Finite Time Averages Research Articles

Dynamical glass in weakly nonintegrable Klein-Gordon chains.

Integrable many-body systems are characterized by a complete set of preserved actions. Close to an integrable limit, a nonintegrable perturbation creates a coupling network in action space which can be short or long ranged. We analyze the dynamics of observables which become the conserved actions in the integrable limit. We compute distributions of their finite time averages and obtain the ergodization time scale T_{E} on which these distributions converge to δ distributions. We relate T_{E} to the statistics of fluctuation times of the observables, which acquire fat-tailed distributions with standard deviations σ_{τ}^{+} dominating the means μ_{τ}^{+} and establish that T_{E}∼(σ_{τ}^{+})^{2}/μ_{τ}^{+}. The Lyapunov time T_{Λ} (the inverse of the largest Lyapunov exponent) is then compared to the above time scales. We use a simple Klein-Gordon chain to emulate long- and short-range coupling networks by tuning its energy density. For long-range coupling networks T_{Λ}≈σ_{τ}^{+}, which indicates that the Lyapunov time sets the ergodization time, with chaos quickly diffusing through the coupling network. For short-range coupling networks we observe a dynamical glass, where T_{E} grows dramatically by many orders of magnitude and greatly exceeds the Lyapunov time, which satisfies T_{Λ}≲μ_{τ}^{+}. This effect arises from the formation of highly fragmented inhomogeneous distributions of chaotic groups of actions, separated by growing volumes of nonchaotic regions. These structures persist up to the ergodization time T_{E}.

Dynamical Glass and Ergodization Times in Classical Josephson Junction Chains.

Models of classical Josephson junction chains turn integrable in the limit of large energy densities or small Josephson energies. Close to these limits the Josephson coupling between the superconducting grains induces a short-range nonintegrable network. We compute distributions of finite-time averages of grain charges and extract the ergodization time T_{E} which controls their convergence to ergodic δ distributions. We relate T_{E} to the statistics of fluctuation times of the charges, which are dominated by fat tails. T_{E} is growing anomalously fast upon approaching the integrable limit, as compared to the Lyapunov time T_{Λ}-the inverse of the largest Lyapunov exponent-reaching astonishing ratios T_{E}/T_{Λ}≥10^{8}. The microscopic reason for the observed dynamical glass is rooted in a growing number of grains evolving over long times in a regular almost integrable fashion due to the low probability of resonant interactions with the nearest neighbors. We conjecture that the observed dynamical glass is a generic property of Josephson junction networks irrespective of their space dimensionality.

Imperfect Amplitude Mediated Chimera States in a Nonlocally Coupled Network

We investigate the dynamical transitions in a network of nonlocally coupled Stuart-Landau oscillators with a combination of attractive and repulsive couplings. The competing interaction between the couplings plays a crucial role in many realistic situations, particularly in neuronal systems. We report that the employed attractive and repulsive couplings induce imperfect amplitude mediated chimera state which emerges as an intermediate between the oscillatory dynamics and the oscillation death state. Each oscillator in the synchronized and desynchronized groups constituting the imperfect amplitude mediated chimera drifts between both the homogeneous and inhomogeneous oscillations as a function of time. To distinguish the homogeneous and inhomogeneous oscillations, we use the finite-time average of each oscillator. The observed distinct dynamical states are further classified by finding the strength of the inhomogeneous oscillators in the corresponding dynamical states. We also find that the number of clusters in the cluster oscillation death states exponentially decays as a function of the coupling range and obeys a power law relation. Finally, we confirm the robustness of the observed amplitude mediated chimera state by introducing a Gaussian white noise in the system.

Ergodicity test of the eddy-covariance technique

Abstract. The ergodic hypothesis is a basic hypothesis typically invoked in atmospheric surface layer (ASL) experiments. The ergodic theorem of stationary random processes is introduced to analyse and verify the ergodicity of atmospheric turbulence measured using the eddy-covariance technique with two sets of field observational data. The results show that the ergodicity of atmospheric turbulence in atmospheric boundary layer (ABL) is relative not only to the atmospheric stratification but also to the eddy scale of atmospheric turbulence. The eddies of atmospheric turbulence, of which the scale is smaller than the scale of the ABL (i.e. the spatial scale is less than 1000 m and temporal scale is shorter than 10 min), effectively satisfy the ergodic theorems. Under these restrictions, a finite time average can be used as a substitute for the ensemble average of atmospheric turbulence, whereas eddies that are larger than ABL scale dissatisfy the mean ergodic theorem. Consequently, when a finite time average is used to substitute for the ensemble average, the eddy-covariance technique incurs large errors due to the loss of low-frequency information associated with larger eddies. A multi-station observation is compared with a single-station observation, and then the scope that satisfies the ergodic theorem is extended from scales smaller than the ABL, approximately 1000 m to scales greater than about 2000 m. Therefore, substituting the finite time average for the ensemble average of atmospheric turbulence is more faithfully approximate the actual values. Regardless of vertical velocity or temperature, the variance of eddies at different scales follows Monin–Obukhov similarity theory (MOST) better if the ergodic theorem can be satisfied; if not it deviates from MOST. The exploration of ergodicity in atmospheric turbulence is doubtlessly helpful in understanding the issues in atmospheric turbulent observations and provides a theoretical basis for overcoming related difficulties.

On the long-time integration of stochastic gradient systems

This article addresses the weak convergence of numerical methods for Brownian dynamics. Typical analyses of numerical methods for stochastic differential equations focus on properties such as the weak order which estimates the asymptotic (stepsize h → 0 ) convergence behaviour of the error of finite-time averages. Recently, it has been demonstrated, by study of Fokker–Planck operators, that a non-Markovian numerical method generates approximations in the long-time limit with higher accuracy order (second order) than would be expected from its weak convergence analysis (finite-time averages are first-order accurate). In this article, we describe the transition from the transient to the steady-state regime of this numerical method by estimating the time-dependency of the coefficients in an asymptotic expansion for the weak error, demonstrating that the convergence to second order is exponentially rapid in time. Moreover, we provide numerical tests of the theory, including comparisons of the efficiencies of the Euler–Maruyama method, the popular second-order Heun method, and the non-Markovian method.

Modeling of slow time-scale behavior of fast molecular dynamic systems

This paper aims to apply a recently developed numerical scheme toward multi-time scale modeling, which we refer to as ‘Practical Time Averaging’ (PTA), to molecular dynamical (MD) systems. In the first part, we investigate the fine-scale dynamics of a one-dimensional chain of identical particles under cyclic loading. Assuming a double-well interatomic potential among adjacent particles leads to a phase transition between two distinct equilibrium states. Specifically, we study the macroscopic stress–strain behavior of the dynamical chain in three settings, i.e. Newtonian MD, Newtonian MD with viscous dissipation and Newtonian MD with thermostat. Rate-independent high frequency oscillations are observed in Newtonian MD due to an instability that is related to the non-convexity of the strain energy. This is stabilized by adding viscosity or a thermostat, which leads to strong hysteresis that is consistent with quasi-static results (i.e. lattice statics). In the second part, we first define coarse variables as finite time averages of phase functions in MD. Then we apply the technique of PTA developed in Tan et al. (2013) to numerically approximate the coarse dynamics for the time averaged quantities. The tested model problems include a two-dimensional lattice made of stoichiometric Nickel–Manganese undergoing detwinning and a three-dimensional atomic chain made of face-centered cubic (FCC) Nickel under uniaxial tension. The macroscopic features (such as space-time averaged strain/stress) are obtained from coarse dynamics. It is also shown that the time savings become significant when the loading rate is small.

Lagrangian descriptors: A method for revealing phase space structures of general time dependent dynamical systems

In this paper we develop new techniques for revealing geometrical structures in phase space that are valid for aperiodically time dependent dynamical systems, which we refer to as Lagrangian descriptors. These quantities are based on the integration, for a finite time, along trajectories of an intrinsic bounded, positive geometrical and/or physical property of the trajectory itself. We discuss a general methodology for constructing Lagrangian descriptors, and we discuss a “heuristic argument” that explains why this method is successful for revealing geometrical structures in the phase space of a dynamical system. We support this argument by explicit calculations on a benchmark problem having a hyperbolic fixed point with stable and unstable manifolds that are known analytically. Several other benchmark examples are considered that allow us the assess the performance of Lagrangian descriptors in revealing invariant tori and regions of shear. Throughout the paper “side-by-side” comparisons of the performance of Lagrangian descriptors with both finite time Lyapunov exponents (FTLEs) and finite time averages of certain components of the vector field (“time averages”) are carried out and discussed. In all cases Lagrangian descriptors are shown to be both more accurate and computationally efficient than these methods. We also perform computations for an explicitly three dimensional, aperiodically time-dependent vector field and an aperiodically time dependent vector field defined as a data set. Comparisons with FTLEs and time averages for these examples are also carried out, with similar conclusions as for the benchmark examples.

Convergence to the Time Average by Stochastic Regularization

In Ergodic Theory it is natural to consider the pointwise convergence of finite time averages of functions with respect to the flow of dynamical systems. Since the pointwise convergence is too weak for applications to Hamiltonian Perturbation Theory, requiring differentiability, we first introduce regularized averages obtained through a stochastic perturbation of an integrable Hamiltonian flow, and then we provide detailed estimates. In particular, for a special vanishing limit of the stochastic perturbation, we obtain convergence even in a Sobolev norm taking into account the derivatives.

Coarse variables of autonomous ODE systems and their evolution

Given an autonomous system of ordinary differential equations (ODE), we consider developing practical models for the deterministic, slow/coarse behavior of the ODE system. Two types of coarse variables are considered. The first type consists of running finite time averages of phase functions. Approaches to construct the coarse evolution equation for this type are discussed and implemented on a ‘Forced’ Lorenz system and a singularly perturbed system whose fast flow does not necessarily converge to an equilibrium. We explore two strategies. In one, we compute (locally) invariant manifolds of the fast dynamics, parameterized by the slow variables. In the other, the choice of our coarse variables automatically guarantees them to be ‘slow’ in a precise sense. This allows their evolution to be phrased in terms of averaging utilizing limit measures (probability distributions) of the fast flow. Coarse evolution equations are constructed based on these approaches and tested against coarse response of the ‘microscopic’ models. The second type of coarse variables are defined as (non-trivial) scalar state functions that are required by design to evolve autonomously, to the extent possible, with the goal of being candidate state functions for unambiguously initializable coarse dynamics. The question motivates a mathematical restatement in terms of a first-order PDE. A computational approximation is developed and tested on the Lorenz system and the Hald Hamiltonian system.

A PDE APPROACH TO FINITE TIME INDICATORS IN ERGODIC THEORY

For dynamical systems defined by vector fields over a compact invariant set, we introduce a new class of approximated first integrals based on finite time averages and satisfying an explicit first order partial differential equation. These approximated first integrals can be used as finite time indicators of the dynamics. On the one hand, they provide the same results on applications than other popular indicators; on the other hand, their PDE based definition — that we show robust under suitable perturbations — allows one to study them using the traditional tools of PDE environment. In particular, we formulate this approximating device in the Lyapunov exponents framework and we compare the operative use of them to the common use of the Fast Lyapunov Indicators to detect the phase space structure of quasi-integrable systems.

Ergodicité, collage et transport anomal

We consider the problem of convergence towards spatial ergodic average of the time average of an observable defined for a one and a half degree of freedom Hamiltonian flow with mixed phase space. The analysis is performed by analysing the evolution of the distribution of finite-time averages. An exponent characterising the “speed of convergence” is defined. Results indicate that for the considered mixed case, the rate of convergence goes as t α , with α = 0.45 while it goes as t 1 / 2 when the full phase space is chaotic. Moreover a formula linking this characteristic exponent to the one corresponding to transport properties β is proposed α = 1 − β / 2 and good agreement is found for the considered cases. To cite this article: X. Leoncini et al., C. R. Mecanique 336 (2008).

Kraichnan Turbulence via Finite Time Averages

Relations central to Kraichnan’s theory of fully developed two-dimensional turbulence are rigorously established for finite time averages. In particular, we prove that if the ratio of the averages of palinstrophy to enstrophy is large, then a large inertial range displaying an enstrophy cascade exists. Moreover, if this ratio is comparable (up to a logarithm) to the dissipation wave number (a necessary condition for turbulence), then the power law for the energy spectrum, until now derived only heuristically, is rigorously shown to provide (up to a logarithm) an upper bound for the energy spectrum. Finally we show that, deep in the dissipation range, the palinstrophy contributed by eddies smaller than a specified length decays exponentially in the corresponding wave number. The averaging times needed for these relations are bounded in terms of the generalized Grashof number, independent of the solution for which the time averages are taken. Solutions are not assumed to be on the global attractor, merely within the absorbing ball, an easily verified condition.

Finite temperature phase transition in the three-dimensional Heisenberg ±J spin glass model

The three-dimensional Heisenberg spin glass model with ±J interactions is studied using an over-relaxed Monte Carlo algorithm. We have measured the correlation length ξ, and using the crossing of ξ∕L for different L find a finite temperature spin glass transition, with TSG=0.220(5). In addition, we have varied the number of Monte Carlo steps used prior to, and during, thermal averaging to control the effects of finite time averaging. We find that the over-relaxation algorithm allows for an accurate measurements using as few as 300 Monte Carlo steps.

Exploiting regenerative structure to estimate finite time averages via simulation

We propose nonstandard simulation estimators of expected time averages over finite intervals [0, t ], seeking to enhance estimation efficiency. We make three key assumptions: (i) the underlying stochastic process has regenerative structure, (ii) the time average approaches a known limit as time t increases and (iii) time 0 is a regeneration time. To exploit those properties, we propose a residual-cycle estimator , based on data from the regenerative cycle in progress at time t , using only the data after time t . We prove that the residual-cycle estimator is unbiased and more efficient than the standard estimator for all sufficiently large t . Since the relative efficiency increases in t , the method is ideally suited to use when applying simulation to study the rate of convergence to the known limit. We also consider two other simulation techniques to be used with the residual-cycle estimator. The first involves overlapping cycles , paralleling the technique of overlapping batch means in steady-state estimation; multiple observations are taken from each replication, starting a new observation each time the initial regenerative state is revisited. The other technique is splitting , which involves independent replications of the terminal period after time t , for each simulation up to time t . We demonstrate that these alternative estimators provide efficiency improvement by conducting simulations of queueing models.

Fluctuation theorem for constrained equilibrium systems

We discuss the fluctuation properties of equilibrium chaotic systems with constraints such as isokinetic and Nosé-Hoover thermostats. Although the dynamics of these systems does not typically preserve phase-space volumes, the average phase-space contraction rate vanishes, so that the stationary states are smooth. Nevertheless, finite-time averages of the phase-space contraction rate have nontrivial fluctuations which we show satisfy a simple version of the Gallavotti-Cohen fluctuation theorem, complementary to the usual fluctuation theorem for nonequilibrium stationary states and appropriate to constrained equilibrium states. Moreover, we show that these fluctuations are distributed according to a Gaussian curve for long enough times. Three different systems are considered here: namely, (i) a fluid composed of particles interacting with Lennard-Jones potentials, (ii) a harmonic oscillator with Nosé-Hoover thermostatting, and (iii) a simple hyperbolic two-dimensional map.

Kolmogorov theory via finite-time averages

Several relations from the Kolmogorov theory of fully-developed three-dimensional turbulence are rigorously established for finite-time averages over Leray–Hopf weak solutions of the Navier–Stokes equations. The Navier–Stokes equations are considered with periodic boundary conditions and an external forcing term. The main parameter is the Grashof number associated with the forcing term. The relations rigorously proved in this article include estimates for the energy dissipation rate, the Kolmogorov wavenumber, the Taylor wavenumber, the Reynolds number, and the energy cascade process. For some estimates the averaging time depends on the macroscale wavenumber and the kinematic viscosity alone, while for others such as the Kolmogorov energy dissipation law and the energy cascade, the estimates depend also on the Grashof number. As compared with earlier works by some of the authors the more physical concept of finite-time average is replacing the concept of infinite-time average used before.

Quantitative characterization of spatiotemporal patterns II

Disorderness of spatiotemporal patterns which are obtained by nonlinear partial differential equations is characterized quantitatively. The mean Lyapunov exponent for a nonlinear partial differential equation is given. The local Lyapunov exponent which is a finite time average of the mean Lyapunov exponent is shown to have close relation to the spatiotemporal patterns. It is suggested that the systems which are described by nonlinear partial differential equations are characterized statistically through the probability distribution function of the local Lyapunov exponent.

A Second-Order Closure Model for the Effect of Averaging Time on Turbulent Plume Dispersion

A practical model for the effect of averaging time on the turbulent dispersion of a continuous plume is presented. The model is based on a second-order turbulence closure scheme, but is applied to the integrated spatial moments of the plume to provide a Gaussian spread prediction. Velocity fluctuation variances are used directly by the closure model to predict the dispersion, and are partitioned into meandering and diffusive scales based on the instantaneous spread of the plume. Finite time averaging is represented by a simple estimate of the turbulent energy spectrum. The model is compared with short-duration atmospheric measurements for dispersing clouds.

The potential long‐range predictability of temperature over New Zealand

Estimates of climate noise are made from temperature records of 20 New Zealand stations. The climate noise is variability of finite time averages, which is due to day-to-day fluctuations in weather, and it is unpredictable at long range (exceeding deterministic predictability limits of about 2 weeks). This unpredictable part of seasonal or monthly averages is compared with the actual variance of these averages. The amount by which the actual variance exceeds the noise variance is taken to be a measure of the potential long‒range predictability. We find that about 50 per cent of the variance of seasonally averaged temperatures is potentially predictable except for winter (June–July–August), when it is less. © 1997 by the Royal Meteorological Society. Int. J. Climatol. 17: 483–495, 1997.

The potential long-range predictability of temperature over New Zealand

Estimates of climate noise are made from temperature records of 20 New Zealand stations. The climate noise is variability of finite time averages, which is due to day-to-day fluctuations in weather, and it is unpredictable at long range (exceeding deterministic predictability limits of about 2 weeks). This unpredictable part of seasonal or monthly averages is compared with the actual variance of these averages. The amount by which the actual variance exceeds the noise variance is taken to be a measure of the potential long-range predictability. We find that about 50 per cent of the variance of seasonally averaged temperatures is potentially predictable except for winter (June-July-August), when it is less.