Uncertainty principle limits on the cosmological constant

The identification of the cosmological constant term with the energy density of the vacuum enables lower and upper limits to be placed on its value. The upper limit arises from the constraint that the total zero-point energy, which also gravitates, should not dominate cosmological dynamics, while the lower limit can result from the operational requirement that the vacuum-energy shifts over atomic or nuclear scales be at least measurable over Hubble time scales.     

随机过程动态自适应小波独立网格多尺度模拟

在随机过程数值仿真中,由多项式混沌展开谱方法得到求解展开系数的确定性偶合方程组。该方程组比相应的确定性仿真时增大许多。并且当多项式展开阶数和随机空间维数提高时,方程维数急剧增加。由于待求未知分量为表征不同尺度波动的混沌展开模,形成节点意义下的的多尺度问题,传统的网格细分自适应逼近不再适用。为此我们采用了小波的多尺度离散,并建立基于空间细化的动态自适应系统,让每个求解点上的多个未知分量有各自独立的小波网格。本文以随机对流扩散方程为例,进行了二个算例的数值实验,论证了此方法的优点。     

Chaotic itinerancy, temporal segmentation and spatio-temporal combinatorial codes

We study a deterministic dynamics with two time scales in a continuous state attractor network. To the usual (fast) relaxation dynamics towards point attractors (“patterns”) we add a slow coupling dynamics that makes the visited patterns lose stability, leading to an itinerant behavior in the form of punctuated equilibria. One finds that the transition frequency matrix for transitions between patterns shows non-trivial statistical properties in the chaotic itinerant regime. We show that mixture input patterns can be temporally segmented by the itinerant dynamics. The viability of a combinatorial spatio-temporal neural code is also demonstrated.     

A computational strategy for multiscale systems with applications to Lorenz 96 model

Numerical schemes for systems with multiple spatio-temporal scales are investigated. The multiscale schemes use asymptotic results for this type of systems which guarantee the existence of an effective dynamics for some suitably defined modes varying slowly on the largest scales. The multiscale schemes are analyzed in general, then illustrated on a specific example of a moderately large deterministic system displaying chaotic behavior due to Lorenz. Issues like consistency, accuracy, and efficiency are discussed in detail. The role of possible hidden slow variables as well as additional effects arising on the diffusive time-scale are also investigated. As a byproduct we obtain a rather complete characterization of the effective dynamics in Lorenz model.     

Steady-state conduction in self-similar billiards

The self-similar Lorentz billiard channel is a spatially extended deterministic dynamical system which consists of an infinite one-dimensional sequence of cells whose sizes increase monotonically according to their indices. This special geometry induces a nonequilibrium stationary state with particles flowing steadily from the small to the large scales. The corresponding invariant measure has fractal properties reflected by the phase-space contraction rate of the dynamics restricted to a single cell with appropriate boundary conditions. In the near-equilibrium limit, we find numerical agreement between this quantity and the entropy production rate as specified by thermodynamics.     

Modal Dynamics for Positive Operator Measures

The modal interpretation of quantum mechanics allows one to keep the standard classical definition of realism intact. That is, variables have a definite status for all time and a measurement only tells us which value it had. However, at present modal dynamics are only applicable to situations that are described in the orthodox theory by projective measures. In this paper we extend modal dynamics to include positive operator measures (POMs). That is, for example, rather than using a complete set of orthogonal projectors, we can use an overcomplete set of nonorthogonal projectors. We derive the conditions under which Bell's stochastic modal dynamics for projective measures reduce to deterministic dynamics, showing (incidentally) that Brown and Hiley's generalization of Bohmian mechanics [quant-ph/0005026, (2000)] cannot be thus derived. We then show how deterministic dynamics for positive operators can also be derived. As a simple case, we consider a Harmonic oscillator, and the overcomplete set of coherent state projectors (i.e., the Husimi POM). We show that the modal dynamics for this POM in the classical limit correspond to the classical dynamics, even for the nonclassical number state |n>. This is in contrast to the Bohmian dynamics, which for energy eigenstates, the dynamics are always non-classical.     

Proper orthogonal decomposition of solar photospheric motions

The spatiotemporal dynamics of the solar photosphere is studied by performing a proper orthogonal decomposition (POD) of line of sight velocity fields computed from high resolution data coming from the MDI/SOHO instrument. Using this technique, we are able to identify and characterize the different dynamical regimes acting in the system. Low-frequency oscillations, with frequencies in the range 20-130 microHz, dominate the most energetic POD modes (excluding solar rotation), and are characterized by spatial patterns with typical scales of about 3 Mm. Patterns with larger typical scales of approximately 10 Mm, are associated to p-modes oscillations at frequencies of about 3000 microHz.     

Decay of turbulence in pipe flow

A novel experiment has been devised which provides direct evidence for critical point behavior in the longstanding problem of the transition to turbulence in a pipe. The novelty lies in the quenching of turbulence by reducing the Reynolds number and observing the decay of disordered motion. Divergence of the time scales implies underlying deterministic dynamics which are analogous to those found in boundary crises in dynamical systems. A modulated wave packet emerges from the long term transients and this coherent state provides evidence for connections with recent theoretical developments.     

Implications of recording strategy for estimates of neocortical dynamics with electroencephalography

Neocortical dynamics evidently involves very complex, nonlinear processes including top-down and bottom-up interactions across spatial scales. The dynamics may also be strongly influenced by global (periodic) boundary conditions. The primary experimental measure of human neocortical dynamics at short time scales ( approximately few ms) is the scalp electroencephalogram (EEG). It is shown that different recording and data analysis strategies are sensitive to different parts of the spatial spectrum. Thus experimental measures of system dynamics (e.g., correlation dimension estimates) can generally be expected to depend on experimental method. These ideas are illustrated in two ways: a large scale, quasilinear theory of neocortical dynamics in which standing wave phenomenon occur with predicted frequencies in the general range of EEG, and a relatively simple nonlinear physical system consisting of a linear string with attached nonlinear springs. The string/springs system is integrated numerically to illustrate transitions from periodic to chaotic behavior as mesoscopic nonlinear influences dominate macroscopic linear effects. The implications of these results for new theories of neocortical dynamics, experimental estimates of dynamic properties, and cognitive EEG studies are considered.     

Two time scales and violation of the fluctuation-dissipation theorem in a finite dimensional model for structural glasses

We study the breakdown of fluctuation-dissipation relations between time-dependent density-density correlations and associated responses following a quench in the chemical potential in the frustrated Ising lattice gas. The corresponding slow dynamics is characterized by two well-separated time scales characterized by a constant value of the fluctuation-dissipation ratio. This result is particularly relevant since activated processes dominate the long-time dynamics of the system.     

Surface elasticity and size effect on the vibrational behavior of silicon nanoresonators

Predominance of nano-scale effects observed in material behavior at small scales requires implementation of new simulation methods which are not merely based on classical continuum mechanic. On the other hand, although the atomistic modeling methods are capable of modeling nano-scale effects, due to the computational cost, they are not suitable for dynamic analysis of nano-structures. In this research, we aim to develop a continuum-based model for nano-beam vibrations which is capable of predicting the results of molecular dynamics (MD) simulations with considerably lower computational effort. In this classical-based modeling, the surface and core regions are taken to have different mechanical properties, where core atoms are assumed to have macroscale properties whereas surface layer is showing a different elastic modulus from the core components. By estimating physical parameters of proposed classical model using molecular dynamics results and the genetic algorithm, calibrated classical Euler–Bernoulli and Timoshenko beam models are developed. The results demonstrates that a Timoshenko beam model incorporating surface effects and having calibrated parameters, is able to provide almost the same results as molecular dynamics method which can be used to predict the vibrational behavior of nano-beams at different scales from nano to macro.     

Irregular diffusion in the bouncing ball billiard

We call a system bouncing ball billiard if it consists of a particle that is subject to a constant vertical force and bounces inelastically on a one-dimensional vibrating periodically corrugated floor. Here we choose circular scatterers that are very shallow, hence this billiard is a deterministic diffusive version of the well-known bouncing ball problem on a flat vibrating plate. Computer simulations show that the diffusion coefficient of this system is a highly irregular function of the vibration frequency exhibiting pronounced maxima whenever there are resonances between the vibration frequency and the average time of flight of a particle. In addition, there exist irregularities on finer scales that are due to higher-order dynamical correlations pointing towards a fractal structure of this curve. We analyze the diffusive dynamics by classifying the attracting sets and by working out a simple random walk approximation for diffusion, which is systematically refined by using a Green–Kubo formula.     

Auto-correlated behavior of WTI crude oil volatilities: A multiscale perspective

In this paper, we investigate the long-range auto-correlated behavior of WTI crude oil volatility series employing multifractal detrended fluctuation analysis. Our findings show that the for small time scales, the auto-correlations of volatilities were multifractal while for large time scales, the auto-correlations were nearly monofractal. Based on multiscale analysis, we also investigate the dynamics of auto-correlations for different intervals of time scales and find that several shocks could make significant effects on the auto-correlated behaviors for small time scales. Analyzing the dynamics of multifractality degrees of auto-correlations for small time scales, we find that the stronger auto-correlations were always related to the lower degrees of multifractality. At last, we have discussions on the determination factors of price behavior, the predictive implications of scaling behavior in volatilities for oil markets and the reasons why long-range auto-correlations of volatility were always strong for both small time scales and large time scales. Our results are very important theoretically and practically.     

Scale and space localization in the Kuramoto-Sivashinsky equation

We describe a wavelet-based approach to the investigation of spatiotemporally complex dynamics, and show through extensive numerical studies that the dynamics of the Kuramoto-Sivashinsky equation in the spatiotemporally chaotic regime may be understood in terms of localized dynamics in both space and scale (wave number). A projection onto a spline wavelet basis enables good separation of scales, each with characteristic dynamics. At the large scales, one observes essentially slow Gaussian dynamics; at the active scales, structured "events" reminiscent of traveling waves and heteroclinic cycles appear to dominate; while the strongly damped small scales display intermittent behavior. The separation of scales and their dynamics is invariant as the length of the system increases, providing additional support for the extensivity of the spatiotemporally complex dynamics claimed in earlier works. We show also that the dynamics are spatially localized, discuss various correlation lengths, and demonstrate the existence of a characteristic interaction length for instantaneous influences. Our results motivate and advance the search for localized, low-dimensional models that capture the full behavior of spatially extended chaotic partial differential equations. (c) 1999 American Institute of Physics.     

Dynamical solution to the problem of a small cosmological constant and late-time cosmic acceleration

Increasing evidence suggests that most of the energy density of the universe consists of a dark energy component with negative pressure that causes the cosmic expansion to accelerate. We address why this component comes to dominate the universe only recently. We present a class of theories based on an evolving scalar field where the explanation is based entirely on internal dynamical properties of the solutions. In the theories we consider, the dynamics causes the scalar field to lock automatically into a negative pressure state at the onset of matter domination such that the present epoch is the earliest possible time consistent with nucleosynthesis restrictions when it can start to dominate.     

Permutation complexity of interacting dynamical systems

The coupling complexity index is an information measure introduced within the framework of ordinal symbolic dynamics. This index is used to characterize the complexity of the relationship between dynamical system components. In this work, we clarify the meaning of the coupling complexity by discussing in detail some cases leading to extreme values, and present examples using synthetic data to describe its properties. We also generalize the coupling complexity index to the multivariate case and derive a number of important properties by exploiting the structure of the symmetric group. The applicability of this index to the multivariate case is demonstrated with a real-world data example. Finally, we define the coupling complexity rate of random and deterministic time series. Some formal results about the multivariate coupling complexity index have been collected in an Appendix.     

Rhythms of the brain: an examination of mixed mode oscillation approaches to the analysis of neurophysiological data

In the nervous system many behaviorally relevant dynamical processes are characterized by episodes of complex oscillatory states, whose periodicity may be expressed over multiple temporal and spatial scales. In at least some of these instances the variability in oscillatory amplitude and frequency can be explained in terms of deterministic dynamics, rather than being purely noise-driven. Recently interest has increased in studying the application of mixed-mode oscillations (MMOs) to neurophysiological data. MMOs are complex periodic waveforms where each period is comprised of several maxima and minima of different amplitudes. While MMOs might be expected to occur in brain kinetics, only a few examples have been identified thus far. In this article, we review recent theoretical and experimental findings on brain oscillatory rhythms in relation to MMOs, focusing on examples at the single neuron level but also briefly touching on possible instances of the phenomenon across local and global brain networks.     

Differentiability implies continuity in neuronal dynamics

Recent work has identified nonlinear deterministic structure in neuronal dynamics using periodic orbit theory. Troublesome in this work were the significant periods of time where no periodic orbits were extracted — “dynamically dark” regions. Tests for periodic orbit structure typically require that the underlying dynamics are differentiable. Since continuity of a mathematical function is a necessary but insufficient condition for differentiability, regions of observed differentiability should be fully contained within regions of continuity. We here verify that this fundamental mathematical principle is reflected in observations from mammalian neuronal activity. First, we introduce a null Jacobian transformation to verify the observation of differentiable dynamics when periodic orbits are extracted. Second, we show that a less restrictive test for deterministic structure requiring only continuity demonstrates widespread nonlinear deterministic structure only partially appreciated with previous approaches.     

Some Aspects of Time-Reversal in Chemical Kinetics

Chemical kinetics govern the dynamics of chemical systems leading towards chemical equilibrium. There are several general properties of the dynamics of chemical reactions such as the existence of disparate time scales and the fact that most time scales are dissipative. This causes a transient relaxation to lower dimensional attracting manifolds in composition space. In this work, we discuss this behavior and investigate how a time reversal effects this behavior. For this, both macroscopic chemical systems as well as microscopic chemical systems (elementary reactions) are considered.     

Characterizing pseudoperiodic time series through the complex network approach

Recently a new framework has been proposed to explore the dynamics of pseudoperiodic time series by constructing a complex network [J. Zhang, M. Small, Phys. Rev. Lett. 96 (2006) 238701]. Essentially, this is a transformation from the time domain to the network domain, which allows for the dynamics of the time series to be studied via organization of the network. In this paper, we focus on the deterministic chaotic Rössler time series and stochastic noisy periodic data that yield substantially different structures of networks. In particular, we test an extensive range of network topology statistics, which have not been discussed in previous work, but which are capable of providing a comprehensive statistical characterization of the dynamics from different angles. Our goal is to find out how they reflect and quantify different aspects of specific dynamics, and how they can be used to distinguish different dynamical regimes. For example, we find that the joint degree distribution appears to fundamentally characterize spatial organizations of cycles in phase space, and this is quantified via an assortativity coefficient. We applied network statistics to electrocardiograms of a healthy individual and an arrythmia patient. Such time series are typically pseudoperiodic, but are noisy and nonstationary and degrade traditional phase-space based methods. These time series are, however, better differentiated by our network-based statistics.