Lyapunov Modes in Hard-Disk Systems

We consider simulations of a two-dimensional gas of hard disks in a rectangular container and study the Lyapunov spectrum near the vanishing Lyapunov exponents. To this spectrum are associated eigen-directions, called Lyapunov modes. We carefully analyze these modes and show how they are naturally associated with vector fields over the container. We also show that the Lyapunov exponents, and the coupled dynamics of the modes (where it exists) follow linear laws, whose coefficients only depend on the density of the gas, but not on aspect ratio and very little on the boundary conditions.     

When can one observe good hydrodynamic Lyapunov modes?

Inspired by recent results on differences in fluctuations of finite-time Lyapunov exponents between hard-core and soft-potential systems we surmise that partial domination of the Oseledec splitting (DOS) with respect to subspaces associated with near-zero Lyapunov exponents is essential for observing good hydrodynamic Lyapunov modes (HLMs). Numerical results for coupled map lattices are presented to show the importance of DOS for observing good HLMs. This is achieved by relating splitting parameters to the maximum value of the Lyapunov mode structure factor.     

Universal features of hydrodynamic Lyapunov modes in extended systems with continuous symmetries

Numerical and analytical evidence is presented to show that hydrodynamic Lyapunov modes (HLMs) do exist in lattices of coupled Hamiltonian and dissipative maps. More importantly, we find that HLMs in these two classes of systems are different with respect to their spatial structure and their dynamical behavior. To be concrete, the corresponding dispersion relations of Lyapunov exponent versus wave number are characterized by lambda approximately k and lambda approximately k2, respectively. The HLMs in Hamiltonian systems are propagating, whereas those of dissipative systems show only diffusive motion. Extensive numerical simulations of various systems confirm that the existence of HLMs is a very general feature of extended dynamical systems with continuous symmetries and that the above-mentioned differences between the two classes of systems are universal in large extent.     

Lyapunov modes and time-correlation functions for two-dimensional systems

The relation between the Lyapunov modes (delocalized Lyapunov vectors) and the momentum autocorrelation function is discussed in two-dimensional hard-disk systems. We show numerical evidence that the smallest time-oscillating period of the Lyapunov modes is twice as long as the time-oscillating period of momentum autocorrelation function for both square and rectangular two-dimensional systems with hard-wall boundary conditions.     

Stretching in a model of a turbulent flow

Using a multi-scaled, chaotic flow known as the KS model of turbulence [J.C.H. Fung, J.C.R. Hunt, A. Malik, R.J. Perkins, Kinematic simulation of homogeneous turbulence by unsteady random fourier modes, J. Fluid Mech. 236 (1992) 281-318], we investigate the dependence of Lyapunov exponents on various characteristics of the flow. We show that the KS model yields a power law relation between the Reynolds number and the maximum Lyapunov exponent, which is similar to that for a turbulent flow with the same energy spectrum. Our results show that the Lyapunov exponents are sensitive to the advection of small eddies by large eddies, which can be explained by considering the Lagrangian correlation time of the smallest scales. We also relate the number of stagnation points within a flow to the maximum Lyapunov exponent, and suggest a linear dependence between the two characteristics.     

基于波导、金和氧化锌的随机激光器的研究

通过将金属金嵌入到上下两层波导结构中,与ZnO随机粒子相结合,设计出一种新型结构的随机激光器。采用时域有限差分法(FDTD),数值模拟了该随机激光器系统的光场分布和模式频谱图,并针对金粒子和金薄膜两种情况进行对比分析。结果显示,采用金粒子作夹层时,出射激光模式数量减少,单色性较好;采用金薄膜作夹层时,出射激光光强较大。     

Fluctuations in a fluid under a stationary heat flux. I. General theory

We present the basic formulas for a unified treatment of the correlation functions of the hydrodynamic variables in a fluid between two horizontal plates which is exposed to a stationary heat flux in the presence of a gravity field (Rayleigh-Bénard system). Our analysis is based on fluctuating hydrodynamics. In this paper (I) we show that in the nonequilibrium stationary state the hydrodynamic fluctuations evolve on slow and fast time scales that are widely separated. A time scale perturbation theory is used to diagonalize the hydrodynamic operator partially. This enables us to derive the eigenvalue equations for the nonequilibrium hydrodynamic modes. Therein we take into account the variation of the macroscopic quantities with position. The correlation functions are formally expressed in terms of the nonequilibrium modes. In paper II the slow hydrodynamic modes (viscous and viscoheat modes) will be determined explicitly for ideal heat-conducting plates with stick boundary conditions and used to compute the slow part of the correlation functions; in paper III the fast hydrodynamic modes (sound modes) will be explicitly determined for stick boundary conditions and used to compute the fast part of the correlation functions. In these papers we will also compute the shape and intensity of the lines measured in light scattering experiments.     

Unusual transmission properties of wave in one-dimensional random system containing left-handed-material

Propagation properties of electromagnetic waves in a one-dimension random system containing left-handed-material are studied by the transfer matrix method. The statistics of the Lyapunov exponent and its variance of the transmitted waves are also analyzed. The nonlocalized modes are not only found in such a disordered system, the Anderson localization states with short localization length can also be easily realized due to the existence of low frequency resonant gap. Furthermore, our results also show that a single-parameter scaling is generally inadequate even for the complete random system with negative-n materials when the frequency we consider is located in a gap.     

Fractality of the hydrodynamic modes of diffusion

Transport by normal diffusion can be decomposed into hydrodynamic modes which relax exponentially toward the equilibrium state. In chaotic systems with 2 degrees of freedom, the fine scale structures of these modes are singular and fractal, characterized by a Hausdorff dimension given in terms of Ruelle's topological pressure. For long-wavelength modes, we relate the Hausdorff dimension to the diffusion coefficient and the Lyapunov exponent. This relationship is tested numerically on two Lorentz gases, one with hard repulsive forces, the other with attractive, Yukawa forces. The agreement with theory is excellent.     

The Green’s function in a channel with a sound-absorbing cover in the case of a uniform flow

We study the modal structure of an acoustic field of a point source as function of channel wall admittance in the case of a two-dimensional channel. The characteristic equation for determining the eigen-values corresponding to the boundary problem is studied in the form of this equation??s dependence on the admittance, which varies in the entire complex plane. All modes, without exception, existing in the channel and forming the source field are classified based on the obtained topography of the characteristic equation. The expressions that describe the amplitudes and spatial distribution of the hydrodynamic modes, attenuation rate (for stable modes), or increment (for unstable modes) were obtained as functions of the wall admittance and flow velocity. It is shown that in addition to the hydrodynamic unstable modes existing downstream from the source, hydrodynamic unstable modes exist upstream from the source at any admittance. They appear only when the admittance has an elastic character. It is shown that hydrodynamic modes are induced only in the case when the source is located close to the wall or on the wall. The amplitude of these modes decreases exponentially with distance from the wall.     

一个全局耦合不连续映像格子中的冻结化随机图案模式

本文研究了一类既不连续又不可逆分段线性映像构成的全局耦合映像格子系统中的一类典型集体动力学行为, 即冻结化随机图案模式. 计算了平均同步序参量和最大李雅普诺夫指数随耦合强度的变化. 结果显示, 当耦合强度超过某个阈值后, 在给定动力学变量的初始下, 系统几乎都能达到完全或部分同步状态, 出现冻结化随机图案. 这些现象表明, 耦合映像格子系统中存在着多个共存的吸引子. 因此, 其冻结化图案的结构和分布敏感地依赖于格点动力学变量初始值的选取. 感兴趣地是, 即使当单映像处于混沌状态时, 格点间的耦合仍能将系统调制到规则的运动状态, 这种特征对于混沌控制具有重要的利用价值. 上述丰富动力学行为的出现是由于单映像中不连续性和不可逆性相互作用的结果.     

Upper Quantum Lyapunov Exponent and Anosov Relations for Quantum Systems Driven by a Classical Flow

We generalize the definition of quantum Anosov properties and the related Lyapunov exponents to the case of quantum systems driven by a classical flow, i.e. skew-product systems. We show that the skew Anosov properties can be interpreted as regular Anosov properties in an enlarged Hilbert space, in the framework of a generalized Floquet theory. This extension allows us to describe the hyperbolicity properties of almost-periodic quantum parametric oscillators and we show that their upper Lyapunov exponents are positive and equal to the Lyapunov exponent of the corresponding classical parametric oscillators. As second example, we show that the configurational quantum cat system satisfies quantum Anosov properties.     

Hydrodynamic modes in dense trapped ultracold gases

We consider the hydrodynamic modes for dense trapped ultracold gases, where the interparticle distance is comparable to the scattering length. We show that the experimental determination of the hydrodynamic mode frequencies allows one to obtain quite directly the equation of state of a dense gas. As an example, we investigate the case of two equal fermionic populations in different hyperfine states with attractive interaction.     

Time-oscillating Lyapunov modes and the momentum autocorrelation function

The Lyapunov vectors corresponding to the steps of Lyapunov spectra for many-particle systems show time-oscillating behavior in two types of Lyapunov modes, one associated with time-translational invariance and the other with spatial translational invariance. Our result is that, for each coordinate direction, the longest period of the Lyapunov modes is twice as long as the period of the momentum autocorrelation function. A simple explanation for this relation is proposed and we argue that this result is generally true for many-particle systems. This gives the first quantitative connection between the Lyapunov modes and an experimentally accessible quantity.     

Chaos Control for Coupling of the Double-Well Duffing System Based on Random Phase Disturbance

Non-feedback methods of chaos control are suited for practical applications. For possible practical applications of the control methods, the robustness of the methods in the presence of noise is of special interest. The noise can be in the form of external disturbances to the system or in the form of uncertainties due to inexact model of the system. This paper deals with the effect of random phase disturbance for a class of coupling of the Double-Well Duffing system in the presence of the noise. Lyapunov index is an important indicator to describe chaos. When the sign of the top Lyapunov exponent is positive, the system is chaotic. We compute top Lyapunov exponent by the Khasminskii’s transform formula of spherical coordinate and extension of Wedig’s algorithm based on linear stochastic system. With the change of the average of top Lyapunov exponent sign, we show that random phase can suppress chaos. Finally Poincaré map and phase portraits analysis are studied to confirm the obtained results.     

Dynamic dielectric response of a quantum wire superlattice in interaction with a nonlocal host medium

We study nonlocality effects of a bulk plasmalike dielectric medium on the plasmon spectrum of a one-dimensional (1D) quantum wire superlattice in interaction with the 3D nonlocal host bulk plasma, by carrying out a closed-form analytic determination of the inverse dielectric function κ for the joint nanostructure system within the random phase approximation (RPA), in which we treat nonlocality of the 1D superlattice in the RPA and that of the bulk medium in the hydrodynamic model. By examining the frequency poles of κ (i.e., the dispersions relations), we show analytically that coupled plasmon modes of the interacting 1D superlattice-3D nonlocal host are damped in high frequencies (damping is pronounced near resonance region) and that nonlocality of the host medium introduces nonlocal low frequency (real) modes into the spectrum, which have cutoff frequencies for finite wave vector values. In order to describe the impact of nonlocality effects more clearly, we also examine the spectrum numerically.     

Macroscopic Lyapunov functions for separable stochastic neural networks with detailed balance

We derive macroscopic Lyapunov functions for large, long-range, Ising-spin neural networks with separable symmetric interactions, which evolve in time according to local field alignment. We generalize existing constructions, which correspond todeterministic (zero-temperature) evolution and to specific choices of the interaction structure, to the case ofstochastic evolution and arbitrary separable interaction matrices, for both parallel and sequential spin updating. We find a direct relation between the form of the Lyapunov functions (which describe dynamical processes) and the saddle-point integration that results from performing equilibrium statistical mechanical studies of the present type of model.