Stresses and strains in a deformable fractal medium and in its fractal continuum model

The model of fractal continuum accounting the topological, metric, and dynamic properties of deformable physical fractal medium is suggested. The kinematics of fractal continuum deformation is developed. The corresponding geometric interpretations are provided. The concept of stresses in the fractal continuum is defined. The conservation of linear and angular momentums is established. The mapping of mechanical problems for physical fractal media into the corresponding problems for fractal continuum is discussed.     

Scaling reduction of the contrast of fractal speckles detected with a finite aperture

It is known that speckle patterns with fractal properties, called fractal speckles, are produced by illuminating a diffuser with the coherent light having the intensity distribution obeying a negative power law. One of key properties of fractal speckles is the spatial correlation function obeying a negative power law, which implies that such speckles have scaling properties. In detecting fractal speckles, the effect of the spatial integration is inevitable in most cases since they have speckle grains of various scales including very fine ones. To evaluate this effect, in this paper, the contrast of spatially integrated intensity distributions is investigated theoretically and experimentally for fractal speckles. The results show that the contrast reduction with the size of the detector aperture obeys a negative power function related with the exponent of the intensity correlation coefficient of fractal speckles.     

Fractal structure of equipotential curves on a continuum percolation model

We numerically investigate the electric potential distribution over a two-dimensional continuum percolation model between the electrodes. The model consists of overlapped conductive particles on the background with an infinitesimal conductivity. Using the finite difference method, we solve the generalized Laplace equation and show that in the potential distribution, there appear quasi-equipotential clusters   which approximately and locally have the same values as steps and stairs. Since the quasi-equipotential clusters have the fractal structure, we compute the fractal dimension of equipotential curves and its dependence on the volume fraction over [0,1]$[0,1]$. The fractal dimension in [1.00, 1.246] has a peak at the percolation threshold _pc$p_c$.     

Self-consistent development and fractal structure of leader discharges along a water surface

Ideas regarding the development of single-and multichannel leader discharges over water surfaces are generalized on the basis of experimental data. The Ohmic conductivity of water is found to be manifested in the self consistency of their dynamics and fractal structure. The fractal dimensionality of a single-channel leader discharge is found to be 0.96±0.05 and that of a multichannel discharge to be 1.85±0.05. Mechanisms are proposed for the branching of leader discharge channels and for the development of bifurcations from branches and channels. Zh. Tekh. Fiz. 69, 35–40 (April 1999)     

Model simulations of ground-state and finite-temperature properties of disordered magnetic nanostructures

The properties of two-dimensional ensembles of magnetic nanoparticles that interact by magnetic dipole coupling are investigated. The low-temperature magnetic arrangements, the average binding energy E _dip due to dipolar interactions, and its scaling behavior with respect to the particle density C are calculated for different types of structural disorder and particle-size distributions. Many different metastable magnetic states are obtained, which exhibit strong noncollinearities and are reminiscent of a spin-glass behavior. For a given C, |E _dip| increases with increasing disorder of the particle positions. For random distributions at low particle densities C0.2, E _dip is dominated by the contributions of short interparticle distances. Thus, it scales as | E _dip| C with an unusually small exponent = 0.85–1. The straightforward scaling of the dipole interaction, 3/2, is obtained only for C0.5 or for nearly periodic ensembles. The finite temperature behavior of these disordered interacting nanomagnets is explored. The specific heat and magnetic susceptibility are calculated by performing Monte Carlo simulations. The onset of long-range magnetic order is discussed. In addition we determine hysteresis loops at finite temperatures and compare the results for different degrees of disorder.     

Scaling properties of the regular dynamics for a dissipative bouncing ball model

Some scaling properties of the regular dynamics for a dissipative version of the one-dimensional Fermi accelerator model are studied. The dynamics of the model is given in terms of a two-dimensional nonlinear area contracting map. Our results show that the velocities of saddle fixed points (saddle velocities) can be described using scaling arguments for different values of the control parameter.     

Self-consistent electronic structure of quantum wells by invariant embedding method

A new approach based on the invariant embedding method for the self-consistent calculation of electronic structure of quantum wells is presented and is applied to both neutral quantum well and parabolic quantum well. Numerical results obtained for these structures agree very well with those of previous theoretical and experiment studies. The present approach is expected to lead to a more efficient and stable scheme for the calculation of electronic band structure of quantum structures. Realistic boundary conditions are naturally taken into account in the present calculation which provides a convenient way for studying boundary effects. Received 21 September     

Numerical simulations of single and multiple scattering by fractal ice clusters

We consider the scattering model in the form of a vertically and horizontally homogeneous particulate slab of an arbitrary optical thickness composed of widely separated fractal aggregates built of small spherical ice monomers. The aggregates are generated by applying three different approaches, including simulated cluster-cluster aggregation (CCA) and diffusion-limited aggregation (DLA) procedures. Having in mind radar remote-sensing applications, we report and analyze the results of computations of the backscattering circular polarization ratio obtained using efficient superposition T-matrix and vector radiative-transfer codes. The computations have been performed at a wavelength of 12.6 cm for fractal aggregates with the following characteristics: monomer refractive index m=1.78+i0.003, monomer radius r=1 cm, monomer packing density p=0.2, overall aggregate radii R in the range 4≤R≤10 cm and fractal dimensions D_f=2.5 and 3.We show that for aggregates generated with simulated CCA and DLA procedures, the respective values of the backscattering circular polarization ratio differ weakly for D_f=2.5, but the differences can increase somewhat for D_f=3, especially in case of an optically semi-infinite medium. For aggregates with a spheroidal overall shape, the dependence of the circular polarization ratio on the cluster morphology can be quite significant and increases with increasing the aspect ratio of the circumscribing spheroid.     

Effects of ramification and connectivity degree on site percolation threshold on regular lattices and fractal networks

This Letter is focused on the impact of network topology on the site percolation. Specifically, we study how the site percolation threshold depends on the network dimensions (topological d and fractal D), degree of connectivity (quantified by the mean coordination number $Z_{\infty }$), and arrangement of bonds (characterized by the connectivity index Q also called the ramification exponent). Using the Fisher's containment principle, we established exact inequalities between percolation thresholds on fractal networks contained in the square lattice. The values of site percolation thresholds on some fractal lattices were found by numerical simulations. Our findings suggest that the most relevant parameters to describe properly the values of site percolation thresholds on fractal networks contained in square lattice (Sierpiński carpets and Cantor tartans) and based on the square lattice (weighted planar stochastic fractal and Cantor lattices) are the mean coordination number and ramification exponent, but not the fractal dimension. Accordingly, we propose an empirical formula providing a good approximation for the site percolation thresholds on these networks. We also put forward an empirical formula for the site percolation thresholds on d-dimensional simple hypercubic lattices.     

The study of fractal correlation method in the displacement measurement and its application

The classical digital speckle, or digital image, correlation method of deformation measurement is based on gray level correlation between unformed and deformed digital images. The pattern of artificial random speckles and the natural textures on some object's surfaces have fractal characteristics, and their fractal dimensions represent both gray and morph information. Furthermore, the fractal dimensions are stable feature parameters of the patterns. The digitized images of the patterns are confirmed to be also fractals. By this fact, a new method of displacement measurement is developed in the paper, based on the fractal dimensions correlation. The in-plane displacement fields of a body can be acquired. In order to verify the validity of the new method, an experiment has been designed and the results have been compared with those obtained from the classical digital image correlation method. The validity of the new method is not less than that of classical method. Further discussions about the traits and the developing vista of the method are given at the end.     

Quantum waveguide theory of a fractal structure

The electronic transport properties of fractal quantum waveguide networks in the presence of a magnetic field are studied. A Generalized Eigen-function Method (GEM) is used to calculate the transmission and reflection coefficients of the studied systems unto the fourth generation Sierpinski fractal network with node number N=123$N=123$. The relationship among the transmission coefficient T, magnetic flux Φ and wave vector k is investigated in detail. The numerical results are shown by the three-dimensional plots and contour maps. Some resonant-transmission features and the symmetry of the transmission coefficient T to flux Φ are observed and discussed, and compared with the results of the tight-binding model.     

Extended calculation of polarization and intensity of fractal aggregates based on rigorous method for light scattering simulations with numerical orientation averaging

The intensity and polarization of fractal aggregates have been investigated using both rigorous and approximate methods for light scattering simulations. However, previous studies using the analytical orientation averaging version of the rigorous method were generally limited to a few hundred monomers when the monomer size parameter was around 1.7. In this study, we propose using numerical orientation averaging instead of analytical orientation averaging. The numerical averaging is performed together with a fixed orientation version of the rigorous T-matrix method for clusters of spheres. This approach enables increasing the number of monomers by a factor of 2-7 or the size of monomers by a factor of 8-10 compared to the analytical orientation averaging version.We investigated the influence of monomer size and the number of monomers on the light scattering of silicate aggregates (refractive index m=1.68+0.03i) for incident light with a wavelength of . We considered ballistic particle-cluster aggregates (BPCA) and ballistic cluster-cluster aggregates (BCCA) composed of 128, 256, 512, and 1024 monomers with radii between 0.11 and .Our results show that the size of monomers plays an important role in reproducing the negative polarization branch for all the BPCA and BCCA. Silicate aggregates with the monomer radius of less than contribute to reproducing the negative polarization branch, while aggregates with monomers larger than do not have the negative polarization branch. Polarization oscillation with scattering angle occurs for larger monomers (i.e., monomer radius ).The maximum polarization decreases for increasing monomer radius between 0.11 and . However, the negative polarization branch is generally enhanced for monomer radii up to around , and reduced for further increase of monomer size.The number of monomers also has a large influence on the negative polarization branch in the case of BPCA. The increase in the number of monomers from 128 to 1024 shifts the scattering angle of minimum polarization to larger angles for BPCA. In addition, the increase in the number of monomers reduces the values of negative polarization for BPCA while the variation with the number of monomers for BCCA is small and is not monotonic.     

Multifractality of some random and disordered systems as a thermodynamics in fractal phase space

It is shown that multifractal properties of some random and disordered systems can be simulated using thermodynamics of a generalized ideal monoatomic gas in a fractal phase space. Received 25 November 1998 and Received in final form 16 December 1998     

Noteworthy fractal features and transport properties of Cantor tartans

This Letter is focused on the impact of fractal topology on the transport processes governed by different kinds of random walks on Cantor tartans. We establish that the spectral dimension of the infinitely ramified Cantor tartan $d_s$ is equal to its fractal (self-similarity) dimension D. Consequently, the random walk on the Cantor tartan leads to a normal diffusion. On the other hand, the fractal geometry of Cantor tartans allows for a natural definition of power-law distributions of the waiting times and step lengths of random walkers. These distributions are Lévy stable if $D>1.5$. Accordingly, we found that the random walk with rests leads to sub-diffusion, whereas the Lévy walk leads to ballistic diffusion. The Lévy walk with rests leads to super-diffusion, if $D>\sqrt{3}$, or sub-diffusion, if $1.5<D<\sqrt{3}$.     

A study of radiative properties of fractal soot aggregates using the superposition T-matrix method

We employ the numerically exact superposition T-matrix method to perform extensive computations of scattering and absorption properties of soot aggregates with varying state of compactness and size. The fractal dimension, D_f, is used to quantify the geometrical mass dispersion of the clusters. The optical properties of soot aggregates for a given fractal dimension are complex functions of the refractive index of the material m, the number of monomers N_S, and the monomer radius a. It is shown that for smaller values of a, the absorption cross section tends to be relatively constant when D_f<2 but increases rapidly when D_f>2. However, a systematic reduction in light absorption with D_f is observed for clusters with sufficiently large N_S, m, and a. The scattering cross section and single-scattering albedo increase monotonically as fractals evolve from chain-like to more densely packed morphologies, which is a strong manifestation of the increasing importance of scattering interaction among spherules. Overall, the results for soot fractals differ profoundly from those calculated for the respective volume-equivalent soot spheres as well as for the respective external mixtures of soot monomers under the assumption that there are no electromagnetic interactions between the monomers. The climate-research implications of our results are discussed.     

Fractal structure of the crystalline-nuclei boundaries in 2D colloidal crystallization: Computer simulations

By performing 2D kinetic Monte Carlo simulations of colloidal crystallization we found that the boundaries of the crystalline nuclei are not only rough, as obtained by experimentalists, but fractal, whose value (_df$d_f$) we calculated. The corresponding boundary for the crystals, above the critical size (_Nc$N_c$), is also fractal but smoother. A knowledge of the particles coordinates during the crystallization process allows us to calculate the _Nc$N_c$, the line tension (γ) and the chemical potential difference (Δμ) between the two phases. However, different from the experimentalists procedure, we found that the boundary fractalities are needed to derive γ and Δμ.     

Magnetic properties of disordered ferrite and ilmenite-hematite thin films

We have synthesized thin films of disordered zinc ferrite (ZnFe₂O₄) and ilmenite-hematite (FeTiO₃-Fe₂O₃) solid solution, the former and the latter of which are interesting from the viewpoints of magnetooptics and spintronics, respectively, by utilizing sputtering and pulsed laser deposition methods, and have explored their magnetic, magnetooptical, and electrical properties. Although ZnFe₂O₄ possesses a normal spinel structure as its stable phase, some of the Fe³⁺ ions occupy the tetrahedral as well as the octahedral sites in ZnFe₂O₄ of which the sputtered thin film is composed. Consequently, the as-deposited thin film manifests large magnetization even at room temperature although the magnetic phase transition temperature of the stable phase of ZnFe₂O₄ is as low as 10 K. Also, the thin film exhibits a cluster spin glass transition at a temperature as high as 325 K. Furthermore, the ZnFe₂O₄ thin films exhibit large Faraday effects at a wavelength of 400 nm or so. The ilmenite-hematite solid solution is one of the ferrimagnetic semiconductors. Most of the compositions possess Curie temperatures higher than room temperature, and the type of carrier can be tuned only by changing the composition. We have succeeded in synthesizing solid-solution thin films of various compositions grown epitaxially on sapphire substrates with a (0 0 0 1) plane, and have shown that the thin films are ferrimagnetic semiconductors.     

Low-temperature thermodynamic properties of a one-dimensional disordered electron lattice system

A new method of low-temperature thermodynamic calculation of a one-dimensional generalized Wigner crystal on a disordered host-lattice is proposed. This method is based on the system statistical sum presentation in terms of modified transfer-matrixes. A gapless structure of the elementary excitations spectrum of the system under consideration is found.     

Statistical properties of a dissipative kicked system: Critical exponents and scaling invariance

A new universal empirical function that depends on a single critical exponent (acceleration exponent) is proposed to describe the scaling behavior in a dissipative kicked rotator. The scaling formalism is used to describe two regimes of dissipation: (i) strong dissipation and (ii) weak dissipation. For case (i) the model exhibits a route to chaos known as period doubling and the Feigenbaum constant along the bifurcations is obtained. When weak dissipation is considered the average action as well as its standard deviation are described using scaling arguments with critical exponents. The universal empirical function describes remarkably well a phase transition from limited to unlimited growth of the average action.     

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.