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DOE OSTI2023

Simultaneous Measurements of Noncommuting Observables: Positive Transformations and Instrumental Lie Groups

We formulate a general program for describing and analyzing continuous, differential weak, simultaneous measurements of noncommuting observables, which focuses on describing the measuring instrument autonomously, without states. The Kraus operators of such measuring processes are time-ordered products of fundamental differential positive transformations, which generate nonunitary transformation groups that we call instrumental Lie groups. The temporal evolution of the instrument is equivalent to the diffusion of a Kraus-operator distribution function, defined relative to the invariant measure of the instrumental Lie group. This diffusion can be analyzed using Wiener path integration, stochastic differential equations, or a Fokker-Planck-Kolmogorov equation. This way of considering instrument evolution we call the Instrument Manifold Program. We relate the Instrument Manifold Program to state-based stochastic master equations. We then explain how the Instrument Manifold Program can be used to describe instrument evolution in terms of a universal cover that we call the universal instrumental Lie group, which is independent not just of states, but also of Hilbert space. The universal instrument is generically infinite dimensional, in which case the instrument’s evolution is chaotic. Special simultaneous measurements have a finite-dimensional universal instrument, in which case the instrument is considered principal, and it can be analyzed within the differential geometry of the universal instrumental Lie group. Principal instruments belong at the foundation of quantum mechanics. We consider the three most fundamental examples: measurement of a single observable, position and momentum, and the three components of angular momentum. As these measurements are performed continuously, they converge to strong simultaneous measurements. For a single observable, this results in the standard decay of coherence between inequivalent irreducible representations. For the latter two cases, it leads to a collapse within each irreducible representation onto the classical or spherical phase space, with the phase space located at the boundary of these instrumental Lie groups.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

NASA NTRS1992

Turbulence and deterministic chaos

Deissler, Robert G.↗

NASA NTRS1994

Generation of electrostatic waves as a consequence of chaotic ion acceleration in the neutral sheet

The stability of ion distributions with velocity space holes caused by chaotic ion acceleration in the vicinity of the field reversal region in the Earth's magnetotail is investigated. It is shown that such distributions are unstable to electrostatic waves with wave vector directed along the Sun-Earth line. The solution of the quasi-linear equation shows that the saturation of the instability occurs due to velocity space trapping of the low-energy background ions and results in a flattening of the reduced ion distribution function in the region of the original velocity space hole. The energy level of the excited waves in the vicinity of the Earth's midplane is estimated as a function of the initial size of the empty velocity space region.

Burinskaya, T.↗

NASA NTRS1997

Nonguiding Center Motion and Substorm Effects in the Magnetotail

Thick and thin models of the middle magnetotail were developed using a consistent orbit tracing technique. It was found that currents carried near the equator by groups of ions with anisotropic distribution functions are not well approximated by the guiding center expressions. The guiding center equations fail primarily because the calculated pressure tensor is not magnetic field aligned. The pressure tensor becomes field aligned as one moves away from the equator, but here there is a small region in which the guiding center equations remain inadequate because the two perpendicular components of the pressure tensor are unequal. The significance of nonguiding center motion to substorm processes then was examined. One mechanism that may disrupt a thin cross-tail current sheet involves field changes that cause ions to begin following chaotic orbits. The lowest-altitude chaotic region, characterized by an adiabaticity parameter kappa approx. equal to 0.8, is especially important. The average cross-tail particle drift is slow, and we were unable to generate a thin current sheet using such ions. Therefore, any process that tends to create a thin current sheet in a region with kappa approaching 0.8 may cause the cross-tail current to get so low that it becomes insufficient to support the lobes. A different limit may be important in resonant orbit regions of a thin current sheet because particles reach a maximum cross-tail drift velocity. If the number of ions per unit length decreases as the tail is stretched, this part of the plasma sheet also may become unable to carry the cross-tail current needed to support the lobes. Thin sheets are needed for both resonant and chaotic orbit mechanisms because the distribution function must be highly structured. A description of current continuity is included to show how field aligned currents can evolve during the transition from a two-dimensional (2-D) to a 3-D configuration.

Kaufmann, Richard L.↗

DOE OSTI2022

Approximating the linear response of physical chaos

Abstract Parametric derivatives of statistics are highly desired quantities in prediction, design optimization and uncertainty quantification. In the presence of chaos, the rigorous computation of these quantities is certainly possible, but mathematically complicated and computationally expensive. Based on Ruelle’s formalism, this paper shows that the sophisticated linear response algorithm can be dramatically simplified in higher-dimensional systems featuring statistical homogeneity in the physical space. We argue that the contribution of the SRB (Sinai–Ruelle–Bowen) measure gradient, which is an integral yet the most cumbersome part of the full algorithm, is negligible if the objective function is appropriately aligned with unstable manifolds. This abstract condition could potentially be satisfied by a vast family of real-world chaotic systems, regardless of the physical meaning and mathematical form of the objective function and perturbed parameter. We demonstrate several numerical examples that support these conclusions and that present the use and performance of a simplified linear response algorithm. In the numerical experiments, we consider physical models described by differential equations, including Lorenz 96 and Kuramoto–Sivashinsky.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

NASA NTRS1980

Nonlinear dynamics; Proceedings of the International Conference, New York, NY, December 17-21, 1979

Papers were presented on turbulence, ergodic and integrable behavior, chaotic maps and flows, chemical and fully developed turbulence, and strange attractors. Specific attention was given to measures describing a turbulent flow, stochastization and collapse of vortex systems, a subharmonic route to turbulent convection, and weakly nonlinear turbulence in a rotating convection layer. The Korteweg-de Vries and Hill equations, plasma transport in three dimensions, a horseshoe in the dynamics of a forced beam, and the explosion of strange attractors exhibited by Duffing's equation were also considered.

Helleman, R. H. G.↗

NASA NTRS1995

Chaotic obliquity and the nature of the Martian climate

Recent calculations of the Martian obliquity suggests that it varies chaotically on timescales longer than about 10(exp 7) years and varies between about 0 and 60 deg. We examine the seasonal water behavior at obliquities between 40 and 60 deg. Up to several tens of centimeters of water may sublime from the polar caps each year, and possibly move to the equator, where it is more stable. The CO2 frost and CO2-H2O clathrate hydrate are stable in thepolar deposits below a few tens of meters depth, so that the polar cap could contain a significant CO2 reservoir. If CO2 is present, it could be left over from the early history of Mars; also, it could be released into the atmosphere during periods of high obliquity, causing occasional periods of more-clement climate.

Jakosky, Bruce M.↗

NASA NTRS1993

Transonic panel flutter

FEM is here used to ascertain the stability and aeroelastic response of thin, 2D panels subjected to Mach 0.8-2.5 flows. In the absence of shocks, it is found that the Euler equations used to represent the unsteady flowfield dynamics predict response behaviors resembling those obtained via potential flow methods. Where shocks do play a significant role in the overall motion of the panel, divergence and limit cycle flutter are observed. In the Mach 1.4-1.5 range, flutter involved the higher modes of the panel, tending toward possible chaotic motion.

Davis, Gary A.↗

NASA NTRS1989

On the Nature of Navier-stokes Turbulence

Several turbulent and nonturbulent solutions of the Navier-Stokes equations are obtained. The unaveraged equations are used numerically in conjunction with tools and concepts from nonlinear dynamics, including time series, phase portraits, Poincare sections, largest Liapunov exponents, power spectra, and strange attractors. Initially neighboring solutions for a low-Reynolds-number fully developed turbulence are compared. The solutions, separate exponentially with time, having a positive Liapunov exponent. Thus the turbulence is characterized as chaotic. In a search for solutions which contrast with the turbulent ones, the Reynolds number is reduced. Several qualitatively different flows are noted. These are, fully chaotic, complex period, weakly chaotic, simple periodic, and fixed-point. Of these, only the fully chaotic flows are classified as turbulent. Those flows have both a positive Liapunov exponent and Poincare sections without pattern. By contrast, the weakly chaotic flows have some pattern in their Poincare sections. The fixed-point and periodic flows are nonturbulent, since turbulence, is both time-dependent and aperiodic. Turbulent solutions are obtained in which energy cascades from large to small-scale motions. In general, the spectral energy transfer takes place between wavenumber bands that are considerably separated. The special transfer can occur either as a result of nonlinear turbulence self-interaction or by interaction of turbulence with mean gradients. Turbulent systems are compared with those studied in kinetic theory. The two types of systems are fundamentally different (continuous and dissipative as opposed to discrete and conservative), but there are similarities. For instance, both are nonlinear and show sensitive dependence on initial conditions. Also, the turbulent and molecular stress tensors are identical if the macroscopic velocities for the turbulent stress are replaced by molecular velocities.

Deissler, Robert G.↗

NASA NTRS2013

Circular-Orbit Maintenance Strategies for Primitive Body Orbiters

For missions to smaller primitive bodies, solar radiation pressure (SRP) is a significant perturbation to Keplerian dynamics. For most orbits, SRP drives large oscillations in orbit eccentricity, which leads to large perturbations from the irregular gravity field at periapsis. Ultimately, chaotic motion results that often escapes or impacts that body. This paper presents an orbit maintenance strategy to keep the orbit eccentricity small, thus avoiding the destabilizing secondary interaction with the gravity field. An estimate of the frequency and magnitude of the required maneuvers as a function of the orbit and body parameters is derived from the analytic perturbation equations.

comet↗

DOE OSTI2023

Temporal Subsampling Diminishes Small Spatial Scales in Recurrent Neural Network Emulators of Geophysical Turbulence

The immense computational cost of traditional numerical weather and climate models has sparked the development of machine learning (ML) based emulators. Because ML methods benefit from long records of training data, it is common to use data sets that are temporally subsampled relative to the time steps required for the numerical integration of differential equations. Here, we investigate how this often overlooked processing step affects the quality of an emulator's predictions. We implement two ML architectures from a class of methods called reservoir computing: (a) a form of Nonlinear Vector Autoregression (NVAR), and (b) an Echo State Network (ESN). Despite their simplicity, it is well documented that these architectures excel at predicting low dimensional chaotic dynamics. We are therefore motivated to test these architectures in an idealized setting of predicting high dimensional geophysical turbulence as represented by Surface Quasi-Geostrophic dynamics. In all cases, subsampling the training data consistently leads to an increased bias at small spatial scales that resembles numerical diffusion. Interestingly, the NVAR architecture becomes unstable when the temporal resolution is increased, indicating that the polynomial based interactions are insufficient at capturing the detailed nonlinearities of the turbulent flow. The ESN architecture is found to be more robust, suggesting a benefit to the more expensive but more general structure. Spectral errors are reduced by including a penalty on the kinetic energy density spectrum during training, although the subsampling related errors persist. Future work is warranted to understand how the temporal resolution of training data affects other ML architectures.

58 GEOSCIENCES↗

NASA NTRS1987

Lyapunov exponents for infinite dimensional dynamical systems

Classically it was held that solutions to deterministic partial differential equations (i.e., ones with smooth coefficients and boundary data) could become random only through one mechanism, namely by the activation of more and more of the infinite number of degrees of freedom that are available to such a system. It is only recently that researchers have come to suspect that many infinite dimensional nonlinear systems may in fact possess finite dimensional chaotic attractors. Lyapunov exponents provide a tool for probing the nature of these attractors. This paper examines how these exponents might be measured for infinite dimensional systems.

Mhuiris, Nessan Mac Giolla↗

NASA NTRS1996

Heat and Momentum Transfer Studies in High Reynolds Number Wavy Films at Normal and Reduced Gravity Conditions

We examined the effect of the gas flow on the liquid film when the gas flows in the countercurrent direction in a vertical pipe at normal gravity conditions. The most dramatic effect of the simultaneous flow of gas and liquid in pipes is the greatly increased transport rates of heat, mass, and momentum. In practical situations this enhancement can be a benefit or it can result in serious operational problems. For example, gas-liquid flow always results in substantially higher pressure drop and this is usually undesirable. However, much higher heat transfer coefficients can be expected and this can obviously be of benefit for purposes of design. Unfortunately, designers know so little of the behavior of such two phase systems and as a result these advantages are not utilized. Due to the complexity of the second order boundary model as well as the fact that the pressure variation across the film is small compared to the imposed gas phase pressure, the countercurrent gas flow affect was studied for the standard boundary layer model. A different stream function that can compensate the shear stress affect was developed and this stream function also can predict periodic solutions. The discretized model equations were transformed to a traveling wave coordinate system. A stability analysis of these sets of equations showed the presence of a Hopf bifurcation for certain values of the traveling wave velocity and the shear stress. The Hopf celerity was increased due to the countercurrent shear. For low flow rate the increases of celerity are more than for the high flow rate, which was also observed in experiments. Numerical integration of a traveling wave simplification of the model also predicts the existence of chaotic large amplitude, nonperiodic waves as observed in the experiments. The film thickness was increased by the shear.

Balakotaiah, V.↗

DOE OSTI2021

Parallel Multigrid in Time and Space for Extreme-Scale Computational Science

The coming massive parallelism of exascale computing presents a pressing challenge for the many DOE simulations of time-dependent partial differential equations, which typically use traditional sequential time stepping methods. Since this traditional approach is inherently serial, it presents a sequential bottleneck when moving to exascale computing, because future performance gains will come through greater concurrency, not faster clock speeds. Thus, the goal of this work is to research parallelism in time, i.e., methods that compute multiple time values simultaneously, not sequentially. The focus will be on hyperbolic and chaotic problems of programmatic interest to DOE, with the goal of enabling scalable simulations of time-dependent hyperbolic and chaotic problems on future architectures.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2021

Using Disorder to Overcome Disorder: A Mechanism for Frequency and Phase Synchronization of Diode Laser Arrays

Noise and disorder are known, in certain circumstances and for certain systems, to improve the level of coherence over that of the noise-free system. Examples include cases in which disorder enhances response to periodic signals, and those where it suppresses chaotic behavior. We report a new type of disorder-enhancing mechanism, observed in a model that describes the dynamics of external cavity-coupled semiconductor laser arrays, where disorder of one type mitigates (and overcomes) the desynchronization effects due to a different disorder source. Here, we demonstrate stabilization of dynamical states due to frequency locking and subsequently frequency locking-induced phase locking. Here, we have reduced the equations to a potential model that illustrates the mechanism behind the misalignment-induced frequency and phase synchronization.

42 ENGINEERING↗

NASA NTRS1990

The fast kinematic magnetic dynamo and the dissipationless limit

The evolution of the magnetic field in models that incorporate chaotic field line stretching, field cancellation, and finite magnetic Reynolds number is examined analytically and numerically. Although the models used here are highly idealized, it is claimed that they display and illustrate typical behavior relevant to fast magnetic dynamic behavior. It is shown, in particular, that consideration of magnetic flux through a finite fixed surface provides a simple and effective way of deducing fast dynamo behavior from the zero resistivity equation. Certain aspects of the fast dynamo problem can thus be reduced to a study of nonlinear dynamic properties of the underlying flow.

Finn, John M.↗

DOE OSTI2022

Discovering causal structure with reproducing-kernel Hilbert space ε -machines

We merge computational mechanics’ definition of causal states (predictively equivalent histories) with reproducing-kernel Hilbert space (RKHS) representation inference. The result is a widely applicable method that infers causal structure directly from observations of a system’s behaviors whether they are over discrete or continuous events or time. A structural representation—a finite- or infinite-state kernel ϵ-machine—is extracted by a reduced-dimension transform that gives an efficient representation of causal states and their topology. In this way, the system dynamics are represented by a stochastic (ordinary or partial) differential equation that acts on causal states. We introduce an algorithm to estimate the associated evolution operator. Paralleling the Fokker–Planck equation, it efficiently evolves causal-state distributions and makes predictions in the original data space via an RKHS functional mapping. We demonstrate these techniques, together with their predictive abilities, on discrete-time, discrete-value infinite Markov-order processes generated by finite-state hidden Markov models with (i) finite or (ii) uncountably infinite causal states and (iii) continuous-time, continuous-value processes generated by thermally driven chaotic flows. The method robustly estimates causal structure in the presence of varying external and measurement noise levels and for very high-dimensional data.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

NASA NTRS1996

Turbulent Fluid Motion 6: Turbulence, Nonlinear Dynamics, and Deterministic Chaos

Deissler, Robert G.↗