Self-similarity and fractals in soliton-supporting systems

We describe a principle that can be used to generate self-similarity and fractals in almost any nonlinear system in nature that supports solitons, given that some proper nonadiabatic conditions are met. We illustrate our idea on a particular optics example that also theoretically demonstrates fractals in nonlinear optics.     

Spontaneous formation of indium-rich nanostructures on InGaN(0001) surfaces

We show how a nonlinear system that supports solitons can be driven to generate exact (regular) Cantor set fractals. As an example, we use numerical simulations to demonstrate the formation of Cantor set fractals by temporal optical solitons. This fractal formation occurs in a cascade of nonlinear optical fibers through the dynamical evolution from a single input soliton.     

Transition density estimates for Brownian motion on affine nested fractals

A class of affine nested fractals is introduced which have different scale factors for different similitudes but still have the symmetry assumptions of nested fractals. For these fractals estimates on the transition density for the Brownian motion are obtained using the associated Dirichlet form. An upper bound for the diagonal can be found using a Nash-type inequality, then probabilistic techniques are used to obtain the off-diagonal bound. The approach differs from previous treatments as it uses only the Dirichlet form and no estimates on the resolvent. The bounds obtained are expressed in terms of an intrinsic metric on the fractal.     

液态In-80wt%Sn合金结构的分形分析

在X射线衍射实验的基础上，把分形理论用于分析In-80wt%Sn在300℃时的液态结构，发现液态结构是一种多度域分形结构，并提出液态结构多度域分形模型.通过分形分析，液态合金的整体结构被清晰地呈现出来.推测多度域两分形之间过渡区的大小和液态合金性质有关，过渡区的变化可能反映了液态合金性质的变化. 关键词： 液态合金结构 分形理论 多度域分形     

Fractal reconstruction in the presence of background events

An analysis of a data set containing fractals and background events is carried out using the method of the equation system of P-adic coverings (SePaC) and by the box-counting (BC) method. The peculiarities of these methods applied to the search for fractals in sets containing only fractals and background events are studied. Procedures allowing one to establish the presence of fractals, estimate their number in the initial set, separate fractals, and evaluate the portion of background events in the extracted set are suggested. A comparison of the result of an analysis of mixed events by these methods is carried out.     

About Maxwell’s equations on fractal subsets of ℝ3

In this paper we have generalized $F^{\bar \xi }$ -calculus for fractals embedding in ?³. $F^{\bar \xi }$ -calculus is a fractional local derivative on fractals. It is an algorithm which may be used for computer programs and is more applicable than using measure theory. In this Calculus staircase functions for fractals has important role. $F^{\bar \xi }$ -fractional differential form is introduced such that it can help us to derive the physical equation. Furthermore, using the $F^{\bar \xi }$ -fractional differential form of Maxwell’s equations on fractals has been suggested.     

The Universality of the Critical Dynamics of the Kinetic Ising Model on the Regular Fractals

The form of the universal scaling law of the critical dynamic exponent, z = D_ƒ + 2/υ, is found on a family of regular fractals by the exact TDRG method. Here, we generate a regular fractal by an anisotropic growing process. Identifying the growing probabilities as the interactions between Ising spins on the fractals, we map the growing probability clouds as a group of the anisotropic Ising Hamiltonians. Applying the RG transformations, we find that the systems of this group of Ising Hamiltonians can be described by two universal static correlation exponents υ₀ = ∞ and υ = 1. So, the growing processes proposed by us capture the essential features in the directed DLA simulations. The studies about their critical dynamic behaviours reveal that unlike the one-dimensional chain the critical dynamics of the kinetic Ising model on the regular fractals is universal. The further discussions show that there is a universal scaling law form of the critical dynamic exponent of the kinetic Ising model, z = D_ƒ + R_max/2υ, on the site models of the regular fractals with R_min = 2. Meanwhile, we discuss Daniel Kandal's correction to the formula of the,critical dynamic exponent in the TDRG method and show that our TDRG calculations are exact.     

Analysis of fractals with dependent branching by box counting, P-adic coverages, and systems of equations of P-adic coverages

Concept of the dimension of space-time in the general relativity theory and quantum theory is discussed. It is emphasized that the dimension of a discrete space can be defined based on the Hausdorff measure. The noninteger dimension is a typical characteristic of a fractal. The process of hadron formation in interactions between high-energy particles and nuclei is supposed to possess fractal properties. The following methods for analyzing fractals are considered: box counting (BC), method of P-adic coverages (PaC), and method of systems of equations of P-adic coverages (SePaC), for determining the fractal dimension. A comparative analysis of fractals with dependent branching is performed using these methods. We determine the optimum values of parameters permitting one to determine the fractal dimension D _F , number of levels N _lev, and the fractal structure with maximal efficiency. It is noted that the SePaC method has advantages in analyzing fractals with dependent branching.     

凝聚态物理中的分形

本文回顾了近年来与凝聚态物理相关的分形研究的一些主要工作,叙述了该领域的一些新进展。着重介绍了涉及分维和分形的一些基本概念、物理思想和研究方法,较详细地讨论了分形网络上的振动和弹性性质、磁序问题、生长和熔化、多分形以及周期分形等课题。     

Simulation model of the fractal patterns in ionic conducting polymer films

Normally polymer electrolyte membranes are prepared and studied for applications in electrochemical devices. In this work, polymer electrolyte membranes have been used as the media to culture fractals. In order to simulate the growth patterns and stages of the fractals, a model has been identified based on the Brownian motion theory. A computer coding has been developed for the model to simulate and visualize the fractal growth. This computer program has been successful in simulating the growth of the fractal and in calculating the fractal dimension of each of the simulated fractal patterns. The fractal dimensions of the simulated fractals are comparable with the values obtained in the original fractals observed in the polymer electrolyte membrane. This indicates that the model developed in the present work is within acceptable conformity with the original fractal.     

Chaos, fractals, and atomic flights in cavities

A semiclassical study is carried out of the nonlinear interaction dynamics between two-level atoms and a standing-wave field in a high-finesse cavity. As a result of atomic movement or wave amplitude modulation, a dynamic local instability occurs in a strongly coupled atom-field system. The appearance of dynamical Hamiltonian chaos, fractals, and Lévy flights is demonstrated for the models of two experimental devices: a (micro)maser with thermal Rydberg atoms and a microlaser with cold atoms. Numerical simulation showed that the manifestations of classical chaos, atomic fractals, and flights can be observed in the appropriate real experiments. Attention is drawn to the prospects provided by work on the atom-field systems in the coupling-modulated high-finesse cavities for further investigation of the quantum-classical correspondence, quantum chaos, and decoherence.     

Koch fractals in physical optics and their Fraunhofer diffraction patterns

This paper gives an overview of the utilization of fractals in physical optics, especially of Koch fractals and their diffractals. The term fractal itself is defined and some basic characteristics of fractals are mentioned. Constructions of the most typical Koch curves are also depicted. Laser diffraction experiments using regular, random and modified Koch curves are described and the corresponding diffraction patterns (intensity distributions of diffractals) are shown. Some interesting properties of these diffraction patterns are discussed.     

Influence of gas pressure on the pattern evolutions of dust clusters and fractals in a dusty plasma system

Dusty plasma has been produced through chemical reaction in a capacitively coupled radio frequency (rf) discharge system. Dust clusters with a few particles and dust fractals are observed. As gas pressure is increased, the suspended height of dust particles descends and the average interparticle distance decreases accordingly. The influence of gas pressure on the pattern evolutions is investigated. Dust clusters or fractals not only can evolve regularly on a horizontal plane, but also can evolve from a horizontal plane to a vertical line array. Under appropriate conditions, the evolutions are reversible. When the evolution is from a symmetrical pattern with a centre particle to another pattern, the centre particle will first show its unsteadiness.     

Physical model of dimensional regularization

We explicitly construct fractals of dimension \(4{-}\varepsilon \) on which dimensional regularization approximates scalar-field-only quantum-field theory amplitudes. The construction does not require fractals to be Lorentz-invariant in any sense, and we argue that there probably is no Lorentz-invariant fractal of dimension greater than 2. We derive dimensional regularization’s power-law screening first for fractals obtained by removing voids from 3-dimensional Euclidean space. The derivation applies techniques from elementary dielectric theory. Surprisingly, fractal geometry by itself does not guarantee the appropriate power-law behavior; boundary conditions at fractal voids also play an important role. We then extend the derivation to 4-dimensional Minkowski space. We comment on generalization to non-scalar fields, and speculate about implications for quantum gravity.     

Wavelet transforms and order-two densities of fractals

We highlight a correspondence between order-two densities and wavelet-like transforms of certain fractal measures. We use a variant of the ergodic theorem to demonstrate that these densities and transforms are well-behaved for a large class of quasi-self-similar fractals. We show that parallel ideas can be used to study the local behavior of certain fractal functions.     

Carbon fractals grown from carbon nanotips by plasma-enhanced hot filament chemical vapor deposition

Carbon nanomaterials with different structures were prepared in a custom-designed plasma-enhanced hot filament chemical vapor deposition system using methane, hydrogen and nitrogen. They were investigated by scanning electron microscopy (SEM) and micro-Raman spectroscopy. The SEM images show that the smooth carbon nanotips are formed under a high bias current and the carbon fractals can grow from the tips of the carbon nanotips under a low bias current. The results of micro-Raman spectroscopy indicate that the graphitization of the carbon nanomaterials was improved by ion bombardment. Combined the ion bombardment, electric field enhancement and electron emission mechanisms, the formation model of the carbon fractals was suggested.     

Percolation on infinitely ramified fractals

We present a family of exact fractals with a wide range of fractal and fracton dimensionalities. This includes the case of the fracton dimensionality of 2, which is critical for diffusion. This is achieved by adjusting the scaling factor as well as an internal geometrical parameter of the fractal. These fractals include the cases of finite and infinite ramification characterized by a ramification exponentp. The infinite ramification makes the problem of percolation on these lattices a nontrivial one. We give numerical evidence for a percolation transition on these fractals. This transition is tudied by a real-space renormalization group technique on lattices with fractal dimensionality ¯d between 1 and 2. The critical exponents for percolation depend strongly on the geometry of the fractals.     

一个关于分形的教学实验 --对大学物理实验课程创新的探索

设计了一个分形（粘性指进）的教学实验，让学生通过动手操作和计算机模拟得出各种粘性指进分形图形，并测量分形维数．从而使他们对目前活跃、有特色的科学概念——分形有所认识，有所学习．     

Fractals,coherent states and self-similarity induced noncommutative geometry

The self-similarity properties of fractals are studied in the framework of the theory of entire analytical functions and the q-deformed algebra of coherent states. Self-similar structures are related to dissipation and to noncommutative geometry in the plane. The examples of the Koch curve and logarithmic spiral are considered in detail. It is suggested that the dynamical formation of fractals originates from the coherent boson condensation induced by the generators of the squeezed coherent states, whose (fractal) geometrical properties thus become manifest. The macroscopic nature of fractals appears to emerge from microscopic coherent local deformation processes.     

Piecewise deterministic quantum dynamics and quantum fractals on the Poincaré disk

It is shown that piecewise deterministic dissipative quantum dynamics in a vector space withindefinite metric can lead to well defined, positive probabilities. The case of quantum jumps on the Poincar'e disk is studied in detail including results of numerical simulations of quantum fractals.