Boundary integral equations in Kinetic Plasma Theory

In this thesis, we use boundary integral equations (BIE) as a powerful tool to gain new insights into the dynamics of plasmas. On the theoretical side, our work provides new results regarding the oscillation of bounded plasmas. With the analytical computation of the frequencies for a general ellipsoid we contribute a new benchmark for numerical methods. Our results are validated by an extensive numerical study of several three-dimensional problems, including a particle accelerator with complex geometry and mixed boundary conditions. The use of Boundary Element Methods (BEM) reduces the dimension of the problem from three to two, thus drastically reducing the number of unknowns. By employing hierarchical methods for the computation of the occurring nonlocal sums and integral operators, our method scales linearly with the number of particles and the number of surface triangles, where the error decays exponentially in the expansion parameter. Furthermore, our method allows the pointwise evaluation of the electric field without loss of convergence order. As we are able to compute the occurring boundary integrals analytically, we can precisely predict the electric field near the boundary. This property makes our method exceptionally well suited for the numerical simulation of plasma sheaths near irregular boundaries or of plasma-surface interaction such as etching of semiconductors.In der vorliegenden Arbeit nutzen wir Randintegralgleichungen als ein mächtiges Werkzeug, um neue Einsichten in die Dynamik von Plasmen zu gewinnen. Auf theoretischer Seite entwickelt diese Arbeit neue Resultate bezüglich der Oszillation beschränkter Plasmen. Durch die ana- lytische Berechnung der Frequenzen im Fall eines allgemeinen Ellipsoids stellen wir ein neues Testbeispiel für numerische Methoden bereit. Unsere Resultate werden durch umfangreiche numerische Untersuchen dreidimensionaler Beispiele validiert, etwa einen Partikelbeschleuniger mit komplexer Geometrie und gemischten Randwerten. Mithilfe der Randelementmethode reduziert sich die Dimension des Problems von drei auf zwei, womit sich die Anzahl der Un- bekannten drastisch reduziert. Dank der Nutzung hierarchischer Methoden zur Berechnung der auftauchenden nichtlokalen Summen und Integraloperatoren skaliert unsere Methode linear mit der Anzahl der Partikel und der Anzahl der Oberflächendreiecken, wobei der Fehler exponen- tiell im Entwicklungsparameter abfällt. Des Weiteren erlaubt unsere Methode die Berechnung des elektrischen Felds ohne Verringerung der Konvergenzordnung. Da wir die auftretenden Randintegrale analytisch berechnen können, können wir präzise Aussagen über das elektrische Feld nahe des Rands treffen. Dank dieser Eigenschaft ist unsere Methode außergewöhnlich gut geeignet, um Plasmaränder nahe irregulärer Ränder oder Plasma-Oberflächen-Interaktionen, etwa das Ätzen von Halbleitern, zu simulieren

On the Efficient Computation of Large Scale Singular Sums with Applications to Long-Range Forces in Crystal Lattices

We develop a new expansion for representing singular sums in terms of integrals and vice versa. This method provides a powerful tool for the efficient computation of large singular sums that appear in long-range interacting systems in condensed matter and quantum physics. It also offers a generalised trapezoidal rule for the precise computation of singular integrals. In both cases, the difference between sum and integral is approximated by derivatives of the non-singular factor of the summand function, where the coefficients in turn depend on the singularity. We show that for a physically meaningful set of functions, the error decays exponentially with the expansion order. For a fixed expansion order, the error decays alge braically both with the grid size, if the method is used for quadrature, or the characteristic length scale of the summand function in case the sum over a fixed grid is approximated by an integral. In absence of a singularity, the method reduces to the Euler–Maclaurin summation formula. We demonstrate the numerical performance of our new expansion by applying it to the computation of the full nonlinear long-range forces inside a domain wall in a macro scopic one-dimensional crystal with 2 × 1010 particles. The code of our implementation in Mathematica is provided online. For particles that interact via the Coulomb repulsion, we demonstrate that finite size effects remain relevant even in the thermodynamic limit of macro scopic particle numbers. Our results show that widely-used continuum limits in condensed matter physics are not applicable for quantitative predictions in this case

Singular Euler-Maclaurin expansion on multidimensional lattices

We extend the classical Euler-Maclaurin expansion to sums over multidimensional lattices that involve functions with algebraic singularities. This offers a tool for the precise quantification of the effect of microscopic discreteness on macroscopic properties of a system. First, the Euler-Maclaurin summation formula is generalised to lattices in higher dimensions, assuming a sufficiently regular summand function. We then develop this new expansion further and construct the singular Euler-Maclaurin (SEM) expansion in higher dimensions, an extension of our previous work in one dimension, which remains applicable and useful even if the summand function includes a singular function factor. We connect our method to analytical number theory and show that all operator coefficients can be efficiently computed from derivatives of the Epstein zeta function. Finally we demonstrate the numerical performance of the expansion and efficiently compute singular lattice sums in infinite two-dimensional lattices, which are of high relevance in solid state and quantum physics. An implementation in Mathematica is provided online along with this article

Multidisciplinary benchmarks of a conservative spectral solver for the nonlinear Boltzmann equation

The Boltzmann equation describes the evolution of the phase-space probability distribution of classical particles under binary collisions. Approximations to it underlie the basis for several scholarly fields, including aerodynamics and plasma physics. While these approximations are appropriate in their respective domains, they can be violated in niche but diverse applications which require direct numerical solution of the original nonlinear Boltzmann equation. An expanded implementation of the Galerkin-Petrov conservative spectral algorithm is employed to study a wide variety of physical problems. Enabled by distributed precomputation, solutions of the spatially homogeneous Boltzmann equation can be achieved in seconds on modern personal hardware, while spatially-inhomogeneous problems are solvable in minutes. Several benchmarks against both analytic theoretical predictions and comparisons to other Boltzmann solvers are presented in the context of several domains including weakly ionized plasma, gaseous fluids, and atomic-plasma interaction.Comment: 17 pages, 5 figures, 1 tabl

Potential der Biolandwirtschaft zur Steigerung der ökologischen Nachhaltigkeit des Agrarsektors in Luxemburg

Organic agriculture is often hailed as an environmentally friendly food production system. The aim of this study was to analyse the effect of the management system (organic (org.)/conventional (conv.)) on the sustainability performance of farms and derive the possible environmental impact of a 100% conversion to organic agriculture in Luxembourg. During a sustainability assessment at farm level using the SMART-Farm Tool, org. farms achieved significantly higher goal achievements in 13 of the 14 sub-themes of the sustainability dimension “Environmental Integrity”. Thus, org. agriculture shows promise for improvement of the Luxembourgish agricultural sector. However, some differences in goal achievement between the org. and conv. farms, especially in the sub-theme Greenhouse Gases, are relatively small and show that org. agriculture also still has a large potential for improvement when we want to tackle environmental challenges such as climate change.SustEATabl

Potential der Biolandwirtschaft zur Steigerung der ökologischen Nachhaltigkeit des Agrarsektors in Luxemburg

Die Biolandwirtschaft wird oft als umweltfreundlicher gepriesen. Die Ergebnisse der SMART-Analyse zeigen, dass eine 100%ige Umstellung auf Biolandbau in Luxemburg die ökologische Nachhaltigkeit des Sektors verbessern kann, auch wenn mehr getan werden muss, um dem Klimawandel gerecht zu werden

Perioperative Chemotherapy in Gastroesophageal Cancer. A Retrospective Monocenter Evaluation of 42 Cases

Background: Perioperative chemotherapy increases the overall and progression-free survival of patients suffering from resectable adenocarcinomas of the lower esophagus, gastroesophageal junction and stomach (GEC). Comparing different chemotherapy regimens platin-based protocols with 5-fluorouracil (5-FU)/calcium folinate (CF) or oral fluoropyrimidines were favorable in terms of efficacy and side-effects. However, there is no consensus which regimen is the most efficacious. Methods: 42 consecutive patients with resectable GEC (UICC II and III) were treated with 3 pre- and postoperative chemotherapy cycles each consisting of epirubicin, oxaliplatin and capecitabine (EOX). We analyzed the overall survival, progression-free survival and toxicity retrospectively in comparison to published data. Results: The median overall survival in our cohort was 29 months and the progression-free survival was 17 months. The most frequent grade 3 and 4 toxicities during preoperative chemotherapy were diarrhea (16.7%), leukocytopenia (9.5%) and nausea (9.5%); overall 38.1% of our patients suffered from grade 3 or 4 toxicity. Surgery was carried out in 83% of our patients, 69% of those achieved R0 resection. Conclusion: Comparing our data with the results of previously published randomized trials EOX is at least non-inferior with regard to overall survival, progression-free survival and toxicity. In conclusion, EOX is an appropriate perioperative therapy for patients with resectable GEC