Effects of Monomer Size Distribution on the fractal dimensionality of diffusion-limited aggregates

Computer simulations of diffusion-limited aggregation (DLA) for monomers to investigate the effects of size and of lognormal distribution on the fractal dimensionality of the aggregates were conducted on a two-dimensional lattice. The results show the DLA clusters posses multifractal characteristics. For clusters consisting of monodisperse monomers, the bifurcation point on the graph of the pair correlation function (PCF) for each cluster is located right at the monomers size under investigation The textural dimension (D_f1) has a stable value of about 1.65, whereas the structural dimension (D_f2) decreased with increase in monomer size. For the cases with monomers in log-normal distributions, the textural dimension is around 1.67; however, the structural dimension decreases with increasing polydispersity of monomer size.     

湍流大气中的准直激光束III.分形结构与相位不连续点饶瑞中

 利用数值方法研究了不同起伏条件下准直激光束在湍流大气中的传播，分析了光斑的分形维数以及相位不连续点数目的统计特征。结果表明：随着起伏条件的增大，光斑的分形维数以及相位不连续点数目增大；光斑的分形维数与锐度（描述光斑质量的参量）有一定的关系，但相位不连续点数目与光斑质量不存在确定的关系。在本文的计算条件下，分形维数一直随起伏条件的增大而增大，没有出现类似于闪烁饱和的现象。在一定的起伏条件下，相位不连续点数目具有一定的统计分布，而不是一个确定的值，并且具有相当的发散性。     

The influence of fractal size distribution of covers on radon exhalation from uranium mill tailings

Tailings produced during mining and milling of uranium ores represent potentially large volumes of low level radioactive materials. A typical environmental problem associated with mill tailings is radon emanation. Covering tailings is widely applied to reduce radon exhalation rate. In this paper, the fractal theories and field covering tests are used to study the fractal characters of size distribution of six types of covering materials, including waste rock, sand, laterite, kaolin, mixture of sand and laterite, and mixture of waste rock and laterite, and their influences on radon exhalation. The size distributions of uranium tailings and the six aforementioned covering materials all exhibit a good fractal structure. The contents of fine grain increase with the increasing value of fractal dimension. The results of field radon measurement show that the radon emanation rate of tailings without covers is 14.7–18.6 Bq/m² s. Covering tests were carried out of the six abovementioned covering materials with thickness of 0.4 m, 0.8 m, 1.2 m, 1.6 m and 2.0 m. The results indicate that the application of these materials for cover layers can decrease the radon exhalation rate markedly. The effectiveness of a cover layer in reducing radon exhalation is related to its fractal texture of size distribution. Under the same thickness conditions, the attenuation coefficient of radon exhalation rate increases with the increasing fractal dimension of size distribution of covers. The empirical expressions of the attenuation coefficients in relation to fractal dimension D of size distribution and thickness x of covers is obtained for evaluating the effectiveness of final covers for uranium tailings impoundments.     

Electromagnetic radiation of charges with fractal distribution

The radiation of charges with fractal distributions is studied in the framework of classical electrodynamics. The properties of the form factork(x) of the density of the fractal distribution are investigated. It is shown that the nature of the decrease of the averaged square of the form factor modulus with increasing wave vectorx of the radiation is determined by the dimension D of the fractal distribution of the charge: ¦k(x)¦^2–D .Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 111–116, January, 1992.The author thanks Professor V. G. Bagrov for a helpful discussion of the questions considered in this paper.     

Universal critical properties of the Eulerian bond-cubic model

We investigate the Eulerian bond-cubic model on the square lattice by means of Monte Carlo simulations,using an efficient cluster algorithm and a finite-size scaling analysis.The critical points and four critical exponents of the model are determined for several values of n.Two of the exponents are fractal dimensions,which are obtained numerically for the first time.Our results are consistent with the Coulomb gas predictions for the critical O(n) branch for n < 2 and the results obtained by previous transfer matrix calculations.For n=2,we find that the thermal exponent,the magnetic exponent and the fractal dimension of the largest critical Eulerian bond component are different from those of the critical O(2) loop model.These results confirm that the cubic anisotropy is marginal at n=2 but irrelevant for n < 2.     

The absence of phase transition for the classical XY-model on Sierpiński pyramid with fractal dimension D=2

For the spin models with continuous symmetry on regular lattices and finite range of interactions, the lower critical dimension is d?=?2. In two dimensions the classical XY-model displays Berezinskii–Kosterlitz–Thouless (BKT) transition associated with unbinding of topological defects (vortices and antivortices). We perform a Monte Carlo study of the classical XY-model on Sierpiński pyramids (SPs) whose fractal dimension is D = log?4/log?2?=?2 and the average coordination number per site is ≈ 7. The specific heat does not depend on the system size which indicates the absence of a long-range order. From the dependence of the helicity modulus on the cluster size and on boundary conditions, we draw a conclusion that in the thermodynamic limit there is no BKT transition at any finite temperature. This conclusion is also supported by our results for linear magnetic susceptibility. The lack of finite temperature phase transition is presumably caused by the finite order of ramification of SP.     

Approach to an irregular time series on the basis of the fractal theory

We present a technique to measure the fractal dimension of the set of points (t, f(t)) forming the graph of a function f defined on the unit interval. First we apply it to a fractional Brownian function [1] which has a property of self-similarity for all scales, and we can get the stable and precise fractal dimension. This technique is also applied to the observational data of natural phenomena. It does not show self-similarity all over the scale but has a different self-similarity across the characteristic time scale. The present method gives us a stable characteristic time scale as well as the fractal dimension.     

Wavefront Sets in Algebraic Quantum Field Theory

The investigation of wavefront sets of n-point distributions in quantum field theory has recently acquired some attention stimulated by results obtained with the help of concepts from microlocal analysis in quantum field theory in curved spacetime. In the present paper, the notion of wavefront set of a distribution is generalized so as to be applicable to states and linear functionals on nets of operator algebras carrying a covariant action of the translation group in arbitrary dimension. In the case where one is given a quantum field theory in the operator algebraic framework, this generalized notion of wavefront set, called “asymptotic correlation spectrum”, is further investigated and several of its properties for physical states are derived. We also investigate the connection between the asymptotic correlation spectrum of a physical state and the wavefront sets of the corresponding Wightman distributions if there is a Wightman field affiliated to the local operator algebras. Finally we present a new result (generalizing known facts) which shows that certain spacetime points must be contained in the singular supports of the 2n-point distributions of a non-trivial Wightman field. Received: 27 July 1998 / Accepted: 3 March 1999     

Fluctuations in the Curie-Weiss version of the random field Ising model

The fluctuations of the order parameter in the Curie-Weiss version of the Ising model with random magnetic field are computed. Away from criticality or at first-order critical points they have a Gaussian distribution with random (i. e.,sample-dependent) mean, thermal fluctuations contributing in same order as the fluctuations of the field; at second- or higher-order critical points, non-Gaussian sample-dependent distributions appear, and the fluctuations of the fields are enhanced, dominating over the thermal ones.     

Catastrophes in the multi-fractal dynamics of social-economic systems

In the present paper, the concept of multi-fractal dynamics is developed. The problem concerning catastrophes in this dynamics is studied in detail. In the framework of the concept of fractal curve as a thick curve, it is proved that the cell approach to measuring the fractal dimension D is equivalent to measuring the dependence of the length L of the line on the scope δ. The introduction of a fractal scale of temperatures T ^f is suggested.     

Resistive state of superconducting structures with fractal clusters of a normal phase

The effect of morphologic factors on magnetic flux trapping and critical currents in a superconducting structure, which presents a type II percolation superconductor with pinning centers, is considered. The role of pinning centers is played by fractal clusters of the normal phase. The properties of these clusters are analyzed in detail: their statistics is studied, the distribution of critical currents of depinning is found, and the depen-dences of the main statistical parameters on the fractal dimension are obtained. The effect of fractal clusters of the normal phase on the electric field caused by the motion of the magnetic flux after the vortices have been broken away from pinning centers is considered. The current-voltage characteristics of superconducting structures in a resistive state are obtained for an arbitrary fractal dimension. It is found that the fractality of the boundaries of normal-phase clusters forces magnetic flux trapping, thereby increasing the critical current.     

基于分形多孔介质三维网络模型的非混溶两相流驱替数值模拟

提出了一个描述多孔介质孔隙尺寸分布的三维分形网络模型,利用该模型对多孔介质中的非混溶两相流驱替进行了数值模拟,研究了孔隙尺寸分布分维D_f和两相流黏滞比M对驱替前沿指进型的影响,结果表明指进型容量维数D_h随着孔隙尺寸分布分维D_f以及黏滞比M的增大而减少,并通过曲线拟合得到了它们之间的定量关系. 关键词： 多孔介质 三维网络 黏滞指进 非混溶两相流     

Parametric excitation of spin waves in ferromagnets by longitudinal pumping localized in space

The equations of motion for the slowly varying complex amplitudes of spin waves parametrically excited by a localized pumping magnetic field have been derived. A solution of these equations satisfying given boundary and initial conditions has been obtained. The energy dissipated by spin waves decreases with the pumping intensity beyond a certain pumping power, which can be termed the regeneration threshold. The losses vanish and change sign at the instability threshold. Both thresholds depend heavily on the linear dimension L of the pumping zone, increasing with decreasing L. Owing to the regeneration process, the dissipation length of spin waves increases without bound as the pumping power approaches the instability threshold. Consequently, perturbations of a uniform state due to the boundary penetrate throughout the pumping zone, regardless of the dimension L. As a result, the full pattern of parametric instability is strongly affected by the zone boundary: 1) the spatial distribution of wave amplitudes becomes nonuniform everywhere inside the zone; 2) the amplitude growth rate in the unstable regime decreases at all points when perturbations due to the boundary reach these points; 3) the instability threshold is independent of the spin-wave frequency offset from the parametric resonance frequency. The calculated minimum instability threshold as a function of the bias magnetic field (the “butterfly” curve) changes shape with L, in agreement with the available experimental data. Zh. éksp. Teor. Fiz. 111, 199–219 (January 1997)     

Estimating the sizes of surface cracks based on Hall element measurements of the leakage magnetic field and a dipole model of a crack

A method is proposed for estimating the sizes of surface cracks in magnetic materials. The method is based on applying a magnetic field, then determining the leakage magnetic field in the vicinity of a crack by moving a Hall element on the surface of the material along one or two scanning lines crossing the crack, and measuring the corresponding Hall voltage distribution. A dipole model of a crack is utilized, in which a surface crack is considered as being full of magnetic dipoles aligned parallel to the applied field, and whose density varies linearly along the depth of the crack. Analytical expressions are derived for the z-component of the intensity of the leakage magnetic field, and for the measured Hall voltage in the vicinity of a crack with an arbitrary cross-section along its long axis when it is perpendicular to the applied field. The crack sizes and the parameters of the distribution of magnetic dipoles along the crack depth are computed by crack inversion, which represents a regression for the Hall voltage distribution. A variable theoretical Hall voltage distribution is fitted to the measured Hall voltage distribution by minimizing the corresponding RMS error, which gives the unknown parameters at the end of the minimization. Hall voltage distributions are measured on ferromagnetic steel samples containing one artificial surface crack. Some crack inversions are performed for estimating the maximum crack depth and the crack width of cracks with rectangular and isosceles triangular cross-sections along the long crack axis. The accuracy of these crack inversions increases by utilizing either Hall voltage distributions measured along only one of the scanning lines, instead of along both scanning lines, or by using more precisely measured Hall voltage distributions. The fast and accurate estimation of the maximum crack depth and the crack width by such crack inversions could be important for pipeline inspection. Other crack inversions are performed for determining the cross-section along the long axis of the investigated cracks with satisfactory results. Received: 2 March 2001 / Accepted: 10 April 2001 / Published online: 25 July 2001     

The scattering of electromagnetic waves in fractal media

Abstract

The scattering of electromagnetic waves in fractal media is studied. The fractal dimension is naturally involved in the formulation of two physical problems studied in this paper. The general theory of multiple scattering of electromagnetic wave in fractal media is developed by modifying Twersky's theory. Statistical quantities, such as the average field and average intensity of the multiple scattered wave, are studied for a wave propagating in a fractal medium. The scattering cross section of the medium is deduced. The backscattering of electromagnetic waves is also studied. The results showing the range of dependence of the backscattered signals are in agreement with numerical simulations by Rastogi and Scheucher (1990). It also suggests a method of measuring the fractal dimension of the fractal embedded media using radar sounding. The theory developed in this paper can also be used for problems related to multiple scattering of other kinds of waves, such as acoustic waves, elastic waves etc, in fractal media.     

Nanoparticle chain-like formation in electrical double-layered magnetic fluids evidenced by small-angle X-ray scattering

Small-angle X-ray scattering (SAXS) was performed on a series of Electric Double-Layered Magnetic Fluids (EDL-MF) composed of ferrite type-- CoFe₂O₄, MnFe₂O₄, ZnFe₂O₄, NiFe₂O₄ and CuFe₂O₄--nanoparticles of different crystalline sizes ( D _XR ranging from 40 to 139?, as determined by X-ray diffraction). The information concerning the scattering objects was obtained through the analysis of the distance distribution function p(r) and of the size distribution function D(R), both retrieved from SAXS data. The results show that EDL-MF, in the absence of an applied magnetic field, are composed of small magnetic particle aggregates in solution. These agglomerates are elongated in one direction (chain-like) with the longest dimension varying from 240 to 330?. The cross-section size is of the order of D _XR. The data also demonstrate that the maximum dimension of these aggregates is independent of the ferrite type. On the other hand, the number of aggregated magnetic particles is nanoparticle-size-dependent. Accordingly, larger ferrite-type nanoparticles as those with D _XR = 139? form aggregates composed of 2-3 magnetic particles, whereas smaller ones with D _XR 40? form agglomerates of about 6 magnetic particles in solution. As the nanoparticle size is reduced, it might increase the particle surface defects. Such occurrence would affect the particle surface charge density, which could reduce the electrostatic screening, favoring the agglomeration phenomenon. Received 28 February 2000 and Received in final form 28 August 2000     

Richardson's pair diffusion and the stagnation point structure of turbulence

DNS and laboratory experiments show that the spatial distribution of straining stagnation points in homogeneous isotropic 3D turbulence has a fractal structure with dimension D(s)=2. In kinematic simulations the exponent gamma in Richardson's law and the fractal dimension D(s) are related by gamma=6/D(s). The Richardson constant is found to be an increasing function of the number density of straining stagnation points in agreement with pair diffusion occurring in bursts when pairs meet such points in the flow.     

Reverse phenomena of magnetic field effects and time-resolved EPR spectra in the photogenerated biradical from intramolecular electron-transfer in a phenothiazine–C60 linked compound with a semi-rigid spacer

Photoinduced electron-transfer reactions and magnetic field effects (MFEs) on the decay rates of the photogenerated biradical in a phenothiazine (Ph)–C₆₀ linked compound with a biphenyl group (Ph(BP)C₆₀) were examined in benzonitrile and benzene. Fluorescence and transient absorption spectra indicate that the intramolecular electron-transfer for Ph(BP)C₆₀ from the Ph to the singlet or triplet excited state of C₆₀ was suppressed by the biphenyl group. The decay rates of the photogenerated biradical decreased in the 0–0.2 T magnetic field range and increased in the 0.2–1 T magnetic field range. The reverse phenomena of the MFEs in Ph(BP)C₆₀ were strongly enhanced with increasing temperature and similar to those in Ph(n)C₆₀ (n = 6?12). The MFEs in Ph(BP)C₆₀ can be governed by spin-lattice relaxation and/or spin-spin relaxation mechanisms as observed in Ph(n)C₆₀ (n = 6?12). Time-resolved EPR spectra of Ph(BP)C₆₀ showed absorption, emission, absorption and emission patterns, and are quite different from those in Ph(n)C₆₀ (n = 4?12). The result indicates that the magnitude and distribution of the exchange interaction |2J| in Ph(BP)C₆₀ are smaller than those in Ph(n)C₆₀ (n = 4?12) and charge recombination occurs in the inverted region because the sign of the J is positive.     

Elastic–plastic transition in three-dimensional random materials: massively parallel simulations,fractal morphogenesis and scaling functions

Elastic–plastic transitions were investigated in three-dimensional (3D) macroscopically homogeneous materials, with microscale randomness in constitutive properties, subjected to monotonically increasing, macroscopically uniform loadings. The materials are cubic-shaped domains (of up to 100?×?100?×?100 grains), each grain being cubic-shaped, homogeneous, isotropic and exhibiting elastic–plastic hardening with a J ₂ flow rule. The spatial assignment of the grains’ elastic moduli and/or plastic properties is a strict-white-noise random field. Using massively parallel simulations, we find the set of plastic grains to grow in a partially space-filling fractal pattern with the fractal dimension reaching 3, whereby the sharp kink in the stress–strain curve of individual grains is replaced by a smooth transition in the macroscopically effective stress–strain curve. The randomness in material yield limits is found to have a stronger effect than that in elastic moduli. The elastic–plastic transitions in 3D simulations are observed to progress faster than those in 2D models. By analogy to the scaling analysis of phase transitions in condensed matter physics, we recognize the fully plastic state as a critical point and, upon defining three order parameters (the ‘reduced von-Mises stress’, ‘reduced plastic volume fraction’ and ‘reduced fractal dimension’), three scaling functions are introduced to unify the responses of different materials. The critical exponents are universal regardless of the randomness in various constitutive properties and their random noise levels.     

Dynamic Scaling and Fractal Behavior of Phase Dispersion in Multicomponent Polymer Blends

Dynamic scaling and fractal behavior of phase dispersion in a binary polymer mixture of polypropylene with poly(cis-butadiene) rubber during processing in a batch mixer was studied by means of a back small-angle laser scattering (BSALS) online system. In the late stage of phase dispersion, a special fractal behavior was found; different overlapped line groups existed simultaneously and the temporal distribution was stochastic. We defined the special behavior as quasi-self-similarity and calculated the fractal dimension, D _p, by using the power spectrum method. The results show that the evolution of different fractal dimensions with time is consistent with average chord length of the dispersed phase.