24 research outputs found

    Collisions in compact star clusters and formation of massive black holes

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    A multiphysics and multiscale software environment for modeling astrophysical systems

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    We present MUSE, a software framework for combining existing computational tools for different astrophysical domains into a single multiphysics, multiscale application. MUSE facilitates the coupling of existing codes written in different languages by providing inter-language tools and by specifying an interface between each module and the framework that represents a balance between generality and computational efficiency. This approach allows scientists to use combinations of codes to solve highly-coupled problems without the need to write new codes for other domains or significantly alter their existing codes. MUSE currently incorporates the domains of stellar dynamics, stellar evolution and stellar hydrodynamics for studying generalized stellar systems. We have now reached a "Noah's Ark" milestone, with (at least) two available numerical solvers for each domain. MUSE can treat multi-scale and multi-physics systems in which the time- and size-scales are well separated, like simulating the evolution of planetary systems, small stellar associations, dense stellar clusters, galaxies and galactic nuclei. In this paper we describe three examples calculated using MUSE: the merger of two galaxies, the merger of two evolving stars, and a hybrid N-body simulation. In addition, we demonstrate an implementation of MUSE on a distributed computer which may also include special-purpose hardware, such as GRAPEs or GPUs, to accelerate computations. The current MUSE code base is publicly available as open source at http://muse.liComment: 24 pages, To appear in New Astronomy Source code available at http://muse.l

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

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    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

    Coherent state-based approaches to quantum dynamics: application to thermalization in finite systems

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    We investigate thermalization in finite quantum systems using coherent state-based approaches to solve the time-dependent Schr\'odinger equation. Earlier, a lot of work has been done in the quantum realm, to study thermalization in spin systems, but not for the case of continuous systems. Here, we focus on continuous systems. We study the zero temperature thermalization i.e., we consider the ground states of the bath oscillators (environment). In order to study the quantum dynamics of a system under investigation, we require numerical methods to solve the time-dependent Schr\'odinger equation. We describe different numerical methods like the split-operator fast fourier transform, coupled coherent states, static grid of coherent states, semiclassical Herman-Kluk propagator and the linearized semiclassical initial value representation to study the quantum dynamics. We also give a comprehensive comparison of the most widely used coherent state based methods. Starting from the fully variational coherent states method, after a first approximation, the coupled coherent states method can be derived, whereas an additional approximation leads to the semiclassical Herman-Kluk method. We numerically compare the different methods with another one, based on a static rectangular grid of coherent states, by applying all of them to the revival dynamics in a one-dimensional Morse oscillator, with a special focus on the number of basis states (for the coupled coherent states and Herman-Kluk methods the number of classical trajectories) needed for convergence. We also extend the Husimi (coherent state) based version of linearized semiclassical theories for the calculation of correlation functions to the case of survival probabilities. This is a case that could be dealt with before only by use of the Wigner version of linearized semiclassical theory. Numerical comparisons of the Husimi and the Wigner case with full quantum results as well as with full semiclassical ones is given for the revival dynamics in a Morse oscillator with and without coupling to an additional harmonic degree of freedom. From this, we see the quantum to classical transition of the system dynamics due to the coupling to the environment (bath harmonic oscillator), which then can lead ultimately to our final goal of thermalization for long-time dynamics. In regard to thermalization in quantum systems, we address the following questions--- is it enough to increase the interaction strength between the different degrees of freedom in order to fully develop chaos which is the classical prerequisite for thermalization, or if, in addition, the number of those degrees of freedom has to be increased (possibly all the way to the thermodynamic limit) in order to observe thermalization. We study the ``toppling pencil'' model, i.e., an excited initial state on top of the barrier of a symmetric quartic double well to investigate thermalization. We apply the method of coupled coherent states to study the long-time dynamics of this system. We investigate if the coupling of the central quartic double well to a finite, environmental bath of harmonic oscillators in their ground states will let the central system evolve towards its uncoupled ground state. This amounts to thermalization i.e., a cooling down to the bath ``temperature'' (strictly only defined in the thermodynamic limit) of the central system. It is shown that thermalization can be achieved in finite quantum system with continuous variables using coherent state-based methods to solve the time-dependent Schr\'odinger equation. Also, here we witness thermalization by coupling the system to a bath of only few oscillators (less than ten), which until now has been seen for more than ten to twenty bath oscillators

    The impact of the stellar evolution of single and binary stars on the global, dynamical evolution of dense star clusters across cosmic time

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    Sternhaufen im Universum stellen dichte, selbstgravitierende und typischerweise dynamisch kollidierende Umgebungen dar, die aus Tausenden bis Millionen von Sternen bestehen. Sie bevölkern galaktische Scheiben, Halos und sogar galaktische Zentren im gesamten Kosmos und bilden eine grundlegende Einheit in einer Hierarchie der kosmischen Strukturbildung. Außerdem sind sie in der Regel viel dichter als ihre Wirtsgalaxie, was sie zu unglaublich faszinierenden astronomischen Objekten macht. Anders als ihre Umgebung erleben Sterne und kompakte Objekte in Sternhaufen häufige dynamische Streuungen, bilden dynamische Doppelsterne, verschmelzen unter Aussendung von Gravitationswellen, werden durch Dreikörperdynamik herausgeschleudert und stoßen in seltenen Fällen sogar direkt zusammen. Infolgedessen sind Sternhaufen Fabriken aller exotischen Doppelsterne, von z.B. Thorne-Zytkow-Objekten und kataklysmischen Variablen bis hin zu kompakten Doppelsternen, beispielsweise Doppelsterne, die aus schwarzen Löchern und Neutronensternen bestehen. Darüber hinaus fangen mit zunehmender Teilchenzahl einzigartige Gravitationseffekte von kollidierenden Vielteilchensystemen an die frühe Entwicklung des Haufens zu dominieren, die zu zusammenziehenden und zunehmend schneller rotierenden Kernen der Sternhaufen führen, die bevorzugt massereiche Sterne und kompakte Objeckte sowie Doppelsterne enthalten, und einem sich ausdehnenden Halo aus Sternen und kompakten Objekten geringerer Masse. Sternhaufen sind daher nicht nur ein Labor für die Gravitationsvielteilchenphysik, sondern auch für die Sternentwicklung von Einzel- und Doppelsternen sowie hierarchischen Sternensystemen höherer Ordnung. Alle diese physikalischen Prozesse können nicht isoliert betrachtet werden - sie verstärken sich in Sternhaufen gegenseitig und viele passieren auf ähnlichen Zeitskalen. In dieser Arbeit möchte ich den Einfluss der Sternentwicklung auf die globale Dynamik von Sternhaufen mit Hilfe von direkten gravitativen N-Körper und Hénon-Typ Monte-Carlo Simulationen von Sternhaufen genauer studieren. Ich konzentriere mich auf die Entwicklung von metallarmen Sternpopulationen (Population II), die in Kugelsternhaufen und extrem metallarme Sternpopulationen (Population III), die die ältesten Sternpopulationen im Universum bilden

    MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

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    Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described

    Generalized averaged Gaussian quadrature and applications

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    A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal

    The modelling of electronic effects in molecular dynamics simulations

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    This thesis describes the development and applications of the continuum-atomistic molecular dynamics (MD) model in the context of radiation damage. By extending the classical MD method to incorporate the electronic excitations represented as an electron fluid and coupled to the ions in the two-temperature (2T) formalism, we have been able to correctly capture the physics governing the atomistic dynamics under huge electronic excitations. The integrated 2T-MD model has been specifically adapted to study three types of non-equilibrium scenarios: laser excitations, swift heavy ion impacts and large-scale high energy collision cascades. Using the 2T-MD model we have estimated the impact of the electron-phonon coupling and the electronic stopping power on the primary radiation damage yield in bcc iron. We have found that the cascade dynamics and the resultant damage from 50-100 keV primary knock-on atom impacts is highly sensitive to the electronic stopping treatment at low projectile velocities, which represents the first rigorous study of this type. By examining the temporal evolution of the structure factor of laser-irradiated gold thin films, we have been able to directly compare the 2T-MD results with Bragg peaks measured by ultrafast electron diffraction and have achieved an excellent agreement between theory and experiment with no fitting parameters. This has enabled us to elucidate the melting dynamics following laser irradiation at a picosecond resolution for the first time and also to validate the two-temperature approach. To simulate semiconductors under electronic excitations, the continuum part of the 2T-MD model, which represents electrons, has been replaced by two continuum equations: one for carrier density and one for their energy, to account for the finite band-gap effects. We have applied such extended method to simulate ion tracks, which result from swift heavy ion impacts. We have achieved a very good agreement with the experimental results on the ion track radii, provided that we are free to adjust the strength of the electron-phonon coupling. We propose future studies in the field of non-equilibrium atomistic modelling. In particular, we discuss ab initio methods and further improvements to hybrid MD to study the effects of the interatomic potential changes in response to high electronic excitations

    Particle Physics Reference Library

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    This third open access volume of the handbook series deals with accelerator physics, design, technology and operations, as well as with beam optics, dynamics and diagnostics. A joint CERN-Springer initiative, the “Particle Physics Reference Library” provides revised and updated contributions based on previously published material in the well-known Landolt-Boernstein series on particle physics, accelerators and detectors (volumes 21A,B1,B2,C), which took stock of the field approximately one decade ago. Central to this new initiative is publication under full open acces
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