Fractional Power Law Research Articles

Nonequilibrium ferroelectric-ferroelastic 10 nm nanodomains: wrinkles, period-doubling, and power-law relaxation

Since the 1935 work of Landau–Lifshitz and of Kittel in 1946 all ferromagnetic, ferroelectric, and ferroelastic domains have been thought to be straight-sided with domain widths proportional to the square root of the sample thickness. We show in the present work that this is not true. We also discover period doubling domains predicted by Metaxas et al (2008 Phys. Rev. Lett. 99 217208) and modeled by Wang and Zhao (2015 Sci. Rep. 5 8887). We examine non-equilibrium ferroic domain structures in perovskite oxides with respect to folding, wrinkling, and relaxation and suggest that structures are kinetically limited and in the viscous flow regime predicted by Metaxas et al in 2008 but never observed experimentally. Comparisons are made with liquid crystals and hydrodynamic instabilities, including chevrons, and fractional power-law relaxation. As Shin et al (2016 Soft Matter 12 3502) recently emphasized: ‘An understanding of how these folds initiate, propagate, and interact with each other is still lacking’. Inside each ferroelastic domain are ferroelectric 90° nano-domains with 10 nm widths and periodicity in agreement with the 10 nm theoretical minima predicted by Feigl et al (2014 Nat. Commun. 5 4677). Evidence is presented for domain-width period doubling, which is common in polymer films but unknown in ferroic domains. A discussion of the folding-to-period doubling phase transition model of Wang and Zhao is included.

Minimum‐cost optimization of nonlinear fluid viscous dampers and their supporting members for seismic retrofitting

SummaryThis paper presents an effective approach for achieving minimum‐cost designs for seismic retrofitting using nonlinear fluid viscous dampers. The damping coefficients of the dampers and the stiffness coefficients of the supporting braces are designed by an optimization algorithm. A realistic retrofitting cost function is minimized subject to constraints on inter‐story drifts at the peripheries of frame structures. The cost function accounts for costs related to both the topology and the sizes of the dampers. The behavior of each damper‐brace element is defined by the Maxwell model, where the force–velocity relation of the nonlinear dampers is formulated with a fractional power law. The optimization problem is first posed and solved as a mixed integer problem. For the reduction of the computational effort required in the optimization, the problem is then reformulated with continuous variables only and solved with a gradient‐based algorithm. Material interpolation techniques, which have been successfully applied in topology optimization and in multi‐material optimization, play a key role in achieving practical final design solutions with a reasonable computational effort. Promising results attained for 3‐D irregular frames are presented and discussed. Copyright © 2017 John Wiley & Sons, Ltd.

Unsteady Marangoni convection heat transfer of fractional Maxwell fluid with Cattaneo heat flux

Fractional shear stress and Cattaneo heat flux models are introduced in characterizing unsteady Marangoni convection heat transfer of viscoelastic Maxwell fluid over a flat surface. Governing equations and boundary condition are formulated firstly via the balance between the surface tension and shear stress. Numerical solutions are obtained by new developed numerical technique and some novel phenomena are found. Results shown that the fractional derivative parameters, Marangoni number and power law exponent have significant influence on characteristics velocity and temperature fields. As fractional derivative parameters increase, the temperature profiles rise remarkably and the viscoelastic effects of the fluid enhance with delayed response to surface tension, however the temperature profiles decline significantly with a thinner thickness of thermal boundary layer with the increase of Marangoni number. The average skin friction coefficient increases with the augment of Marangoni number, while the average Nusselt number decreases for larger values of power law exponent.

The Dirichlet problem of a conformable advection-diffusion equation

The fractional advection-diffusion equations are obtained from a fractional power law for the matter flux. Diffusion processes in special types of porous media which has fractal geometry can be modelled accurately by using these equations. However, the existing nonlocal fractional derivatives seem complicated and also lose some basic properties satisfied by usual derivatives. For these reasons, local fractional calculus has recently been emerged to simplify the complexities of fractional models defined by nonlocal fractional operators. In this work, the conformable, a local, well-behaved and limit-based definition, is used to obtain a local generalized form of advection-diffusion equation. In addition, this study is devoted to give a local generalized description to the combination of diffusive flux governed by Fick?s law and the advection flux associated with the velocity field. As a result, the constitutive conformable advection-diffusion equation can be easily achieved. A Dirichlet problem for conformable advection-diffusion equation is derived by applying fractional Laplace transform with respect to time t and finite sin-Fourier transform with respect to spatial coordinate x. Two illustrative examples are presented to show the behaviours of this new local generalized model. The dependence of the solution on the fractional order of conformable derivative and the changing values of problem parameters are validated using graphics held by MATLcodes.

An autoregressive approach to modeling commodity prices as a quasi-fractional Brownian motion

Purpose– The purpose of this paper is to argue that a stationary-differenced autoregressive (AR) process with lag greater than 1, AR(q> 1), has certain properties that are consistent with a fractional Brownian motion (fBm). What the authors are interested in is the investigation of approaches to identifying the existence of persistent memory of one form or another for the purposes of simulating commodity (and other asset) prices. The authors show in theory, and with application to agricultural commodity prices the relationship between AR(q) and quasi-fBm.Design/methodology/approach– In this paper the authors develop mathematical relationships in support of using AR(q> 1) processes for simulating quasi-fBm.Findings– From theory the authors show that any AR(q) process is a stationary, self-similar process, with a lag structure that captures the essential elements of scaling and a fractional power law. The authors illustrate through various means the approach, and apply the quasi-fractional AR(q) process to agricultural commodity prices.Research limitations/implications– While the results can be applied to most time series of commodity prices, the authors limit the evaluation to the Gaussian case. Thus the approach does not apply to infinite-variance models.Practical implications– The approach to using the structure of an AR(q> 1) model to simulate quasi-fBm is a simple approach that can be applied with ease using conventional Monte Carlo methods.Originality/value– The authors believe that the approach to simulating quasi-fBm using standard AR(q> 1) models is original. The approach is intuitive and can be applied easily.

Spatial Dispersion of Elastic Waves in a Bar Characterized by Tempered Nonlocal Elasticity

Abstract We apply the framework of tempered fractional calculus to investigate the spatial dispersion of elastic waves in a one-dimensional elastic bar characterized by range-dependent nonlocal interactions. The measure of the interaction is given by the attenuation kernel present in the constitutive stress-strain relation of the bar, which follows from the Kröner-Eringen’s model of nonlocal elasticity. We employ a fractional power-law attenuation kernel and spatially temper it, to make the model physically valid and mathematically consistent. The spatial dispersion relation is derived, but it turns out to be difficult to solve, both analytically and numerically. Consequently, we use numerical techniques to extract the real and imaginary parts of the complex wavenumber for a wide range of frequency values. From the dispersion plots, it is found that the phase velocity dispersion of elastic waves in the tempered nonlocal elastic bar is similar to that from the time-fractional Zener model. Further, we also examine the unusual attenuation pattern obtained for the elastic wave propagation in the bar.

Onto resolving spurious wave reflection problem with changing nonlocality among various length scales

In this work an effective method was proposed in order to resolve the wave reflection problems between local/nonlocal models as well as multiple nonlocal models with varying nonlocality. Spurious wave reflection has been a primary concern in developing a robust multiscale–multiresolution model. In the current work a power-law based nonlocal peridynamic model has been proposed in order to mitigate this issue in a versatile manner. The fractional power-law eliminates the spurious wave reflection at the interfaces between local/nonlocal regions or regions with different nonlocalities. By controlling the exponent of the power-law it is possible to vary the frequency components of short or long waves without requiring a large handshake region. Using this underlying idea, 1|x−x′|1+α,∀0<α<2 can be used as a kernel function in order to define nonlocal interaction between x and x′. It was shown that by controlling α it is possible to change the nature of nonlocal interaction within any given cutoff range. Besides power law, Gaussian kernel is another good choice in minimizing the wave reflection issue. However, Gaussian function has some limitations with large variation in nonlocality or waves with higher frequency. In that context the proposed model demonstrated its effectiveness by removing any spurious wave reflections originated in various cases.

On the temperature dependence of fast electron transport in crystal lattices

Building upon the findings of Muto et al. [Phys. Lett. A 136, 33 (1989)] and Marchesoni and Lucheroni [Phys. Rev. E 44, 5303 (1991)] about the growth of the number of (anharmonic) lattice solitons with increasing temperature and using a recent transport theory developed by the present authors [A.P. Chetverikov, W. Ebeling, G. Ropke, M.G. Velarde, Eur. Phys. J. B 87, 153 (2014)] here we provide the fractional power law of the temperature dependence of resistivity in a rather general model for one-dimensional crystal lattices as, e.g., conducting polymers. We also show that the determining factor for the transport is the possibility of forming electron-soliton bound states (in short solectrons) with a most significant contribution arising from the (bosonic) bound state of two electrons to a soliton (in short bisolectrons).

Universal far-from-equilibrium dynamics of a holographic superconductor.

Symmetry-breaking phase transitions are an example of non-equilibrium processes that require real-time treatment, a major challenge in strongly coupled systems without long-lived quasiparticles. Holographic duality provides such an approach by mapping strongly coupled field theories in D dimensions into weakly coupled quantum gravity in D+1 anti-de Sitter spacetime. Here we use holographic duality to study the formation of topological defects-winding numbers-in the course of a superconducting transition in a strongly coupled theory in a 1D ring. When the system undergoes the transition on a given quench time, the condensate builds up with a delay that can be deduced using the Kibble-Zurek mechanism from the quench time and the universality class of the theory, as determined from the quasinormal mode spectrum of the dual model. Typical winding numbers deposited in the ring exhibit a universal fractional power law dependence on the quench time, also predicted by the Kibble-Zurek Mechanism.

Nonlinear analysis of energy harvesting systems with fractional order physical properties

An electromechanical energy harvesting system with a fractional order current–voltage relationship for the electrical circuit and fractional power law in the restoring force of its mechanical part is considered to act as an energy harvester. Our results show that, under a single-well potential configuration, for a small amplitude of the perturbation, as the order of derivative increases, the resonant amplitude of mechanical vibration decreases while the bending degree (hardening case) remains fairly constant. For a large amplitude of the perturbation, the output power is increased due to the hardening effects. Under a double-well configuration, the fractional power stiffness \(\alpha \) strongly affects the crossing well dynamics (large amplitude motion) and consequently the output electrical power. The harvested electric power appears to be maximal for deterministic and random excitation for small \(\alpha \). High-level noise intensity is found to reduce the output power in the region of resonance and surprisingly increases the output in other regions of \(\alpha \). For sufficiently large amplitude of harmonic excitation, this effect is realized in a stochastic resonance.

A fractional approach to wicking fluid flow in nonwoven porous fabric

Purpose – The purpose of this paper is to employ a fractional approach to predict the permeability of nonwoven fabrics by simulating diffusion process. Design/methodology/approach – The method described here follows a similar approach to anomalous diffusion process. The relationship between viscous hydraulic permeability and electrical conductivity of porous material is applied in the derivation of fractional power law of permeability. Findings – The presented power law predicted by fractional method is validated by the results obtained from simulation of fluid flow around a 3D nonwoven porous material by using the lattice-Boltzmann approach. A relation between the fluid permeability and the fluid content (filling fraction), namely, following the power law of the form, was derived via a scaling argument. The exponent n is predominantly a function of pore-size distribution dimension and random walk dimension of the fluid. Originality/value – The fractional scheme by simulating diffusion process presented in this paper is a new method to predict wicking fluid flow through nonwoven fabrics. The forecast approach can be applied to the prediction of the permeability of other porous materials.

Viscosity dependence of passage through a fluctuating bottleneck.

We generalize the model of a rate process involving the passage of an object through a fluctuating bottleneck. The rate of passage is considered to be proportional to a power function of the radius of the bottleneck with exponent α > 0. The fluctuations of the bottleneck are coupled to the motion of the surrounding medium and governed by Langevin dynamics. We show numerically and also explain analytically that for slow bottleneck fluctuations the long time decay rate of the process has a fractional power law dependence on the solvent viscosity with exponent α/(α + 2). The results are consistent with the experimental data on ligand binding to myoglobin, and might also be relevant to other reactions for which exponents between 0 and 1 were reported.

Vanadium oxide nanourchins: Hydrothermal synthesis, characterization, frequency dependent electrical properties

Vanadium oxide nanourchins (4-phenylbutylamine/vanadium oxide) have been synthesized via one-step hydrothermal treatment. The electrical and dielectric properties dependences on temperature and on frequency of the nanocomposite have been reported. It has been found that the dielectric constant and the dielectric loss decrease when frequency increases. However they increase when temperature increases over the studied ranges. Furthermore, the low frequency dielectric response followed a universal fractional power law. This was ascribed to the quasi-dc conductivity. At low frequencies the ac conductivity tends to be constant while at higher frequencies it becomes frequency dependent, which can be explained in terms of Jonscher׳s law. Values of the exponents decrease with increasing temperature and the experimental data reveal that the hopping model is the rate determining mechanism.

Universal microstructure and conductivity relaxation of polymer-conductor composites across the percolation threshold

Micro-structural and impedance analysis of series of insulating polymer/conductor composites (PCC) as a function of frequency and volume fractions of the conductor (fcon) have been studied. Evidences of conductivity relaxation have been noticed with a correlation with the sample micro-structure. This has been understood and explained in terms of equivalent electrical circuit model of the material established through complex impedance spectroscopy (CIS) across the percolation threshold (fc) for all the PCC. CIS analysis confirmed that PCC with fcon ≥ fc, exhibit conductivity/interfacial relaxation due to polarization of Maxwell–Wagner–Sillars (MWS) type at fc and the relaxation frequency increases with increase of fcon. The modulus spectroscopy analysis suggests the presence of two types of relaxations in different frequency ranges; (i) dipolar relaxation associated with the flipping of dipoles present in the pure polymer for fcon < fc and (ii) the conductivity/interfacial relaxation due to the formation of artificial MWS dipoles at the interface of the two components. A long range dc conductivity appears at fcon ≥ fc and Jonschers universal fractional power law is satisfied for both the regions of fcon < fc and fcon ≥ fc in all PCC.

Ultralow frequency dielectric dispersion in PZT + PFS ferroelectric ceramic

The depolarization decaying currents were measured for the PZT compound with morphotropic composition modified by the Fe 1/3 Sb 2/3 cations, i.e., Pb [( Fe 1/3 Sb 2/3)x Ti y Zr z] O 3 with x + y + z = 1, x = 0.1, y = 0.44 (morphotropic boundary) and y = 0.47. The measurements were performed at room temperature and a few temperatures above (up to 473 K), as well as at a few lower temperatures (down to 77 K). The samples were poling at fields 0.02 kV/cm and 0.2 kV/cm at higher and lower temperatures, respectively. In the high temperature range, the time dependence depolarization current follows the fractional power law with two different exponents (n &lt; 1 and m &gt; 1), while in the low temperature range with a single exponent (n). The appropriate procedure has been used to transform the results obtained in the time domain into the frequency domain in order to calculate real and imaginary parts of the dielectric permittivity in the frequency range 5 × 10-5-0.1 Hz. The possible dielectric relaxation mechanisms have been discussed.

Dielectric spectroscopy investigations of nanostructured silicon

Dielectric spectroscopy measurements were performed on planar silicon nanostructures buried within a crystalline Si that form a nanoscale Si-layered system. An insight into the specific behavior of the free-carrier population confined in the surface potential well was then made possible. It was found that the presence or the absence of the SiO2 passivation modifies considerably relaxation responses of the studied structures. A clear differentiation of two dielectric responses: from the same sample with and without electronic passivation allowed determination of the conduction behavior in the surface c-Si delimited by the nanoscale Si-layered system. The sample with a 100 nm thick SiO2 layer (and an excellent quality of the SiO2/c-Si interface) exhibits a fractional power-law dielectric response, corresponding clearly to the generalized Mittag–Leffler pattern. Simultaneously, the dielectric response of a bare sample (after the total RIE of the previously deposited SiO2 layer, about 10 nm native SiO2 layer and poor quality of the SiO2/c-Si interface) is dominated by the conductivity term.

Universal fractional noncubic power law for density of metallic glasses.

As a fundamental property of a material, density is controlled by the interatomic distances and the packing of microscopic constituents. The most prominent atomistic feature in a metallic glass (MG) that can be measured is its principal diffraction peak position (q1) observable by x-ray, electron, or neutron diffraction, which is closely associated with the average interatomic distance in the first shell. Density (and volume) would naturally be expected to vary under compression in proportion to the cube of the one-dimensional interatomic distance. However, by using high pressure as a clean tuning parameter and high-resolution in situ techniques developed specifically for probing the density of amorphous materials, we surprisingly found that the density of a MG varies with the 5/2 power of q1, instead of the expected cubic relationship. Further studies of MGs of different compositions repeatedly produced the same fractional power law of 5/2 in all three MGs we investigated, suggesting a universal feature in MG.

Spin measurements and control of cold atoms using spin-orbit fields

We show that by switching on a spin-orbit interaction in a cold-atom system, experiencing a Zeeman-like coupling to an external field, e.g., in a Bose-Einstein condensate, one can simulate a quantum measurement on a precessing spin. Depending on the realization, the measurement can access both the ergodic and the Zeno regimes, while {the time dependence} of the spin's decoherence may vary from a Gaussian to an inverse fractional power law. Back action of the measurement forms time- and coordinate-dependent profiles of the atoms' density, resulting in its translation, spin-dependent fragmentation, and appearance of interference patterns.

Quantum localization of chaotic eigenstates and the level spacing distribution

The phenomenon of quantum localization in classically chaotic eigenstates is one of the main issues in quantum chaos (or wave chaos), and thus plays an important role in general quantum mechanics or even in general wave mechanics. In this work we propose two different localization measures characterizing the degree of quantum localization, and study their relation to another fundamental aspect of quantum chaos, namely the (energy) spectral statistics. Our approach and method is quite general, and we apply it to billiard systems. One of the signatures of the localization of chaotic eigenstates is a fractional power-law repulsion between the nearest energy levels in the sense that the probability density to find successive levels on a distance S goes like [proportionality]S(β) for small S, where 0≤β≤1, and β=1 corresponds to completely extended states. We show that there is a clear functional relation between the exponent β and the two different localization measures. One is based on the information entropy and the other one on the correlation properties of the Husimi functions. We show that the two definitions are surprisingly linearly equivalent. The approach is applied in the case of a mixed-type billiard system [M. Robnik, J. Phys. A: Math. Gen. 16, 3971 (1983)], in which the separation of regular and chaotic eigenstates is performed.

Onset of dynamic activity in globally coupled excitable and oscillatory units

The onset of dynamic activity is studied in heterogeneous populations of globally coupled units such that each unit is either excitable or oscillatory depending on its parameter. By varying the mean of the parameter distribution as well as coupling strength, we show that all or part of the populations studied have common features: similar phase diagrams, hysteresis near or at the onset of dynamic activity, and a fractional power law obeyed by an order parameter. A simplified model is proposed to explain these results.