Extending Newton's law from nonlocal-in-time kinetic energy

We study a new equation of motion derived from a context of classical Newtonian mechanics by replacing the kinetic energy with a form of nonlocal-in-time kinetic energy. It leads to a hypothetical extension of Newton's second law of motion. In a first stage the obtainable solution form is studied by considering an unknown value for the nonlocality time extent. This is done in relation to higher-order Euler-Lagrange equations and a Hamiltonian framework. In a second stage the free particle case and harmonic oscillator case are studied and compared with quantum mechanical results. For a free particle it is shown that the solution form is a superposition of the classical straight line motion and a Fourier series. We discuss the link with quanta interpretations made in Pais-Uhlenbeck oscillators. The discrete nature emerges from the continuous time setting through application of the least action principle. The harmonic oscillator case leads to energy levels that approximately correspond to the quantum harmonic oscillator levels. The solution to the extended Newton equation also admits a quantization of the nonlocality time extent, which is determined by the classical oscillator frequency. The extended equation suggests a new possible way for understanding the relationship between classical and quantum mechanics.     

Fractality of self-grown nanostructured tungsten by He plasma irradiation

Fractal property of the helium irradiated nanostructured tungsten was investigated from the scanning electron microscope (SEM) micrographs, gas adsorption isotherms, and transmission electron microscope (TEM) micrographs. From the SEM micrographs, fractal dimension and the parameter _dmin$d_{\min }$, which characterizes the SEM texture image, were deduced, and the fractal dimension was compared with the one obtained from the gas adsorption isotherms. It was revealed that the fractal dimension obtained from the top view SEM micrographs was significantly lower than that from the adsorption isotherms. From the cross sectional SEM micrographs, two power law relations were identified in two different scales, and the fractal dimension from the adsorption was in between the two fractal dimensions. From the TEM micrographs, it was found that the porosity distribution also has fractal relation with height of the nanostructures when the nanostructures were sufficiently grown.     

Velocity Profiles of Slow Blood Flow in a Narrow Tube

A fractal model is introduced into the slow blood motion. When blood flows slowly in a narrow tube, red cell aggregation results in the formation of an approximately cylindrical core of red cells. By introducing the fractal model and using the power law relation between area fraction φ and distance from tube axis ρ, rigorous velocity profiles of the fluid ia and outside the aggregated core and of the core itself are obtained analytically for different fractal dimensions. It shows a blunted velocity distribution for a relatively large fractal dimension (D～2), which can be observed in normal blood; a pathological velocity profile for moderate dimension (D = 1), which is similar to the Segre-Silberberg effect; and a parabolic profile for negligible red cell concentration (D = 0), which likes in the Poiseuille flow.     

Constant phase angle response with fractal electrodes

An analytical solution is obtained for the dielectric response of an electrode composed of pores whose cross-sectional areas possess a fractal arrangement. When the surface dimension of the pore system is greater than unity the divergent constant phase angle response of previous computations is recovered. However when the surface dimensionality falls below unity the capacitance saturates and a peak is obtained in the loss component which exhibits a low frequency power law dependence on frequency.     

The microscopic feshbach optical potential for a schematic coupled channel example

By means of a method of expanding the solutions of a system of coupled equations into a set of positive-energy sturmian basis functions, a separable approximation to the microscopic (Feshbach) elastic optical potential is obtained. The properties of this nonlocal potential are analysed in terms of the moments of the potential and of the wronskian of two independent solutions of the single-channel nonlocal Schroedinger equation. A numerical application is given for a schematic coupled channel example, representing a 60 MeV neutron incident on a schematic nucleus with up to 10 channels, coupled by phenomenological surface-peaked potentials.The nature of the nonlocality of this schematic potential is very different from the nonlocality due to exchange effects in that the wronskian is strongly angular momentum dependent and can be larger as well as smaller than unity. The range of the nonlocality is found to be larger than the wavelength of the elastic projectile.     

Flocculation and percolation in reversible cluster-cluster aggregation

Off-lattice dynamic Monte-Carlo simulations were done of reversible cluster-cluster aggregation for spheres that form rigid bonds at contact. The equilibrium properties were found to be determined by the life time of encounters between two particles (t_e). t_e is a function not only of the probability to form or break a bond, but also of the elementary step size of the Brownian motion of the particles. In the flocculation regime the fractal dimension of the clusters is d_f=2.0 and the size distribution has a power law decay with exponent τ=1.5. At larger values of t_e transient gels are formed. Close to the percolation threshold the clusters have a fractal dimension d_f=2.7 and the power law exponent of the size distribution is τ=2.1. The transition between flocculation and percolation occurs at a characteristic weight average aggregation number that decreases with increasing volume fraction.     

Implications of an extended fractal hydrodynamic model

Considering that the motions of the particles take place on continuous but non-differentiable curves, i.e. on fractals with constant fractal dimension, an extended scale relativity model in its hydrodynamic version is built. In this approach, static (particle in a box and harmonic oscillator) and time-dependent (free particle etc.) systems are analyzed. The static systems can be associated with a coherent fractal fluid (of superconductor or of super-fluid types behavior), whose particles are moving on stationary trajectories. The complex speed field of the fractal fluid proves to be essential: the zero value of the real (differentiable) part specifies the coherence of the fractal fluid, while the non-zero value of the imaginary (non-differentiable or fractal) part selects, through some “quantization” relations, the “stationary” trajectories (that may correspond to the observables from quantum mechanics) of the fractal fluid particles. Moreover, the momentum transfer in the fractal fluid is achieved only through the fractal component of the complex speed field. The free time-dependent systems can be associated with an incoherent fractal fluid, and both the differentiable and fractal components of complex speed field are inhomogeneous in fractal coordinates due to the action of a fractal potential. It exist momentum transfer on both speed components and the “observable” in the form of an uniform motion is generated through a specific mechanism of “vacuum” polarization induced by the same fractal potential. The analysis on the fractal fluid specifies conductive properties in the case of movements synchronization both on differentiable and fractal scales, and convective properties in the absence of synchronization.     

完全非弹性蹦球倍周期运动的分形特征

一个落在振动台面上的完全非弹性球的运动是倍周期的.倍周期分岔过程受约化振动加速度的控制,倍周期分岔图由疏密相间的区域构成.在密集区内,倍周期分岔过程敏感地依赖于控制参数,呈现出复杂的几何结构.分析了密集区的分形特性,并计算了各密集区的分维数.结果表明密集区的分维数是依次增大的,逐渐趋于一个约为1.8的常数. 关键词： 蹦球 倍周期分岔 分形 颗粒物质     

周期受击陀螺系统波函数的分形

研究了周期受击陀螺系统波函数的分形. 发现在打击强度系数较弱时 (即≤ 1时), 相空间是规则的, 分形维接近于1; 随着打击强度系数的增大, 相空间开始变得混沌, 分形维也随之增大; 当打击强度系数达到6时, 相空间完全混沌, 分形维将达到最大值, 此时若继续增大打击强度系数, 分形维保持基本不变. 关键词： 陀螺 波函数 分形维 相空间     

Fractal model of the atom and some properties of the matter through an extended model of scale relativity

The scale relativity model was extended for the motions on fractal curves of fractal dimension D _F and third order terms in the equation of motion of a complex speed field. It results that, in a fractal fluid, the convection, dissipation and dispersion are compensating at any scale (differentiable or non-differentiable), whereas a generalized Schrödinger type equation is obtained for an irrotational movement of the fractal fluid. For D _F = 2 and the dissipative approximation of the motions, the fractal model of atom is build: the real part of the complex speed field describes the electron motion on stationary orbits according to a quantification condition, while the imaginary part of the complex speed field gives the electron energy quantification. For D _F = 3 and the dispersive approximation of motions, some properties of the matter are explained: at the differentiable scale the flowing regimes (non-quasi-autonomous and quasi-autonomous) of the fractal fluids are separated by the experimental “0.7 structure”, while for the non-differentiable scale the fractal potential acts as an energy accumulator and controls through coherence the transport phenomena. Moreover, the compatibility between the differentiable and non-differentiable scales implies a Cantor space-time, and consequently a fractal at any scale. Thus, some properties of the matter (the anomaly of nano-fluids thermal conductivity, the superconductivity etc.) can be explained by this model.     

Special electronic structures of inverse spinels LiMVO<Subscript>4</Subscript>(M = Ni and Cu): a first-principles study

We use the Ulam method to study spectral properties of the Perron-Frobenius operators of dynamical maps in a chaotic regime. For maps with absorption we show numerically that the spectrum is characterized by the fractal Weyl law recently established for nonunitary operators describing poles of quantum chaotic scattering with the Weyl exponent ν = d-1, where d is the fractal dimension of corresponding strange set of trajectories nonescaping in future times. In contrast, for dissipative maps we numerically find the Weyl exponent ν = d/2 where d is the fractal dimension of strange attractor. The Weyl exponent can be also expressed via the relation ν = d₀/2 where d₀ is the fractal dimension of the invariant sets. We also discuss the properties of eigenvalues and eigenvectors of such operators characterized by the fractal Weyl law.     

P-adic coverage method in fractal analysis of showers

Self-similarity in multiple processes at high energies is considered. It is assumed that a parton cascade transforms into a hadron shower with a fractal structure. The box counting (BC) method used to calculate the fractal dimension is analyzed. The parton shower with permissible 1/3 parts of pseudorapidity space, which corresponds to a triadic Cantor set, was used as a test fractal. It was found that there is an optimal set of bins (a parameter of the BC method) that allows one to find the fractal dimension with maximal accuracy. The optimal set of bins is shown to depend on the fractal generation law. The P-adic coverage (PaC) method is proposed and used in the fractal analysis. This method makes it possible to determine the fractal dimension of a shower as accurately as possible, the number of fractal levels and partons at each branching point during the parton shower evolution, the type of cascade (either random or regular), and its structure. It is shown to be applicable to an analysis of the regular and random N-ary cascades with permissible 1/k parts of the space studied.     

电流变液中悬浮粒子的布朗运动与其聚集之间关系

在外电场作用下,电流变液中粒子链的聚集与其布朗运动密切相关。为了具体分析这种关系,采用不分离但有相互运动的双球构成的系统去简化粒子链。通过分析双球系统布朗运动特性,得到其无规运动轨迹维数小于2,大量这样的系统在平面中就不可能均匀分布。这说明由于存在布朗运动,在外电场中电流变液中粒子链必然相互聚集,即形成高密度区与低密度区。 关键词：     

火灾烟雾颗粒凝并分形特性研究

火灾烟雾颗粒因布朗运动由初期的主粒子凝并形成大颗粒的凝团结构，其外形呈现出分形特征，根据分形理论对火灾烟雾颗粒凝团结构进行分析研究，采用场发射扫描电镜对多种材料的烟雾颗粒图像进行测试，通过对烟雾凝团图像进行处理，获得了火灾探测中常见的多种材料的分形维数和分形系数，给出了烟雾颗粒的主粒子粒径，并对其影响因素进行了对比分析，为火灾烟雾探测中颗粒凝并分形特性研究提供有益的探索. 关键词： 火灾 烟雾 颗粒 凝并 分形     

Spatial soliton steering induced by weak nonlocality in optical lattices

We investigate the soliton steering in weakly nonlocal nonlinear media with harmonic modulation of the refractive index. It is shown that the weak nonlocality originated from the nonlocal nonlinear response of medium induces a transverse modulation of the refractive index, which leads to the occurrence of interplay between the weak nonlocality-induced transverse modulation and the harmonic modulation. Thus, one can control the soliton steering in weakly nonlocal medium, such as nematic liquid crystal, by employing the competition between the weak nonlocality parameter and the lattice depth.     

Subcooled pool boiling heat transfer in fractal nanofluids:A novel analytical model

A novel analytical model to determine the heat flux of subcooled pool boiling in fractal nanofluids is developed. The model considers the fractal character of nanofluids in terms of the fractal dimension of nanoparticles and the fractal dimen- sion of active cavities on the heated surfaces; it also takes into account the effect of the Brownian motion of nanoparticles, which has no empirical constant but has parameters with physical meanings. The proposed model is expressed as a function of the subcooling of fluids and the wall superheat. The fractal analytical model is verified by a reasonable agreement with the experimental data and the results obtained from existing models.     

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.     

SIMULATION OF FRACTAL GROWTH OF THIN FILMS AT LOW TEMPERATURE

Fractal growth of thin films at low temperature (50－175 K) is simulated by Monte Carlo method. It is shown that the thin film growth is quite different from the diffusion-limited aggregation (DLA) model when the coverage is larger than 0.1 ML. The average branch width of clusters increases with increasing temperature and it usually larger than the branch width (1.9 atom) in the classic DLA model. The average fractal dimension of clusters increases also with increasing coverage while the fractal dimension of DLA model remains constant. This difference comes from the weak screening effect during the late stage of thin film growth. The relationship between the saturation island number n_s and deposition interval Δt is described in a power law: n_s∝Δt^γ, where γ=-0.332 is very close to the theoretical value -1/3 of rate equations from nucleation theory.     

A dissipative network model with neighboring activation

We study the properties of spectrum and eigenstates of the Google matrix of a directed network formed by the procedure calls in the Linux Kernel. Our results obtained for various versions of the Linux Kernel show that the spectrum is characterized by the fractal Weyl law established recently for systems of quantum chaotic scattering and the Perron-Frobenius operators of dynamical maps. The fractal Weyl exponent is found to be ν ≈ 0.65 that corresponds to the fractal dimension of the network d ≈ 1.3. An independent computation of the fractal dimension by the cluster growing method, generalized for directed networks, gives a close value d ≈ 1.4. The eigenmodes of the Google matrix of Linux Kernel are localized on certain principal nodes. We argue that the fractal Weyl law should be generic for directed networks with the fractal dimension d < 2.     

随机多孔介质的蒙特卡罗重构与渗流特性模拟

根据多孔介质微观结构的分形尺度标度特征,采用蒙特卡罗方法分别重构随机多孔介质的微观颗粒和孔隙结构,并基于分形毛管束模型研究多尺度多孔介质的气体渗流特性,建立多孔介质微观结构和宏观渗流特性的定量关系。结果表明：分形蒙特卡罗重构的多孔介质微细结构接近真实介质结构,气体渗流特性的计算结果与格子玻尔兹曼模拟数据较为吻合; 多孔介质气体渗透率随着克努森数的增加而增大,孔隙分形维数对于气体渗流的微尺度效应具有显著影响,而迂曲度分形维数对于表观渗透率和固有渗透率的比值影响可以忽略。提出的分形蒙特卡罗方法具有收敛速度快且计算误差与维数无关的优点,有利于深入理解多尺度多孔介质的渗流机理。