Moyal Quantum Mechanics: The Semiclassical Heisenberg Dynamics

The Moyal--Weyl description of quantum mechanics provides a comprehensive phase space representation of dynamics. The Weyl symbol image of the Heisenberg picture evolution operator is regular in $\hbar$. Its semiclassical expansion `coefficients,' acting on symbols that represent observables, are simple, globally defined differential operators constructed in terms of the classical flow. Two methods of constructing this expansion are discussed. The first introduces a cluster-graph expansion for the symbol of an exponentiated operator, which extends Groenewold's formula for the Weyl product of symbols. This Poisson bracket based cluster expansion determines the Jacobi equations for the semiclassical expansion of `quantum trajectories.' Their Green function solutions construct the regular $\hbar\downarrow0$ asymptotic series for the Heisenberg--Weyl evolution map. The second method directly substitutes such a series into the Moyal equation of motion and determines the $\hbar$ coefficients recursively. The Heisenberg--Weyl description of evolution involves no essential singularity in $\hbar$, no Hamilton--Jacobi equation to solve for the action, and no multiple trajectories, caustics or Maslov indices.Comment: 50, MANIT-94-0

Variational Method for the Simulation of Systems with Diverse Frequencies

Mechanical Engineerin

Lagrangian description of warm plasmas

Efforts are described to extend the averaged Lagrangian method of describing small signal wave propagation and nonlinear wave interaction, developed by earlier workers for cold plasmas, to the more general conditions of warm collisionless plasmas, and to demonstrate particularly the effectiveness of the method in analyzing wave-wave interactions. The theory is developed for both the microscopic description and the hydrodynamic approximation to plasma behavior. First, a microscopic Lagrangian is formulated rigorously, and expanded in terms of perturbations about equilibrium. Two methods are then described for deriving a hydrodynamic Lagrangian. In the first of these, the Lagrangian is obtained by velocity integration of the exact microscopic Lagrangian. In the second, the expanded hydrodynamic Lagrangian is obtained directly from the expanded microscopic Lagrangian. As applications of the microscopic Lagrangian, the small-signal dispersion relations and the coupled mode equations are derived for all possible waves in a warm infinite, weakly inhomogeneous magnetoplasma, and their interactions are examined

Nonlinear control using linearizing transformations

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mechanical Engineering, 1991.Includes bibliographical references (p. 129-133).by Nazareth Sarkis Bedrossian.Ph.D

Application of the generalized Melnikov method to weakly damped parametrically excited cross waves with surface tension

The Wiggins-Holmes extension of the generalized Melnikov method (GMM) is applied to weakly damped parametrically excited cross waves with surface tension in a long rectangular wave channel in order to determine if these cross waves are chaotic. The Lagrangian density function for surface waves with surface tension is simplified by transforming the volume integrals to surface integrals and by subtracting the zero variation integrals. The Lagrangian is written in terms of the three generalized coordinates (or, equivalently the three degrees of freedom) that are the time-dependent components of the velocity potential. A generalized dissipation function is assumed to be proportional to the Stokes material derivative of the free surface. The generalized momenta are calculated from the Lagrangian and the Hamiltonian is determined from a Legendre transformation of the Lagrangian. The first order ordinary differential equations derived from the Hamiltonian are usually suitable for the application of the GMM. However, the cross wave equations of motion must be transformed in order to obtain a suspended system for the application of the GMM. Only three canonical transformations that preserve the dynamics of the cross wave equations of motion are made because of an extension of the Herglotz algorithm to nonautonomous systems. This extension includes two distinct types of the generalized Herglotz algorithm (GHA). The system of nonlinear nonautonomous evolution equations determined from Hamilton's equations of motion of the second kind are averaged in order to obtain an autonomous system. The unperturbed system is analyzed to determine hyperbolic saddle points that are connected by heteroclinic orbits The perturbed Hamiltonian system that includes surface tension satisfies the KAM nondegeneracy requirements; and the Melnikov integral is calculated to demonstrate that the motion is chaotic. For the perturbed dissipative system with surface tension, the Melnikov integral is identically zero implying that a higher dimensional GMM is necessary in order to demonstrate by the GMM that the motion is chaotic. However, numerical calculations of the largest Liapunov characteristic exponent demonstrate that the perturbed dissipative system with surface tension is also chaotic. A chaos diagram is computed in order to search for possible regions of the damping parameter and the Floquet parametric forcing parameter where chaotic motions may exist

Simulation of Ion Migration with Particle Dynamics and the Heat-Poisson-Nernst-Planck System

In this thesis we study the numerical simulation of ion migration and its coupled thermal effects. Many of the existing mathematical models in this area of research implicitly rely on thermal equilibrium conditions, despite the fact that the physical processes are almost exclusively driven by external influences, which move the ensemble away from equilibrium. For the simulation to be self-sufficient and independent from experimental data for novel materials or structures, we adopt a multiscale approach. On the microscale we regard the dynamics of individual atoms and molecules using meshless particle dynamics methods in the form of non-equilibrium Molecular Dynamics. On the macroscale the ions are no longer considered individually, but as concentration functions, which are driven by an electro-chemical field. The resulting system of partial differential equations is known as the Poisson-Nernst-Planck equation system. The basis of a Molecular Dynamics simulation is formed by the Hamiltonian function, from which conservation properties and the equations of motion for the particles are derived. For the first time we make use of the duality of work performed on a particle and its energy state to derive a formulation of the external energy, which allows for the inclusion of explicit external forces in the Hamiltonian function. The new approach is explicitly designed to also handle periodic boundary conditions and we further demonstrate that it can be combined with other variants of the Hamiltonian, such as those modeling thermostats and barostats. This approach allows for the exact computation of energy exchanged between the ensemble and its exterior, enabling us to compute the heat generated by the external forces on the atomistic scale, permitting the upscaling of a temperature source term to the macroscopic equations. For the measurement of the transferred heat we provide an a priori error estimate based on the transport properties. Measuring the transferred energy also allows for the detection of steady states in conjunction with other external effects such as thermostats. On the macroscale we extend the Poisson-Nernst-Planck equation system by the heat equation, a constellation not present in the literature so far. We analyze the nature of the coupling between the different types of partial differential equations and consequently present a taylored discretization scheme based on the Finite Element method. For the first time we present a numerical solver for the extended Heat-Poisson-Nernst-Planck system with an arbitrary number of concentration functions and dynamic transport coefficients. Our implementation of this system allows for a variety of boundary conditions for all solution functions and the use a separate domain (and finite element space) for the evolution of the temperature. We demonstrate the capabilities of the methods on both scales on a series of numerical experiments. On the microscale we confirm the energy transfer and conservation as well as the consistency with thermostat applications. On the macroscale we determine the convergence rates for uniform, graded and adaptively refined grids. Final experiments include a well matching comparison with experimental results from an industrial application, sensitivity analysis of simulation parameters based on uncertainty quantification methods and a showcase for the solver capabilities on complex geometries.In dieser Arbeit untersuchen wir die numerische Simulation von Ionenmigration und von den daran gekoppelten thermischen Effekten. Viele der bereits existierenden mathematischen Modelle in diesem Forschungsgebiet basieren auf thermischen Gleichgewichtsannahmen, obwohl die zu Grunde liegenden physikalischen Prozesse nahezu vollständig von externen Einflüssen gelenkt werden, die die Prozesszustände vom Equilibrium entfernen. Damit die Simulation selbständig und für neuartige Materialien möglichst unabhängig von experimentellen Daten ist, verwenden wir einen Multiskalenansatz. Auf der Mikroskala betrachten wir das Verhalten individueller Atome und Moleküle mittels gitterlosen Partikelmethoden in Form von Nicht-Equilibriums Moleküldynamik. Auf der Makroskala werden die Ionen nicht mehr individuell, sondern als Konzentrationsfuktionen berücksichtigt, die wiederum von einem elektrochemischen Feld beeinflusst werden. Das resultierende System partieller Differentialgleichungen ist als das Poisson-Nernst-Planck Gleichungssystem bekannt. Der Ausgangspunkt der Moleküldynamik wird von der Hamilton-Funktion gebildet, von der die Erhaltungsgrößen des Systems und die Bewegungsgleichungen der Partikel hergeleitet werden. In dieser Arbeit machen wir uns zum ersten Mal die Dualität des Energiezustandes eines Partikels und der an ihm verrichteten Arbeit zunutze, um eine Formulierung der externen Energie herzuleiten, die es erlaubt, den Einfluss externer Kräfte in die Hamilton-Funktion einzubinden. Dieser neue Ansatz ist explizit so angelegt, dass er mit periodischen Randbedingungen kompatibel ist, Weiterhin demonstrieren wir, dass er sich mit anderen Varianten der Hamilton-Funktion kombinieren lässt, wie solchen, die Thermostate und Barostate modellieren. Der Ansatz erlaubt die exakte Berechnung der Energie, die zwischen dem lokalen Ensemble und seiner Umgebung ausgetauscht wird, so dass wir auf der atomaren Ebene die Wärme, die durch externe Kräfte generiert wird, berechnen und durch Upscaling Methoden auf die Makroskala übertragen können. Für die Messung der transferierten Wärme stellen wir eine a-priori Fehlerschätzung vor, die diese Größe mit anderen Transportkoeffizienten verknüpft. Die Messung der ausgetauschten Energie erlaubt des Weiteren auch stationäre Zustände festzustellen, die sich im Zusammenspiel mit anderen externen Effekten wie Thermostaten ausbilden. Auf der Makroskala erweitern wir das Poisson-Nernst-Planck Gleichungssystem um die Wärmeleitungsgleichung zu einer Konstellation, die in dieser Form noch nicht in der Literatur auftaucht. Wir analysieren die Eigenschaften der Kopplungen, die zwischen den verschiedenen Typen von partiellen Differentialgleichungen herrschen und präsentieren ein entsprechend angepasstes Diskretisierungsschema auf Basis finiter Elemente. Als Neuheit präsentieren wir einen numerischen Löser für das gekoppelte Wärme-Poisson-Nernst-Planck Gleichungssystem mit einer beliebigen Anzahl von Konzentrationsfunktionen und dynamischen Transportkoeffizienten. Unsere Implementierung des Systems erlaubt die Nutzung von variierenden Randbedingungen für jede der Lösungsfunktionen und zusätzlich die Verwendung von separaten Simulationsgeometrien für die zeitliche Änderung der Temperatur. Wir demonstrieren die Fähigkeiten der Methoden auf beiden Skalen durch eine Serie von numerischen Experimenten. Auf der Mikroskala finden wir den Transfer und die Erhaltung der Energiewerte ebenso bestätigt wie die Konsistenz der Methode mit Thermostatvarianten. Auf der Makroskala bestimmen wir die Konvergenzraten für uniforme, gradierte und adaptiv verfeinerte Gitter. Zu guter Letzt präsentieren wir einen erfolgreichen Vergleich mit experimentellen Daten im Rahmen einer Industrieanwendung, Sensitivitätsanalyse der Simulationsparameter mit Methoden aus dem Gebiet der sogenannten Uncertainty Quantification und einen Demonstrator für die Fähigkeiten des Lösers auf komplexen Geometrien

Proceedings of the 3rd Annual Conference on Aerospace Computational Control, volume 1

Conference topics included definition of tool requirements, advanced multibody component representation descriptions, model reduction, parallel computation, real time simulation, control design and analysis software, user interface issues, testing and verification, and applications to spacecraft, robotics, and aircraft