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"nonlinear dynamics"
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Thermalization and its mechanism for generic isolated quantum systems
It is demonstrated that an isolated generic quantum many-body system does relax to a state well described by the standard statistical mechanical prescription. The thermalization happens at the level of individual eigenstates, allowing the computation of thermal averages from knowledge of any eigenstate in the microcanonical energy window.
An understanding of the temporal evolution of isolated many-body quantum systems has long been elusive. Recently, meaningful experimental studies
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,
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of the problem have become possible, stimulating theoretical interest
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,
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,
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,
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,
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. In generic isolated systems, non-equilibrium dynamics is expected
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,
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to result in thermalization: a relaxation to states in which the values of macroscopic quantities are stationary, universal with respect to widely differing initial conditions, and predictable using statistical mechanics. However, it is not obvious what feature of many-body quantum mechanics makes quantum thermalization possible in a sense analogous to that in which dynamical chaos makes classical thermalization possible
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. For example, dynamical chaos itself cannot occur in an isolated quantum system, in which the time evolution is linear and the spectrum is discrete
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. Some recent studies
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,
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even suggest that statistical mechanics may give incorrect predictions for the outcomes of relaxation in such systems. Here we demonstrate that a generic isolated quantum many-body system does relax to a state well described by the standard statistical-mechanical prescription. Moreover, we show that time evolution itself plays a merely auxiliary role in relaxation, and that thermalization instead happens at the level of individual eigenstates, as first proposed by Deutsch
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and Srednicki
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. A striking consequence of this eigenstate-thermalization scenario, confirmed for our system, is that knowledge of a single many-body eigenstate is sufficient to compute thermal averages—any eigenstate in the microcanonical energy window will do, because they all give the same result.
Journal Article
Sparse identification of nonlinear dynamics for model predictive control in the low-data limit
Data-driven discovery of dynamics via machine learning is pushing the frontiers of modelling and control efforts, providing a tremendous opportunity to extend the reach of model predictive control (MPC). However, many leading methods in machine learning, such as neural networks (NN), require large volumes of training data, may not be interpretable, do not easily include known constraints and symmetries, and may not generalize beyond the attractor where models are trained. These factors limit their use for the online identification of a model in the low-data limit, for example following an abrupt change to the system dynamics. In this work, we extend the recent sparse identification of nonlinear dynamics (SINDY) modelling procedure to include the effects of actuation and demonstrate the ability of these models to enhance the performance of MPC, based on limited, noisy data. SINDY models are parsimonious, identifying the fewest terms in the model needed to explain the data, making them interpretable and generalizable. We show that the resulting SINDY-MPC framework has higher performance, requires significantly less data, and is more computationally efficient and robust to noise than NN models, making it viable for online training and execution in response to rapid system changes. SINDY-MPC also shows improved performance over linear data-driven models, although linear models may provide a stopgap until enough data is available for SINDY. SINDY-MPC is demonstrated on a variety of dynamical systems with different challenges, including the chaotic Lorenz system, a simple model for flight control of an F8 aircraft, and an HIV model incorporating drug treatment.
Journal Article
Constrained sparse Galerkin regression
The sparse identification of nonlinear dynamics (SINDy) is a recently proposed data-driven modelling framework that uses sparse regression techniques to identify nonlinear low-order models. With the goal of low-order models of a fluid flow, we combine this approach with dimensionality reduction techniques (e.g. proper orthogonal decomposition) and extend it to enforce physical constraints in the regression, e.g. energy-preserving quadratic nonlinearities. The resulting models, hereafter referred to as Galerkin regression models, incorporate many beneficial aspects of Galerkin projection, but without the need for a high-fidelity solver to project the Navier–Stokes equations. Instead, the most parsimonious nonlinear model is determined that is consistent with observed measurement data and satisfies necessary constraints. Galerkin regression models also readily generalize to include higher-order nonlinear terms that model the effect of truncated modes. The effectiveness of such an approach is demonstrated on two canonical flow configurations: the two-dimensional flow past a circular cylinder and the shear-driven cavity flow. For both cases, the accuracy of the identified models compare favourably against reduced-order models obtained from a standard Galerkin projection procedure. Finally, the entire code base for our constrained sparse Galerkin regression algorithm is freely available online.
Journal Article
Sparse reduced-order modelling: sensor-based dynamics to full-state estimation
We propose a general dynamic reduced-order modelling framework for typical experimental data: time-resolved sensor data and optional non-time-resolved particle image velocimetry (PIV) snapshots. This framework can be decomposed into four building blocks. First, the sensor signals are lifted to a dynamic feature space without false neighbours. Second, we identify a sparse human-interpretable nonlinear dynamical system for the feature state based on the sparse identification of nonlinear dynamics (SINDy). Third, if PIV snapshots are available, a local linear mapping from the feature state to the velocity field is performed to reconstruct the full state of the system. Fourth, a generalized feature-based modal decomposition identifies coherent structures that are most dynamically correlated with the linear and nonlinear interaction terms in the sparse model, adding interpretability. Steps 1 and 2 define a black-box model. Optional steps 3 and 4 lift the black-box dynamics to a grey-box model in terms of the identified coherent structures, if non-time-resolved full-state data are available. This grey-box modelling strategy is successfully applied to the transient and post-transient laminar cylinder wake, and compares favourably with a proper orthogonal decomposition model. We foresee numerous applications of this highly flexible modelling strategy, including estimation, prediction and control. Moreover, the feature space may be based on intrinsic coordinates, which are unaffected by a key challenge of modal expansion: the slow change of low-dimensional coherent structures with changing geometry and varying parameters.
Journal Article
Parameter Space Compression Underlies Emergent Theories and Predictive Models
The microscopically complicated real world exhibits behavior that often yields to simple yet quantitatively accurate descriptions. Predictions are possible despite large uncertainties in microscopic parameters, both in physics and in multiparameter models in other areas of science. We connect the two by analyzing parameter sensitivities in a prototypical continuum theory (diffusion) and at a self-similar critical point (the Ising model). We trace the emergence of an effective theory for long-scale observables to a compression of the parameter space quantified by the eigenvalues of the Fisher Information Matrix. A similar compression appears ubiquitously in models taken from diverse areas of science, suggesting that the parameter space structure underlying effective continuum and universal theories in physics also permits predictive modeling more generally.
Journal Article
Chaos and threshold for irreversibility in sheared suspensions
No turning back
According to the laws of fluid motion, when a simple fluid or suspension of particles is slowly stirred then unstirred — imagine a spoon in a jar of honey — all parts of the system should miraculously return to their starting points. This is a consequence of the time-reversible equations of motion, at least for two-dimensional flows. But in more complex flows, such as those in three-dimensional or rigorously stirred systems, this delicate effect is destroyed. An investigation of a slowly sheared suspension of solid particles now reveals the microscopic processes behind this transition to irreversible behaviour. Beyond a concentration-dependent threshold strain, irreversibility sets in as a result of chaotic collisions between the particles.
Systems governed by time reversible equations of motion often give rise to irreversible behaviour
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,
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,
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. The transition from reversible to irreversible behaviour is fundamental to statistical physics, but has not been observed experimentally in many-body systems. The flow of a newtonian fluid at low Reynolds number can be reversible: for example, if the fluid between concentric cylinders is sheared by boundary motion that is subsequently reversed, then all fluid elements return to their starting positions
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. Similarly, slowly sheared suspensions of solid particles, which occur widely in nature and science
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, are governed by time reversible equations of motion. Here we report an experiment showing precisely how time reversibility
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fails for slowly sheared suspensions. We find that there is a concentration dependent threshold for the deformation or strain beyond which particles do not return to their starting configurations after one or more cycles. Instead, their displacements follow the statistics of an anisotropic random walk
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. By comparing the experimental results with numerical simulations, we demonstrate that the threshold strain is associated with a pronounced growth in the Lyapunov exponent (a measure of the strength of chaotic particle interactions). The comparison illuminates the connections between chaos, reversibility and predictability.
Journal Article
Refined composite variable-step multiscale multimapping dispersion entropy: a nonlinear dynamical index
Nonlinear dynamical index can measure the complexity for a single time scale of the series, and when combined with coarse-grained methods, multiple time scales can be obtained to extract more information. In this study, a new coarse-grained method called refined composite variable-step multiscale (RCVM) is proposed, which obtains more subseries by setting different initial points and step lengths and thus extracts more potential information; moreover, in order to get a nonlinear dynamical index value with stronger stability, this study proposes the multimapping dispersion entropy (MDE) by averaging multiple classes of effective mapping approaches on the basis of dispersion entropy; by combining MDE and RCVM processing, RCVM-MDE is proposed to be used as a new nonlinear dynamical index, which can reflect the complexity of the series at multiple scales. The results of the four classes of chaotic simulated signals show that RCVM-MDE is not only able to detect the series nonlinear dynamic changes, but also has a very high stability; the results of three classes of real-world signals demonstrate the differentiability of RCVM-MDE compared to other commonly used entropies, as well as the best classification effect.
Journal Article
Nonlinear dynamic response and bifurcation of variable thickness sandwich conical shell with internal resonance
This paper explores the nonlinear dynamic responses and bifurcations of the truncated sandwich simply supported porous conical shell with varying thickness under 1:1 internal resonance. Two skins with carbon fiber and a core with porous aluminum foam, which has an exponentially variable thickness along the generator and various porosity distribution types along the core thickness, make up the sandwich shell structure with varying stiffness. The porous shell structure is affected by a combination of the in-plane load, transverse excitation, thermal stress and aerodynamic force, which is formulated by employing first-order piston theory with a modified term for curvature. By way of FSDT, von-Karman geometrical formulations, Hamilton’s principle and Galerkin procedure, the nonlinear dynamic formulations in ordinary differential form for the variable stiffness porous sandwich shell structure are identified. The averaged equations in polar and Cartesian coordinate forms for the sandwich structure under the combined circumstance of 1:1 internal resonance, first-order main resonance and 1/2 subharmonic resonance are determined by multiple-scale technique. The frequency-amplitude and force–amplitude characteristic curves, phase portraits, time history and bifurcation diagrams are exhibited by numerical simulation. The impacts of the damping coefficient, detuning parameters, temperature increment, transverse and in-plane excitations on the nonlinear dynamics and bifurcation behaviors of variable thickness sandwich porous conical shell are demonstrated.
Journal Article
Nonlinear dynamic response of sandwich plates with functionally graded auxetic 3D lattice core
This paper presents full-scale modeling and nonlinear dynamic analysis of sandwich plates with auxetic 3D lattice core, which is further designed to possess three functionally graded (FG) configurations through the plate thickness direction for the first time. The effective Poisson’s ratio (EPR) and fundamental frequencies of auxetic 3D lattice metamaterials are analyzed and verified by static and vibration tests using specimens fabricated by 3D printing. Considering the large deflection nonlinearity of sandwich plates and the accompanying changes in effective properties of lattice microstructures, full-scale FE modeling and nonlinear dynamic thermal–mechanical analysis are performed, with material properties assumed to be temperature dependent. Numerical results revealed that the auxetic core can significantly reduce the dynamic deflections, in comparison with its counterpart with positive EPR. Furthermore, FG configurations have distinct effects on the natural frequencies and dynamic deflection–time curves of sandwich plates, along with EPR–deflection curves in the large deflection region.
Journal Article