Asymptotic Time-scale Research Articles

Chaotic fields out of equilibrium are observable independent

Chaotic dynamics is always characterized by swarms of unstable trajectories, unpredictable individually, and thus generally studied statistically. It is often the case that such phase-space densities relax exponentially fast to a limiting distribution, that rules the long-time average of every observable of interest. Before that asymptotic time scale, the statistics of chaos is generally believed to depend on both the initial conditions and the chosen observable. I show that this is not the case for a widely applicable class of models, that feature a phase-space (‘field’) distribution common to all pushed-forward or integrated observables, while the system is still relaxing towards statistical equilibrium or a stationary state. This universal profile is determined by both leading and first subleading eigenfunctions of the transport operator (Koopman or Perron–Frobenius) that maps phase-space densities forward or backward in time.

Nontrivial anomalous diffusions induced by the harmonic velocity Lévy noise

Moving in the force-free potential and the tilted periodic potential, the diffusion of the under-damped particle subjected to correlated Lévy noise, i.e. the harmonic velocity Lévy noise, is investigated in this paper. We find that the underlying negative time correlation, especially the strong one, distinctly collides with the long-tailed nature of the noise, which results in a multi-diffusive process in the force-free potential. In the tilted periodic potential, strong correlation means that the transport of particle cannot be separated as the typical locked state and running state in coordinate space. Correspondingly, the anomalous diffusion in the tilted periodic potential does not hold anymore, which becomes a normal shape at asymptotic timescale.

Kinetic Monte Carlo modeling of Y2O3 nano-cluster formation in radiation resistant matrices

Abstract As known, Y2O3 nano-clusters considerably increase radiation resistance of reactor construction materials. To model the nano-cluster formation kinetics, we propose the simplest possible mathematical model and perform kinetic Monte Carlo (KMC) simulations. We extended the KMC simulated results to the experimentally relevant times using autoregressive integrated moving average forecasting. Within the model, we have studied prototypical attractive interaction energies and particle concentrations, and compared the simulations with experiments. We have observed the standard Lifshitz-Slyozov-Wagner (LSW) theory, predicting the average cluster radius growth with time, R ‾ ∼ t 1 / p , with p = 3 in the long-time limit, for weak (0.1 eV) mutual particle attraction. However, the respective cluster growth rates in these KMC simulations are overestimated compared to the experiments. The best agreement with experiment is obtained for a medium (0.3 eV) and strong (0.5 eV) attractions, when nano-cluster formation occurs during intermediate asymptotic time scale, where power order p ranges from 5 to 7.6 depending on interaction, without reaching actually the LSW long-time limit. Such a stronger interaction leads also to a more compact {110}–faceted nano-clusters.

Diffusion and Mixing in Globular Clusters

Abstract Collisional relaxation describes the stochastic process with which a self-gravitating system near equilibrium evolves in phase-space due to the fluctuating gravitational field of the system. The characteristic timescale of this process is called the relaxation time. In this paper, we highlight the difference between two measures of the relaxation time in globular clusters: (1) the diffusion time with which the isolating integrals of motion (i.e., energy E and angular momentum magnitude L) of individual stars change stochastically and (2) the asymptotic timescale required for a family of orbits to mix in the cluster. More specifically, the former corresponds to the instantaneous rate of change of a star’s E or L, while the latter corresponds to the timescale for the stars to statistically forget their initial conditions. We show that the diffusion timescales of E and L vary systematically around the commonly used half-mass relaxation time in different regions of the cluster by a factor of ∼10 and ∼100, respectively, for more than 20% of the stars. We define the mixedness of an orbital family at any given time as the correlation coefficient between its E or L probability distribution functions and those of the whole cluster. Using Monte Carlo simulations, we find that mixedness converges asymptotically exponentially with a decay timescale that is ∼10 times the half-mass relaxation time.

Asymptotics, structure, and integration of sound-proof atmospheric flow equations

Relative to the full compressible flow equations, sound-proof models filter acoustic waves while maintaining advection and internal waves. Two well-known sound-proof models, an anelastic model by Bannon and Durran's pseudo-incompressible model, are shown here to be structurally very close to the full compressible flow equations. Essentially, the anelastic model is obtained by suppressing @t in the mass continuity equation and slightly modifying the gravity term, whereas the pseudoincompressible model results from dropping @tp from the pressure equation. For length scales small compared to the density and pressure scale heights, the anelastic model reduces to the Boussinesq approximation, while the pseudo-incompressible model approaches the zero Mach number, variable density flow equations. Thus, for small scales, both models are asymptotically consistent with the full compressible flow equations, yet the pseudo-incompressible model is more general in that it remains valid in the presence of large density variations. For the relatively small density variations found in typical atmosphere-ocean flows, both models are found to yield very similar results, with deviations between models much smaller than deviations obtained when using different numerical schemes for the same model. This in agreement with Smolarkiewicz and Dornbrack (2007). Despite these useful properties, neither model can be derived by a low-Mach number asymptotic expansion for length scales comparable to the pressure scale height, i.e., for the regime they were originally designed for. Derivations of these models via scale analysis ignore an asymptotic time scale separation between advection and internal waves. In fact, only the classical Ogura & Phillips model, which assumes weak stratication of the order of the Mach number squared, can be obtained as a leading-order model from systematic low Mach number asymptotic analysis. Issues of formal asymptotics notwithstanding, the close structural similarity of the anelastic and pseudo-incompressible models to the full compressible flow equations makes them useful limit systems in building computational models for atmospheric flows. In the second part of the paper we propose a second-order finite-volume projection method for the anelastic and pseudo-incompressible models that observes these structural similarities. The method is applied to test problems involving free convection in a neutral atmosphere, the breaking of orographic waves at high altitudes, and the descent of a cold air bubble in the small-scale limit. The scheme is meant to serve as a starting point for the development of a robust compressible atmospheric flow solver in future work.

Symbolic realization of asymptotic time-scale and eigenstructure assignment design method in multivariable control

This paper reports on a symbolic realization of the asymptotic time-scale and eigenstructure assignment (ATEA) state feedback design technique for multivariable control. The resulting state feedback laws are parameterized in a scalar ε. Under these state feedback laws, the closed-loop system possesses a pre-specified time-scale and its eigenstructure approaches a pre-specified one, as the value of the parameter ε approaches zero. By appropriately specifying the time-scale and the eigenstructure, the feedback laws can be obtained to solve various control problems, such as the H 2 and H ∞ suboptimal control, and almost disturbance decoupling problems. We present, in this paper, the software implementation of the ATEA design algorithm using the MATLAB symbolic programming technique. Our m-functions are capable of returning a result, which is explicitly expressed in terms of a symbolic variable epsilon, which represents ε. The controller design for a piezoelectric bimorph actuator is used to illustrate how the symbolic realization works.

Design and Implementation of a Robust Controller for a Free Gyro-Stabilized Mirror System

In this paper, we consider the problem of designing a robust controller for a multivariable servomechanism of a free gyro-stabilized mirror system where there exists cross-coupling between the axes. The nonlinear dynamics of the system are first linearized and reformulated into an H∞ problem. A reduced order output feedback controller is then designed using the asymptotic time-scale and eigenstructure assignment (ATEA) technique. The overall design has been implemented using a personal computer with C++ and tested on the actual gyro-stabilized mirror system. Our simulation and implementation results show that the design is very successful.

Extremum principles for selection in the Saffman–Taylor finger and Taylor–Saffman bubble problems

We consider the Saffman–Taylor problem describing the displacement of one fluid by another having a smaller viscosity, in a porous medium or in a Hele–Shaw configuration, and the Taylor–Saffman problem of a bubble moving in a channel containing moving fluid. Each problem is known to possess a family of traveling wave solutions, the former corresponding to propagating fingers and the latter to propagating bubbles, with each member characterized by its own velocity and each occupying a different fraction of the channel through which it propagates. To select the “correct” member of the family of solutions we propose two related extremum principles. Employing these principles for a symmetric family with zero surface tension (σ=0) selects the solution (finger or bubble, viscous or inviscid) which happens to be the same as that obtained by taking the limit σ→0 of the nonzero isotropic surface tension problem. For other problems, e.g., perturbation by anisotropic surface tension, the fingers selected by the two approaches are not necessarily the same. We claim that the finger selected by our criteria describes what will be observed on an intermediate time scale in which the effect of perturbations is neglected, whereas the finger selected by taking the limit of vanishingly small perturbations describes what will be observed on the asymptotic time scale O(1/σ) for the problem with the perturbation included. The intermediate time scale is much shorter than the asymptotic time scale. For infinitesimal σ the asymptotic time scale may be well beyond any reasonable observational time scale.

An H/sub ∞/ almost disturbance decoupling robust controller design for a piezoelectric bimorph actuator with hysteresis

A robust controller design for a piezoelectric bimorph nonlinear actuator is considered. The nonlinear dynamics of the actuator are first linearized using the stochastic equivalent linearization method and reformulated into a standard almost disturbance decoupling problem. Then a robust controller, which is explicitly parameterized by two tuning parameters, is carried out using a so-called asymptotic time-scale and eigenstructure assignment approach. The parameterized controller can be tuned by adjusting the parameters to achieve disturbance decoupling and other design goals for the problem that we consider. Simulation results of time-domain responses show that the design is very successful in terms of steady-state tracking error and settling time as well as other performances.

Phase Coarsening as a Relaxation Phenomenon

AbstractPhase coarsening, also known as Ostwald ripening, is generally treated as a growth phenomenon in which the average particle size increases during isothermal aging. However, coarsening is a relaxation process driven by a reduction in the excess interfacial energy of a two-phase structure. This viewpoint introduces a single temporal offset parameter that relates the experimental clock to the asymptotic time scale. Analysis of ripening as a relaxation phenomenon permits accurate determination of the kinetics that is less sensitive to particle morphology. Implications of this approach on measurement and interpretation of ripening kinetics will be discussed.

Field-induced quantum barrier crossing: classical chaos and weak localization

Based on a model double-well oscillator driven by a classical field we have carried out a classical and quantum mechanical analysis of field-induced barrier crossing. We have shown that classical chaos significantly enhances the quantum barrier crossing rate as well as the quantum fluctuations. The classical chaos is also manifested through a kind of weak dynamic localization at the barrier top (due to the time reversal symmetry of the Hamiltonian) in the time scale relevant for barrier crossing dynamics, which is much shorter than the asymptotic time scale required to resolve the quasi-energy levels of the system.

Theory of LTR for non-minimum phase systems, recoverable target loops, and recovery in a subspace Part 2. Design

This Part focuses on the design of full order observer based controllers for the recovery of target loop transfer function or sensitivity and complimentary sensitivity functions when the given system is not necessarily left invertible and not necessarily of minimum phase. Four design tasks are considered. The first task concerns with arbitrarily specified target loop transfer functions and develops an observer design which has the capability to shape the inevitable recovery error according to the designer's needs whenever they are feasible. The second task considers observer design for exactly recoverable target loop transfer functions. The third task is similar to the second one in that it makes use of the specific properties of the target loop transfer function, but it considers observer design for asymptotically recoverable target loop transfer functions. The fourth task can be thought of as a generalization of the second and third tasks, and it considers observer design so that the achieved and target sensitivity and complimentary sensitivity functions match each other either exactly or asymptotically over a given subspace of the control space whenever it is possible. For all these tasks, observer design constraints and the available design freedom are reviewed. In view of the available freedom, possible specifications on the time-scale and/or eigenstructure of the observer dynamic matrix are formulated. In the case of first task, the conventional approach of designing observer based controllers by Kalman filter formalism which requires solving algebraic Riccati equations, is shown to have several fundamental limitations. A method of design based on asymptotic time-scale and eigenstructure assignment (ATEA) developed here overcomes these limitations. For the other tasks, no design methods other than the ones developed here are available in the literature. All the developed design methods are implemented in a ‘Matlab’ software package. A bank of examples illustrate that the proposed methods of design are capable of directly exploiting all the available freedom so as to achieve the desired results.