On the basic states of one-dimensional disordered structures

The purpose of this paper is to study a limit probability distribution of the set of the first κ eigenvalues λ₁(?)<λ₂(?)<...<λ_κ(?) (with a fixed κ and ?→∞) of the boundary problem on the interval [0, ?] wherea(t, ω),q(t, ω) are the random stationary processes. Particularly the question of the repulsion between the first eigenvalues (small energetic levels) is studied. It has been proved that in the “divergent” case (q(t, ω)=0,a(t, ω)≠0) levels repulsion exists. As for the “potential” case (a(t, ω)≡1,q(t, ω)≠0) there is not any repulsion at all. This is one of the main differences between these two cases.     

Four-wave mixing and hyper-Raman scattering in CuCl

Non-degenerate four-wave mixing using two non-collinear laser beams with frequencies (wavevectors) ω_p, ω_t (k_p, k_t) respectively is studied in CuCl. Two emission lines at frequencies ω⁽¹⁾=2ω_t-ω_p, and ω⁽²⁾=2ω_p-ω_t are observed. Their excitation spectrum is sharply peaked if the phase-match condition k⁽¹⁾=2k_t-k_p is fulfilled. This is the case, if ω_p coincides with the hyper-Raman lines (R⁺_T, R^-_T) of the laser labelled (t) in a well-defined geometrical configuration.     

Non-adiabatic transitions for random processes in a two-level system

Non-adiabatic transitions in two-level systems are investigated theoretically for a random time dependence of ?ω, the energy difference, between the levels. We assumed that ω = ω(x) and the coordinate x = x(t) is a random function of time. Diffusion and Poisson processes (both homogeneous and with a source) for x(t) were assumed. The cases of linear crossing terms (ω = γx) and non-linear terms (ω = ω _e exp (- αx) + ω₀) were considered. Values of the non-adiabatic transition probability per unit time were obtained by perturbation theory for ω₁τ _c ? 1 where τ _c is the correlation time and ω₁ is the off-diagonal matrix element.     

The exact solution of an elimination problem in kinetic theory: I. The one-dimensional hard sphere gas

A class of initial value problems for a one-dimensional hard sphere gas is considered where a specified particle has a given distribution f⁽¹⁾(z₁; 0) and the rest are in equilibrium at t=0. An exact expansion is obtained for a certain n-particle reduced distribution function f⁽ⁿ⁾(z₁;…;z_n; t) in terms of the 1-particle reduced distribution function f⁽¹⁾(z₁; t) for the specified particle by starting with separate expressions for these functions in terms of f⁽¹⁾(z₁; 0). Expansions for the corresponding cluster functions are first obtained and then graph theoretic methods applied to obtain a solution.     

Overdamped motion of interacting particles in general confining potentials: time-dependent and stationary-state analyses

By comparing numerical and analytical results, it is shown that a system of interacting particles under overdamped motion is very well described by a nonlinear Fokker-Planck equation, which can be associated with nonextensive statistical mechanics. The particle-particle interactions considered are repulsive, motivated by three different physical situations: (i) modified Bessel function, commonly used in vortex-vortex interactions, relevant for the flux-front penetration in disordered type-II superconductors; (ii) Yukawa-like forces, useful for charged particles in plasma, or colloidal suspensions; (iii) derived from a Gaussian potential, common in complex fluids, like polymer chains dispersed in a solvent. Moreover, the system is subjected to a general confining potential, ??(x)?=?(??|x| ^z )/z (???>?0, z?>?1), so that a stationary state is reached after a sufficiently long time. Recent numerical and analytical investigations, considering interactions of type (i) and a harmonic confining potential (z?=?2), have shown strong evidence that a q-Gaussian distribution, P(x,t), with q?=?0, describes appropriately the particle positions during their time evolution, as well as in their stationary state. Herein we reinforce further the connection with nonextensive statistical mechanics, by presenting numerical evidence showing that: (a) in the case z?=?2, different particle-particle interactions only modify the diffusion parameter D of the nonlinear Fokker-Planck equation; (b) for z????2, all cases investigated fit well the analytical stationary solution P _st(x), given in terms of a q-exponential (with the same index q?=?0) of the general external potential ??(x). In this later case, we propose an approximate time-dependent P(x,t) (not known analytically for z????2), which is in very good agreement with the simulations for a large range of times, including the approach to the stationary state. The present work suggests that a wide variety of physical phenomena, characterized by repulsive interacting particles under overdamped motion, present a universal behavior, in the sense that all of them are associated with the same entropic form and nonlinear Fokker-Planck equation.     

Chaotic time series prediction: From one to another

In this Letter, a new local linear prediction model is proposed to predict a chaotic time series of a component x(t) by using the chaotic time series of another component y(t) in the same system with x(t). Our approach is based on the phase space reconstruction coming from the Takens embedding theorem. To illustrate our results, we present an example of Lorenz system and compare with the performance of the original local linear prediction model.     

A study of self-organizing processes of nonlinear stochastic variables

A general theory is given for the time evolution of nonlinear stochastic variables a(t) = {a_i(t)} whose statistical distribution is changing due to the self-organization of “macroscopic” order. The dynamics of a(t) is conveniently expressed by self-consistent equations for the ensemble average x(t) = 〈a(t)〉, the supersystem, and for the deviations ξ(t) = a(t)?x(t), the subsystem; the systems are connected to each other by feedback loops in their dynamics. The time dependence of the variance and the correlation function ofξ(t) are studied in terms of relaxation toward local equilibrium underx(t) and dynamical coupling withx(t). A special example shows that the stochastic motions of subsystems are pulled together by the motion of the supersystem through feedback loops, and that this pull-together phenomenon occurs when symmetry-breaking instability exists in nonlinear systems.     

Change in the shape of a symmetric light pulse passing through a quantum well

A theory for the response of a 2D two-level system to irradiation by a symmetric light pulse is developed. Under certain conditions, such an electron system approximates an ideal solitary quantum well in a zero field or a strong magnetic field H perpendicular to the plane of the well. One of the energy levels is the ground state of the system, while the other is a discrete excited state with energy ?ω₀, which may be an exciton level for H=0 or any level in a strong magnetic field. It is assumed that the effect of other energy levels and the interaction of light with the lattice can be ignored. General formulas are derived for the time dependence of the dimensionless “coefficients” of the reflection ?(t), absorption A(t), and transmission ?(t) for a symmetric light pulse. It is shown that the ?(t), A(t), and ?(t) time dependences have singular points of three types. At points t ₀ of the first type, A(t ₀)=T(t ₀)=0 and total reflection takes place. It is shown that for γ_r?γ, where γ_r and γ are the radiative and nonradiative reciprocal lifetimes, respectively, for the upper energy level of the two-level system, the amplitude and shape of the transmitted pulse can change significantly under the resonance ω_l=ω₀. In the case of a long pulse, when γ_l<γ_r, the pulse is reflected almost completely. (The quantity γ_l characterizes the duration of the exciting pulse.) In the case of an intermediate pulse duration γ_l?γ_r, the reflection, absorption, and transmission are comparable in value and the shape of the transmitted pulse differs considerably from the shape of the exciting pulse: the transmitted pulse has two peaks due to the existence of the point t ₀ of total reflection, at which the transmission is zero. If the carrier frequency ω_l of light differs from the resonance frequency ω₀, the oscillating ?(t), A(t), and ?(t) time dependences are observed at the frequency Δω=ωl?ω₀. Oscillations can be observed most conveniently for Δω?γ_l. The position of the singular points of total absorption, reflection, and transparency is studied for the case when ω_l differs from the resonance frequency.     

An analysis of the Lorenz equations: Stable exact solutions

We prove the existence of stable solutions (x(t), y(t), z(t)) to the Lorenz system, t being the time variable. These solutions are valid for b = 2σ and r > 1, b, r and σ being positive parameters, and fall into two classes C_i and C_d depending on whether (x(t), z(t)) are both monotonically increasing or decreasing functions of t. The analytic structure of the solutions is also discussed.     

[Russian Text Ignored]

The parametric excitation of longitudinal waves in an infinite homogeneous plasma by a pump field E (t) = E ₀(t) sin (ω₀t + φ(t)) is studied on the basis of the Vlasov equation, where the amplitude E ₀(t) and the phase φ(t) are slowly varying compared with ω₀ periodic functions. Firstly it is assumed that ω₀ is much larger than the electron plasma frequency ω_Le. In the second part the parametric instabilities are considered under the resonance condition ω₀ ≈ ω_Le. In both parts the threshold fields for the excitation of the longitudinal waves and their growth rates are calculated. As an example these values are analysed for both a sequence of pump impulses and a phase-moduated pump field. They are compared with the results received for a monochromatic pump field.     

Temperature measurements in small holes drilled in superconducting bulk during pulsed field magnetization

The time dependence of the temperatures T(z, t) has been measured along the thickness direction z in several drilled holes in a superconducting bulk during pulsed field magnetization (PFM) and the heat generation and heat transfer in the bulk have been discussed. In the previous paper [H. Fujishiro, S. Kawaguchi, K. Kakehata, A. Fujiwara, T. Tateiwa, T. Oka, Supercond. Sci. Technol. 19 (2006) S540], we calculated the T(z, t) profiles in the bulk by solving a three-dimensional heat-diffusion equation to reproduce the measured T(t) on the bulk surface; the heat generation took place adiabatically and the calculated T(z, t) was isothermal along the z direction. In this study, the measured T(z, t) at the top surface was higher than that at the bottom surface just after the pulse field application at t < 0.5 s, and then became isothermal with increasing time. These results suggest that the magnetic flux intrudes inhomogeneously into the bulk from the edge of the top surface and the periphery at the early stage. The inhomogeneous magnetic flux intrusion and the flux trap during PFM change depending on the strength of the pulsed field and the pulse number in the successive pulse field application.     

SUPER AND COMBINATORIAL HARMONIC RESPONSE OF FLEXIBLE ELASTIC CABLES WITH SMALL SAG

The paper deals with the analysis of cables in stayed bridges and TV-towers, where the excitation is caused by harmonically varying in-plane motions of the upper support point with the amplitude U. Such cables are characterized by a sag-to-chord-length ratio below &0uml;02, which means that the lowest circular eigenfrequencies for in-plane and out-of-plane eigenvibrations, ω₁and ω₂, are closely separated. The dynamic analysis is performed by a two-degree-of-freedom modal decomposition in the lowest in-plane and out-of-plane eigenmodes. Modal parameters are evaluated based on the eigenmodes for the parabolic approximation to the equilibrium suspension. Superharmonic components of the ordern , supported by the parametric terms of the excitation and the non-linear coupling terms, are registered in the response for circular frequency ω?ω₁/n. At moderate U, the cable response takes place entirely in the static equilibrium plane. At larger amplitudes the in-plane response becomes unstable and a coupled whirling superharmonic component occurs. In the paper a first order perturbation solution to the superharmonic response is performed based on the averaging method. For ω?(m/n)ω₁, m<n, the geometrical non-linear restoring forces gives rise to a substantial combinatorial harmonic component with the circular frequency (n/m)ω. Both entirely in-plane and coupled in-plane and out-of-plane responses occur. Based on an initial frequency analysis of the response, an analytical model for these vibrations is formulated with emphasis on superharmonics of the order n=3 and combinatorial harmonics of the order (n, m)=(3,2). All analytical solutions have been verified by direct numerical integration of the modal equations of motion.     

Development of high vorticity in incompressible 3D Euler equations: Influence of initial conditions

The incompressible three-dimensional ideal flows develop very thin pancake-like regions of increasing vorticity. These regions evolve with the scaling ω_max(t) ∝ l(t)^-2/3 between the vorticity maximum and pancake thickness, and provide the leading contribution to the energy spectrum, where the gradual formation of the Kolmogorov interval E _k ∝ k^?5/3 is observed for some initial flows. With the massive numerical simulations, we study the influence of initial conditions on the processes of pancake formation and the Kolmogorov energy spectrum development.     

Monte Carlo simulation of film growth in a phase separating binary alloy model

Using a kinetic Monte Carlo method, we simulate binary film (A_0.5B_0.5/A) growth on an L×L square lattice with the focus on the domain growth behaviour. We compute the average domain area, A(t), as a measure of domain size. For a sufficiently large system, we find that A(t) grows with a power law in time with A(t)∼t^2/3 after the initial transient time. This implies that the dynamic exponent for domain growth with non-conserved order parameter is z=3, a value which was theoretically predicted for the conserved order parameter case. Further analysis reveals that such a power-law behaviour emerges because the order parameter is approximately conserved after the early stage of growth.     

First passage time of multiple Brownian particles on networks with applications

In this article, we study some theoretical and technological problems with relation to multiple Brownian particles on networks. We are especially interested in the behavior of the first arriving Brownian particle when all the Brownian particles start out from the source s simultaneously and head to the destination h randomly. We analyze the first passage time (FPT) Y_sh(z) and the mean first passage time (MFPT) 〈Y_sh(z)〉 of multiple Brownian particles on complex networks. Equations of Y_sh(z) and 〈Y_sh(z)〉 are obtained. On a variety of commonly encountered networks, we observe first passage properties of multiple Brownian particles from different aspects. We find that 〈Y_sh(z)〉 drops substantially when particle number z increases at the first stage, and converges to d_sh, the distance between the source and the destination when z→∞. The distribution of FPT Prob{Y_sh(z)=t},t=0,1,2,… is also analyzed in these networks. The distribution curve peaks up towards t=d_sh when z increases. Consequently, if particle number z is set appropriately large, the first arriving Brownian particle will go along the shortest or near shortest paths between the source and the destination with high probability. Simulations confirm our analysis. Based on theoretical studies, we also investigate some practical problems using multiple Brownian particles, such as communication on P2P networks, optimal routing in small world networks, phenomenon of asymmetry in scale-free networks, information spreading in social networks, pervasion of viruses on the Internet, and so on. Our analytic and experimental results on multiple Brownian particles provide useful evidence for further understanding and properly tackling these problems.     

A triangle model of criminality

This paper is concerned with a quantitative model describing the interaction of three sociological species, termed as owners, criminals and security guards, and denoted by X, Y and Z respectively. In our model, Y is a predator of the species X, and so is Z with respect to Y. Moreover, Z can also be thought of as a predator of X, since this last population is required to bear the costs of maintaining Z.We propose a system of three ordinary differential equations to account for the time evolution of X(t), Y(t) and Z(t) according to our previous assumptions. Out of the various parameters that appear in that system, we select two of them, denoted by H, and h, which are related with the efficiency of the security forces as a control parameter in our discussion. To begin with, we consider the case of large and constant owners population, which allows us to reduce (3), (4) and (5) to a bidimensional system for Y(t) and Z(t). As a preliminary step, this situation is first discussed under the additional assumption that Y(t)+Z(t) is constant. A bifurcation study is then performed in terms of H and h, which shows the key role played by the rate of casualties in Y and Z, that results particularly in a possible onset of bistability. When the previous restriction is dropped, we observe the appearance of oscillatory behaviours in the full two-dimensional system. We finally provide a exploratory study of the complete model (3), (4) and (5), where a number of bifurcations appear as parameter H changes, and the corresponding solutions behaviours are described.     

A new procedure for investigating three-dimensional stress fields in a thin plate with a through-the-thickness crack

In the paper, a new procedure is proposed to investigate three-dimensional fracture problems of a thin elastic plate with a long through-the-thickness crack under remote uniform tensile loading. The new procedure includes a new analytical method and high accurate finite element simulations. In the part of theoretical analysis, three-dimensional Maxwell stress functions are employed in order to derive three-dimensional crack tip fields. Based on the theoretical analysis, an equation which can describe the relationship among the three-dimensional J-integral J(z), the stress intensity factor K(z) and the tri-axial stress constraint level T _z (z) is derived first. In the part of finite element simulations, a fine mesh including 153360 elements is constructed to compute the stress field near the crack front, J(z) and T _z (z). Numerical results show that in the plane very close to the free surface, the K field solution is still valid for in-plane stresses. Comparison with the numerical results shows that the analytical results are valid.     

Analyses of double exponential decays using modulating functions

Use of modulating functions (t^α(T - t)^β), with numerical integration over only the range of the decay (√^T₀), yields estimates of the decay rates directly. The effects of the choice of α and β, of noise and of truncation on the bias found in the extracted parameters is examined by analyzing simulated decays. It is recommended that such analyses be simply preludes to iterative, non-linear least squares analyses, which require estimated decay times.