Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “Hamiltonian systems”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 55 records · Page 3
All sourcesNASA NTRSDOE OSTI
NASA NTRS2006

Exploring the Connection Between Sampling Problems in Bayesian Inference and Statistical Mechanics

The Bayesian and statistical mechanical communities often share the same objective in their work - estimating and integrating probability distribution functions (pdfs) describing stochastic systems, models or processes. Frequently, these pdfs are complex functions of random variables exhibiting multiple, well separated local minima. Conventional strategies for sampling such pdfs are inefficient, sometimes leading to an apparent non-ergodic behavior. Several recently developed techniques for handling this problem have been successfully applied in statistical mechanics. In the multicanonical and Wang-Landau Monte Carlo (MC) methods, the correct pdfs are recovered from uniform sampling of the parameter space by iteratively establishing proper weighting factors connecting these distributions. Trivial generalizations allow for sampling from any chosen pdf. The closely related transition matrix method relies on estimating transition probabilities between different states. All these methods proved to generate estimates of pdfs with high statistical accuracy. In another MC technique, parallel tempering, several random walks, each corresponding to a different value of a parameter (e.g. "temperature"), are generated and occasionally exchanged using the Metropolis criterion. This method can be considered as a statistically correct version of simulated annealing. An alternative approach is to represent the set of independent variables as a Hamiltonian system. Considerab!e progress has been made in understanding how to ensure that the system obeys the equipartition theorem or, equivalently, that coupling between the variables is correctly described. Then a host of techniques developed for dynamical systems can be used. Among them, probably the most powerful is the Adaptive Biasing Force method, in which thermodynamic integration and biased sampling are combined to yield very efficient estimates of pdfs. The third class of methods deals with transitions between states described by rate constants. These problems are isomorphic with chemical kinetics problems. Recently, several efficient techniques for this purpose have been developed based on the approach originally proposed by Gillespie. Although the utility of the techniques mentioned above for Bayesian problems has not been determined, further research along these lines is warranted

Pohorille, Andrew↗
DOE OSTI2024

Chaotic LLM billiards

Abstract We study null geodesics of the ten-dimensional LLM geometries. In particular, we show that there are a subset of these null geodesics that are confined to the LLM plane. The effective dynamics of these in-plane geodesics is that of a Hamiltonian system with two degrees of freedom (a phase space of dimension 4). We show that these are chaotic. In the two-coloring of the LLM plane, if they start in the empty region, they cannot penetrate the filled region and viceversa. The dynamical problem is therefore very similar to that of a billiards problem with fixed obstacles. We study to what extent LLM geometries with many droplets may be treated as an incipient black hole and draw analogies with the fuzzball proposal. We argue that for in-plane null geodesics deep in the interior of a region with a lot of droplets, in order to exit towards theAdSboundary they will need to undergo a process that resembles diffusion. This mechanism can account for signals getting lost in the putative black hole for a very long time.

Physics↗
DOE OSTI2024

Adaptive Variational Quantum Computing Approaches for Green’s Functions and Nonlinear Susceptibilities

Here, we present and benchmark quantum computing approaches for calculating real-time single-particle Green’s functions and nonlinear susceptibilities of Hamiltonian systems. The approaches leverage adaptive variational quantum algorithms for state preparation and propagation. Using automatically generated compact circuits, the dynamical evolution is performed over sufficiently long times to achieve adequate frequency resolution of the response functions. We showcase accurate Green’s function calculations using a statevector simulator on classical hardware for Fermi-Hubbard chains of 4 and 6 sites, with maximal ansatz circuit depths of 65 and 424 layers, respectively, and for the molecule LiH with a maximal ansatz circuit depth of 81 layers. Additionally, we consider an antiferromagnetic quantum spin-1 model that incorporates the Dzyaloshinskii-Moriya interaction to illustrate calculations of the third-order nonlinear susceptibilities, which can be measured in two-dimensional coherent spectroscopy experiments. These results demonstrate that real-time approaches using adaptive parametrized circuits to evaluate linear and nonlinear response functions can be feasible with near-term quantum processors.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1973

Statistical mechanics of negative temperature states

The dynamics of two-dimensional interacting line vortices is identical to that of the two-dimensional electrostatic guiding center plasma. Both are Hamiltonian systems and are therefore susceptible to statistical mechanical treatments. The predictions of the microcanonical ensemble are explored for this system. Interest focuses primarily on the regime of total positive interaction energy, which should be above the Onsager negative temperature threshold. Calculations of the probability distribution for a component by means of the central limit theorem are carried out in the manner of Khinchin. The probability distribution of a component reduced to the usual Gibbs distribution in the regime of positive temperatures, and is still explicitly calculable for negative temperatures. The negative temperature states are neither quiescent nor spatially uniform. Expressions for the temperature are explicitly provided in terms of the total particle energy and particle number. A BBGKY hierarchy can be derived for both temperature regimes. Numerical simulations involving solutions of the equations of motion of 4008 particles are presented.

Montgomery, D. C.↗
NASA NTRS1984

On the topological stability of magnetostatic equilibria

The topological stability of MHD equilibria is investigated by exploring the formal analogy, in the ideal MHD limit, between the topology of magnetic lines of force in coordinate space and the topology of integral surfaces of one- and two-dimensional Hamiltonian systems in phase space. It is demonstrated that in an astrophysical setting, symmetric magnetostatic equilibria satisfying the ideal MHD equations are exceptional. The principal result of the study is that previous infinitesimal perturbation theory calculations can be generalized to include finite-amplitude and symmetry-breaking effects. The effect of the ergodicity of perturbed symmetric equilibria on heat dispersal in magnetically dominated plasmas is discussed.

Tsinganos, K. C.↗
NASA NTRS1985

Meteorites may follow a chaotic route to earth

A numerical integration was employed to show that the 3/1 Kirkwood gap region between Jupiter and the sun can serve as the source for asteroidal debris that strikes the earth as chondrites. A previous study by Wetherill (1968) considered planetary close encounters, secular resonances, collisional destruction in space and subsequent atmospheric abrasion of the chondrites. The factors would have resulted in chaotic trajectories for asteroids in the asteroid belt, i.e., eccentricities large enough to make the objects earth-crossing. A sampling of trajectories which could reach large eccentricities is integrated in terms of deterministic solutions of Hamiltonian systems of equations which behave erratically and are dependent on the initial conditions. The examinations successfully predicted a trajectory with a peak eccentricity of 0.652, a time interval of 340,000 yr, and a semimajor axis between 2.4-2.53 AU that was earth-crossing.

Wisdom, J.↗
NASA NTRS1985

Pattern formation by particles settling in viscous flows

Results are reported of several simulations of the motions of small objects in two-dimensional cellular flows performed to study the evolution of the distributions of particle flows over time in processes such as sedimentation. Stream functions are defined in the form of a Hamiltonian system for the particle motions, with inertial and particle acceleration being neglected. Conditions are found for which the particles will become trapped in an oscillatory motion or the particle orbits attain an oscillary motion. Various simulation images are provided to illustrate the usefulness of the technique for modeling fractal swarms.

Smith, L. A.↗
NASA NTRS1990

Fractal boundaries in magnetotail particle dynamics

It has been recently established that particle dynamics in the magnetotail geometry can be described as a nonintegrable Hamiltonian system with well-defined entry and exit regions through which stochastic orbits can enter and exit the system after repeatedly crossing the equatorial plane. It is shown that the phase space regions occupied by orbits of different numbers of equatorial crossings or different exit modes are separated by fractal boundaries. The fractal boundaries in an entry region for stochastic orbits are examined and the capacity dimension is determined.

Chen, J.↗
NASA NTRS1991

A family of asymptotically stable control laws for flexible robots based on a passivity approach

A general family of asymptotically stabilizing control laws is introduced for a class of nonlinear Hamiltonian systems. The inherent passivity property of this class of systems and the Passivity Theorem are used to show the closed-loop input/output stability which is then related to the internal state space stability through the stabilizability and detectability condition. Applications of these results include fully actuated robots, flexible joint robots, and robots with link flexibility.

Lanari, Leonardo↗
NASA NTRS1993

Nonlinear modes of the tensor Dirac equation and CPT violation

Recently, it has been shown that Dirac's bispinor equation can be expressed, in an equivalent tensor form, as a constrained Yang-Mills equation in the limit of an infinitely large coupling constant. It was also shown that the free tensor Dirac equation is a completely integrable Hamiltonian system with Lie algebra type Poisson brackets, from which Fermi quantization can be derived directly without using bispinors. The Yang-Mills equation for a finite coupling constant is investigated. It is shown that the nonlinear Yang-Mills equation has exact plane wave solutions in one-to-one correspondence with the plane wave solutions of Dirac's bispinor equation. The theory of nonlinear dispersive waves is applied to establish the existence of wave packets. The CPT violation of these nonlinear wave packets, which could lead to new observable effects consistent with current experimental bounds, is investigated.

Reifler, Frank J.↗
NASA NTRS1994

Modal decomposition of Hamiltonian variational equations

Over any finite arc of trajectory, the variational equations of a Hamiltonian system can be separated into 'normal' modes. This transformation is canonical, and the Lyapunov exponents over the trajectory arc occur as positive/negative pairs for conjugate modes, while the modal vectors remain unit vectors. This decomposition effectively solves the variational equations for any canonical, linear-dependent system. As an example, we study the Voyager I trajectory. In an interplanetary flyby, some of the modal variables increase by very large multiplicative factors, but this means that their conjugate modal variables decrease by those same very large multiplicative vectors. Maneuver strategies for this case are explored, and the minimum delta upsilon maneuver is found.

Wiesel, William E.↗
NASA NTRS2012

Tuning the Sensitivity of an Optical Cavity with Slow and Fast Light

We have measured mode pushing by the dispersion of a rubidium vapor in a Fabry-Perot cavity and have shown that the scale factor and sensitivity of a passive cavity can be strongly enhanced by the presence of such an anomalous dispersion medium. The enhancement is the result of the atom-cavity coupling, which provides a positive feedback to the cavity response. The cavity sensitivity can also be controlled and tuned through a pole by a second, optical pumping, beam applied transverse to the cavity. Alternatively, the sensitivity can be controlled by the introduction of a second counter-propagating input beam that interferes with the first beam, coherently increasing the cavity absorptance. We show that the pole in the sensitivity occurs when the sum of the effective group index and an additional cavity delay factor that accounts for mode reshaping goes to zero, and is an example of an exceptional point, commonly associated with coupled non-Hermitian Hamiltonian systems. Additionally we show that a normal dispersion feature can decrease the cavity scale factor and can be generated through velocity selective optical pumping

Smith, David D.↗
DOE OSTI2026

Operationally induced preferred basis in unitary quantum mechanics

The preferred-basis problem and the definite-outcome aspect of the measurement problem persist even if the detector is modeled unitarily, because experimental data are necessarily represented in a Boolean event algebra of mutually exclusive records whereas the theoretical description is naturally formulated in a noncommutative operator algebra with continuous unitary symmetry. This change of mathematical type constitutes the core of the 'cut': a structurally necessary interface from group-based kinematics to set-based counting. In the presented view the basis relevant for recorded outcomes is not determined by the system Hamiltonian alone; it is induced by the measurement mapping, i.e., by the detector channel together with the coarse-grained readout that defines an instrument. The probabilistic mapping is anchored in symmetry and measure theory: by Gleason-type uniqueness (Gleason for projections in $d>2$ and Busch's extension for Positive Operator-Valued Measures (POVMs) including $d=2$), the trace rule is the unique probability measure consistent with additivity over exclusive events and basis-independence of the unitary sector. A compact qubit--pointer model yields an induced unsharp POVM $E_\pm=\tfrac12(\id\pm η\,σ_z)$ with $η$ fixed by pointer resolution, displaying explicitly how the detector induces the relevant basis. Finally, nested-observer paradoxes are tightened into a non-composability lemma: joint assignment of outcome propositions is obstructed unless a joint instrument exists. This relocates the origin of randomness to the stochasticity of the transition rules.

Pronskikh, Vitaly [Fermilab] (ORCID:00000002518174↗
NASA NTRS1992

Non-Hamiltonian perturbations of integrable systems and resonance trapping

This paper studies general, non-Hamiltonian perturbations of integrable systems with two degrees of freedom and derives conditions for temporary and permanent resonance trapping. The analysis involves a noncanonical transformation of variables near the resonant manifold and averaging with respect to the fast phase to investigate oscillatory behavior on the intermediate timescale. The resulting reduced system is Hamiltonian to leading order and permits, after averaging on the intermediate, or libration, timescale, a canonical transformation to action-angle variables in the oscillation zone. The final system so obtained reveals the possible existence of two- and three-dimensional invariant tori in the vicinity of the resonant manifold. An explicit divergence condition for general perturbations to be dissipative on the slow timescale follows from the analysis. An application of this approach to the problem of resonant trapping and escape is outlined for the restricted problem of three bodies subject to dissipative perturbations with a radial symmetry.

Ghil, M.↗
DOE OSTI2025

Feynman path integrals for discrete-variable systems: Walks on Hamiltonian graphs

We propose a natural, parameter-free, discrete-variable formulation of Feynman path integrals. We show that for discrete-variable quantum systems, Feynman path integrals take the form of walks on the graph whose weighted adjacency matrix is the Hamiltonian. By working out expressions for the partition function and transition amplitudes of discretized versions of continuous-variable quantum systems, and then taking the continuum limit, we explicitly recover Feynman's continuous-variable path integrals. We also discuss the implications of our result.

Feynman diagrams↗
DOE OSTI2026

Learning thermodynamic master equations for open quantum systems

The characterization of Hamiltonians and other components of open quantum dynamical systems plays a crucial role in quantum computing and other applications. Scientific machine learning techniques have been applied to this problem in a variety of ways, including by modeling with deep neural networks. However, the majority of mathematical models describing open quantum systems are linear, and the natural nonlinearities in learnable models have not been incorporated using physical principles. We present a data-driven model for open quantum systems that includes learnable, thermodynamically consistent terms. The trained model is interpretable, as it directly estimates the system Hamiltonian and linear components of coupling to the environment. We validate the model on synthetic two and three-level data, as well as experimental two-level data collected from a quantum device at Lawrence Livermore National Laboratory.

Mathematics and Computing↗