magnetization isotherms in La_0.8Ca_0.2MnO₃ and La_0.8Ca_0.1□_0.1MnO₃ (□ is the calcium deficiency). The critical exponents are estimated by various techniques such as the Modified Arrott plot, Kouvel–Fisher plot and critical isotherm technique. The critical exponents values for La_0.8Ca_0.2MnO₃ are very close to those found out by the 3D-Ising model (β=0.328, γ=1.180, and δ=4.826 at an average T_C=181.676 K). Moreover, the estimated critical exponents of La_0.8Ca_0.1□_0.1MnO₃ are consistent with the prediction of the 3D-Heisenberg model (β=0.357, γ=1.167, and δ=4.802 at an average T_C=178.182 K). We noted that the critical exponents γ are almost similar to the value of the mean-field theory which can be explained by the existence of a long-range dipole–dipole interaction. Following the Harris criterion, we deduced that the disorder in our case is relevant, which can be the cause of the change in the universality class.