Critical behavior studies in ferromagnetic (Nd, K)-Mn-O compounds

The critical scaling behavior of K-doped Nd-Mn-O based double-exchange ferromagnetic compounds was studied by measuring isothermal magnetization of Nd_0.84K_0.16MnO₃ and Nd_0.77K_0.23MnO₃ samples. The critical exponents β, γ and δ corresponding to the spontaneous magnetization, initial susceptibility and isothermal magnetization, respectively, were determined by analyzing the magnetization data in terms of the modified Arrott plot method. The critical exponent values of both samples are found to be comparable to values predicted by a mean field model. The role of ferromagnetic clusters on the scaling behavior is discussed. The critical exponent values are found to be consistent with the Widom scaling relation and the universal scaling hypothesis.     

Multifractal Detrended Cross-Correlation Analysis of sunspot numbers and river flow fluctuations

We use the Detrended Cross-Correlation Analysis (DCCA) to investigate the influence of sun activity represented by sunspot numbers on one of the climate indicators, specifically rivers, represented by river flow fluctuation for Daugava, Holston, Nolichucky and French Broad rivers. The Multifractal Detrended Cross-Correlation Analysis (MF-DXA) shows that there exist some crossovers in the cross-correlation fluctuation function versus time scale of the river flow and sunspot series. One of these crossovers corresponds to the well-known cycle of solar activity demonstrating a universal property of the mentioned rivers. The scaling exponent given by DCCA for original series at intermediate time scale, , is λ=1.17±0.04 which is almost similar for all underlying rivers at 1σ confidence interval showing the second universal behavior of river runoffs. To remove the sinusoidal trends embedded in data sets, we apply the Singular Value Decomposition (SVD) method. Our results show that there exists a long-range cross-correlation between the sunspot numbers and the underlying streamflow records. The magnitude of the scaling exponent and the corresponding cross-correlation exponent are λ∈(0.76,0.85) and γ_×∈(0.30,0.48), respectively. Different values for scaling and cross-correlation exponents may be related to local and external factors such as topography, drainage network morphology, human activity and so on. Multifractal cross-correlation analysis demonstrates that all underlying fluctuations have almost weak multifractal nature which is also a universal property for data series. In addition the empirical relation between scaling exponent derived by DCCA and Detrended Fluctuation Analysis (DFA), is confirmed.     

Phenomenology of local scale invariance: from conformal invariance to dynamical scaling

Statistical systems displaying a strongly anisotropic or dynamical scaling behaviour are characterized by an anisotropy exponent θ or a dynamical exponent z. For a given value of θ (or z), we construct local scale transformations, which can be viewed as scale transformations with a space–time-dependent dilatation factor. Two distinct types of local scale transformations are found. The first type may describe strongly anisotropic scaling of static systems with a given value of θ, whereas the second type may describe dynamical scaling with a dynamical exponent z. Local scale transformations act as a dynamical symmetry group of certain non-local free-field theories. Known special cases of local scale invariance are conformal invariance for θ=1 and Schrödinger invariance for θ=2.The hypothesis of local scale invariance implies that two-point functions of quasiprimary operators satisfy certain linear fractional differential equations, which are constructed from commuting fractional derivatives. The explicit solution of these yields exact expressions for two-point correlators at equilibrium and for two-point response functions out of equilibrium. A particularly simple and general form is found for the two-time autoresponse function. These predictions are explicitly confirmed at the uniaxial Lifshitz points in the ANNNI and ANNNS models and in the aging behaviour of simple ferromagnets such as the kinetic Glauber–Ising model and the kinetic spherical model with a non-conserved order parameter undergoing either phase-ordering kinetics or non-equilibrium critical dynamics.     

Alternative scaling hypothesis forSU(2) andSU(3) lattice gauge theories

High precision data from a variety of sources forSU(2) andSU(3) Wilson action lattice gauge theory are analyzed with respect to the hypothesis of the possible existence of a zero temperature deconfining phase transition, in analogy with theU(1) theory. The internal energy, specific heat, string tension, and Wilson line, fit well to correlation length scaling laws associated with a finite order transition occurring at the weak coupling end of the crossover region for both theories. TheSU(2) theory is consistent with a correlation length exponent ν=2/3 and critical pointβ _c ≈2.47. ForSU(3) the data fit well to ν=1 andβ _c ≈6.69. Additional indirect evidence for the existence of such phase transitions is discussed, as is the possible crucial role of light dynamical fermions in the confinement mechanism.     

Dynamical properties of a dissipative discontinuous map: A scaling investigation

The effects of dissipation on the scaling properties of nonlinear discontinuous maps are investigated by analyzing the behavior of the average squared action 〈I²〉$〈I^2〉$ as a function of the n-th iteration of the map as well as the parameters K and γ  , controlling nonlinearity and dissipation, respectively. We concentrate our efforts to study the case where the nonlinearity is large; i.e., K?1$K?1$. In this regime and for large initial action _I0?K$I_0?K$, we prove that dissipation produces an exponential decay for the average action 〈I〉$〈I〉$. Also, for _I0≅0$I_0\cong 0$, we describe the behavior of 〈I²〉$〈I^2〉$ using a scaling function and analytically obtain critical exponents which are used to overlap different curves of 〈I²〉$〈I^2〉$ onto a universal plot. We complete our study with the analysis of the scaling properties of the deviation around the average action ω.     

Critical behavior of a stochastic anisotropic Bak–Sneppen model

In this paper we present our study on the critical behavior of a stochastic anisotropic Bak–Sneppen (saBS) model, in which a parameter α is introduced to describe the interaction strength among nearest species. We estimate the threshold fitness f _c and the critical exponent τ _r by numerically integrating a master equation for the distribution of avalanche spatial sizes. Other critical exponents are then evaluated from previously known scaling relations. The numerical results are in good agreement with the counterparts yielded by the Monte Carlo simulations. Our results indicate that all saBS models with nonzero interaction strength exhibit self-organized criticality, and fall into the same universality class, by sharing the universal critical exponents.     

Temporal variations of long-term correlations in seismic activity fluctuations

The scaling behavior of the 1998-2009 seismicity in Guerrero, southern Mexico, was studied by means of the detrended fluctuation analysis (DFA). We found that inter-seismic periods are correlated with a transition in the scaling behavior at about 200 seismic events. Correlations are relatively weak for small time scales. However, for large time scales, correlations are associated with a 1/f fractional process, indicating that the seismicity pattern emerges from a self-organized critical state. Temporal variations of the scaling exponent along years computed from the DFA indicate the presence of a quasi-biennial cycle in the seismicity correlations. This cyclic behavior was apparently triggered by the large 2001-2002 slow slip event in the Guerrero seismic gap. Besides, the significant seismic events (M_w>5) originate, on the average, at deeper regions in each cycle.     

Aging in the glass phase of a two-dimensional random periodic elastic system

Using the renormalization group method we investigate the nonequilibrium relaxation of the (Cardy-Ostlund) 2D random sine-Gordon model, which describes pinned arrays of lines. Its statics exhibit a marginal (theta = 0) glass phase for T < Tg described by a line of fixed points. We obtain the universal scaling functions for two-time dynamical response and correlations near Tg for various initial conditions, as well as the autocorrelation exponent. The fluctuation dissipation ratio is found to be nontrivial and continuously dependent on T.     

Finite-size scaling of correlation functions in finite systems

We propose the finite-size scaling of correlation functions in finite systems near their critical points.At a distance r in a ddimensional finite system of size L,the correlation function can be written as the product of|r|~(-(d-2+η))and a finite-size scaling function of the variables r/L and tL~(1/ν),where t=(T-T_c)/T_c,ηis the critical exponent of correlation function,andνis the critical exponent of correlation length.The correlation function only has a sigificant directional dependence when|r|is compariable to L.We then confirm this finite-size scaling by calculating the correlation functions of the two-dimensional Ising model and the bond percolation in two-dimensional lattices using Monte Carlo simulations.We can use the finite-size scaling of the correlation function to determine the critical point and the critical exponentη.     

Classical spin systems with long-range interactions: universal reduction of mixing

We present the numerical study of chaos in a classical model of N coupled rotators on a lattice, in dimensions d=2,3. The coupling constants decay with distance as r_ij^−α (α⩾0). The thermodynamics of the model is extensive if α/d>1 and nonextensive otherwise. For energies above a critical threshold U_c the largest Lyapunov exponent scales as N^−κ, where κ is a universal function of α/d. The function κ decreases from 1/3 to 0 when α/d increases from 0 to 1, and vanishes above 1. We conjecture that this scaling law is related to the nonextensivity of the model, through a power-law sensitivity to initial conditions (weak mixing).     

Roughness evolution of Si surfaces upon Ar ion erosion

We studied the roughness evolution of Si surfaces upon Ar ion erosion in real time. Following the theory of surface kinetic roughening, a model proposed by Majaniemi was used to obtain the value of the dynamic scaling exponent β from our data. The model was found to explain both the observed roughening and the smoothening of the surfaces. The values of the scaling exponents α and β, important for establishing a universal model for ion erosion of (Si) surfaces, have been determined. The value of β proved to increase with decreasing ion energy, while the static scaling exponent α was found to be ion energy independent.     

二层Ising模型的短时临界动力学性质

采用动力学Monte Carlo 方法研究了二层Ising模型的临界性质及早期动力学标度行为.结果表明层间耦合不为零时也存在临界点;计算了早期动力学临界指数θ;估计了传统的临界指数1/νz.其结果支持临界线存在的猜想,并表明此模型很可能是一种弱普适模型. 关键词：     

Extension of scaling to the description of crystals with point defects

The concept of scaling is extended to the case of crystals with point defects. In contrast to scaling for defect-free crystals, in order to describe the critical behavior of a crystal with defects, three independent parameters and not two are required. It is shown that the width of the smoothing region is determined by the relation δ ? X ^α/ν , whereν is the critical exponent describing the temperature dependence of the correlation radius;α is some constant; X is the concentration of point defects. Some new relations are obtained between the critical exponents. These relations are obtained without calling upon the statistical similarity hypothesis.     

Monte Carlo analysis of critical properties of the two-dimensional randomly site-diluted Ising model via Wang-Landau algorithm

The influence of random site dilution on the critical properties of the two-dimensional Ising model on a square lattice was explored by Monte Carlo simulations with the Wang-Landau sampling. The lattice linear size was L=20-120 and the concentration of diluted sites q=0.1,0.2,0.3. Its pure version displays a second-order phase transition with a vanishing specific heat critical exponent α, thus, the Harris criterion is inconclusive, in that disorder is a relevant or irrelevant perturbation for the critical behaviour of the pure system. The main effort was focused on the specific heat and magnetic susceptibility. We have also looked at the probability distribution of susceptibility, pseudocritical temperatures and specific heat for assessing self-averaging. The study was carried out in appropriate restricted but dominant energy subspaces. By applying the finite-size scaling analysis, the correlation length exponent ν was found to be greater than one, whereas the ratio of the critical exponents (α/ν) is negative and (γ/ν) retains its pure Ising model value supporting weak universality.     

Absence of hyperscaling violations for phase transitions with positive specific heat exponent

Finite size scaling theory and hyperscaling are analyzed in the ensemble limit which differs from the finite size scaling limit. Different scaling limits are discussed. Hyperscaling relations are related to the identification of thermodynamics as the infinite volume limit of statistical mechanics. This identification combined with finite ensemble scaling leads to the conclusion that hyperscaling relations cannot be violated for phase transitions with strictly positive specific heat exponent. The ensemble limit allows to derive analytical expressions for the universal part of the finite size scaling functions at the critical point. The analytical expressions are given in terms of generalH-functions, scaling dimensions and a new universal shape parameter. The universal shape parameter is found to characterize the type of boundary conditions, symmetry and other universal influences on critical behaviour. The critical finite size scaling functions for the order parameter distribution are evaluated numerically for the cases =3, =5 and =15 where is the equation of state exponent. Using a tentative assignment of periodic boundary conditions to the universal shape parameter yields good agreement between the analytical prediction and Monte-Carlo simulations for the two dimensional Ising model. Analytical expressions for critical amplitude ratios are derived in terms of critical exponents and the universal shape parameters. The paper offers an explanation for the numerical discrepancies and the pathological behaviour of the renormalized coupling constant in mean field theory. Low order moment ratios of difference variables are proposed and calculated which are independent of boundary conditions, and allow to extract estimates for a critical exponent.     

Trap-size scaling of finite Bose systems within an exact canonical ensemble

Based on an exact canonical partition function, we investigate the trap-size scaling for ideal Bose gases with a finite number of particles N confined in a cubic box or in a harmonic trap. We study the trap-size scaling behaviors of condensate fraction 〈n₀〉/N and specific heat C_N around the transition temperature T_c (i.e., t = T/T_c − 1 → 0) for the three different traps, where a trap exponent θ in dependence of the trapping potential and the universality class of transition are introduced. In the box trap with periodic and Dirichlet boundary conditions, where θ → 1, we find that the scaling functions governing the various critical behaviors are universal but respective of the boundary conditions. The calculated critical exponents are in nice agreement with analytical scaling predictions. The borders of universality validity are obtained numerically. In the case of the harmonic trap, the critical behavior of the system is also found to be universal, and the trap exponent is obtained as θ ? 0.068.     

Finite-size scaling studies of one-dimensional reaction-diffusion systems. Part II. Numerical methods

The scaling exponent and the scaling function for the 1D single-species coagulation model (A+AA) are shown to be universal, i.e., they are not influenced by the value of the coagulation rate. They are independent of the initial conditions as well. Two different numerical methods are used to compute the scaling properties of the concentration: Monte Carlo simulations and extrapolations of exact finite-lattice data. These methods are tested in a case where analytical results are available. To obtain reliable results from finite-size extrapolations, numerical data for lattices up to ten sites are sufficient.     

Scaling in quantum gravity

The 2-point function is the natural object in quantum gravity for extracting critical behavior: The exponential falloff of the 2-point function with geodesic distance determines the fractal dimension d_H of space-time. The integral of the 2-point function determines the entropy exponent γ, i.e. the fractal structure related to baby universes, while the short distance behavior of the 2-point function connects γ and d_H by a quantum gravity version of Fisher's scaling relation. We verify this behavior in the case of 2d gravity by explicit calculation.     

Ferrimagnetic to Paramagnetic Transition in Magnetite: Mössbauer versus Monte Carlo

Powder magnetite was analyzed in situ via Mössbauer with temperatures ranging from 170 K up to 900 K. Hyperfine fields of the tetrahedral and octahedral sites of magnetite as well as the corresponding average field were followed as a function of temperature in order to elucidate the critical behavior of magnetite at around the Curie temperature. Results evidence a progressive collapse of the Mössbauer spectra onto a singlet-type line at a critical temperature of around 870 K characterized by a critical exponent β = 0.28(2) for the hyperfine field. In order to describe such temperature dependence of the hyperfine field, a Monte Carlo-Metropolis simulation based on a stoichiometric magnetite and an Ising model with nearest magnetic neighbor interactions was also carried out. In the model, we have taken into account antiferromagnetic and ferromagnetic interactions depending on the involved ions. A discussion about the critical behavior of magnetite and a comparison between the hyperfine field obtained via Mössbauer and the magnetization obtained via Monte Carlo is finally presented.     

Long time tail correlations in discrete chaotic dynamics

A scaling relation is derived connecting the exponent of the algebraically decaying correlation and response functions with the degree of intermittency and the order of the maximum. It is universal, i.e. within a large class independent of the correlated variables. This implies universal 1/f-like spectra. The corrections to scaling are investigated, too.