Existence of solutions and asymtptotic limits of the Euler-Poisson and the quantum drift-diffusion systems

My work concerns two different systems of equations used in the mathematical modeling of semiconductors and plasmas: the Euler-Poisson system and the quantum drift-diffusion system. The first is given by the Euler equations for the conservation of mass and momentum, with a Poisson equation for the electrostatic potential. The second one takes into account the physical effects due to the smallness of the devices (quantum effects). It is a simple extension of the classical drift-diffusion model which consists of two continuity equations for the charge densities, with a Poisson equation for the electrostatic potential. Using an asymptotic expansion method, we study (in the steady-state case for a potential flow) the limit to zero of the three physical parameters which arise in the Euler-Poisson system: the electron mass, the relaxation time and the Debye length. For each limit, we prove the existence and uniqueness of profiles to the asymptotic expansion and some error estimates. For a vanishing electron mass or a vanishing relaxation time, this method gives us a new approach in the convergence of the Euler-Poisson system to the incompressible Euler equations. For a vanishing Debye length (also called quasineutral limit), we obtain a new approach in the existence of solutions when boundary layers can appear (i.e. when no compatibility condition is assumed). Moreover, using an iterative method, and a finite volume scheme or a penalized mixed finite volume scheme, we numerically show the smallness condition on the electron mass needed in the existence of solutions to the system, condition which has already been shown in the literature. In the quantum drift-diffusion model for the transient bipolar case in one-space dimension, we show, by using a time discretization and energy estimates, the existence of solutions (for a general doping profile). We also prove rigorously the quasineutral limit (for a vanishing doping profile). Finally, using a new time discretization and an algorithmic construction of entropies, we prove some regularity properties for the solutions of the equation obtained in the quasineutral limit (for a vanishing pressure). This new regularity permits us to prove the positivity of solutions to this equation for at least times large enough.

Statistical problems related to excitation threshold and reset value of membrane potentials

Die vorliegende Arbeit ist motiviert durch biologische Fragestellungen bezüglich des Verhaltens von Membranpotentialen in Neuronen. Ein vielfach betrachtetes Modell für spikende Neuronen ist das Folgende. Zwischen den Spikes verhält sich das Membranpotential wie ein Diffusionsprozess X der durch die SDGL dX_t= beta(X_t) dt+ sigma(X_t) dB_t gegeben ist, wobei (B_t) eine Standard-Brown'sche Bewegung bezeichnet. Spikes erklärt man wie folgt. Sobald das Potential X eine gewisse Exzitationsschwelle S überschreitet entsteht ein Spike. Danach wird das Potential wieder auf einen bestimmten Wert x_0 zurückgesetzt. In Anwendungen ist es manchmal möglich, einen Diffusionsprozess X zwischen den Spikes zu beobachten und die Koeffizienten der SDGL beta() und sigma() zu schätzen. Dennoch ist es nötig, die Schwellen x_0 und S zu bestimmen um das Modell festzulegen. Eine Möglichkeit, dieses Problem anzugehen, ist x_0 und S als Parameter eines statistischen Modells aufzufassen und diese zu schätzen. In der vorliegenden Arbeit werden vier verschiedene Fälle diskutiert, in denen wir jeweils annehmen, dass das Membranpotential X zwischen den Spikes eine Brown'sche Bewegung mit Drift, eine geometrische Brown'sche Bewegung, ein Ornstein-Uhlenbeck Prozess oder ein Cox-Ingersoll-Ross Prozess ist. Darüber hinaus beobachten wir die Zeiten zwischen aufeinander folgenden Spikes, die wir als iid Treffzeiten der Schwelle S von X gestartet in x_0 auffassen. Die ersten beiden Fälle ähneln sich sehr und man kann jeweils den Maximum-Likelihood-Schätzer explizit angeben. Darüber hinaus wird, unter Verwendung der LAN-Theorie, die Optimalität dieser Schätzer gezeigt. In den Fällen OU- und CIR-Prozess wählen wir eine Minimum-Distanz-Methode, die auf dem Vergleich von empirischer und wahrer Laplace-Transformation bezüglich einer Hilbertraumnorm beruht. Wir werden beweisen, dass alle Schätzer stark konsistent und asymptotisch normalverteilt sind. Im letzten Kapitel werden wir die Effizienz der Minimum-Distanz-Schätzer anhand simulierter Daten überprüfen. Ferner, werden Anwendungen auf reale Datensätze und deren Resultate ausführlich diskutiert.

Nuclear charge radius determination of 7,10 Be and the one-neutron halo nucleus 11 Be

The only nuclear model independent method for the determination of nuclear charge radii of short-lived radioactive isotopes is the measurement of the isotope shift. For light elements (Z < 10) extremely high accuracy in experiment and theory is required and was only reached for He and Li so far. The nuclear charge radii of the lightest elements are of great interest because they have isotopes which exhibit so-called halo nuclei. Those nuclei are characterized by a a very exotic nuclear structure: They have a compact core and an area of less dense nuclear matter that extends far from this core. Examples for halo nuclei are 6^He, 8^He, 11^Li and 11^Be that is investigated in this thesis. Furthermore these isotopes are of interest because up to now only for such systems with a few nucleons the nuclear structure can be calculated ab-initio. In the Institut für Kernchemie at the Johannes Gutenberg-Universität Mainz two approaches with different accuracy were developed. The goal of these approaches was the measurement of the isotope shifts between (7,10,11)^Be^+ and 9^Be^+ in the D1 line. The …first approach is laser spectroscopy on laser cooled Be^+ ions that are trapped in a linear Paul trap. The accessible accuracy should be in the order of some 100 kHz. In this thesis two types of linear Paul traps were developed for this purpose. Moreover, the peripheral experimental setup was simulated and constructed. It allows the efficient deceleration of fast ions with an initial energy of 60 keV down to some eV and an effcient transport into the ion trap. For one of the Paul traps the ion trapping could already be demonstrated, while the optical detection of captured 9^Be^+ ions could not be completed, because the development work was delayed by the second approach. The second approach uses the technique of collinear laser spectroscopy that was already applied in the last 30 years for measuring isotope shifts of plenty of heavier isotopes. For light elements (Z < 10), it was so far not possible to reach the accuracy that is required to extract information about nuclear charge radii. The combination of collinear laser spectroscopy with the most modern methods of frequency metrology …finally permitted the …first-time determination of the nuclear charge radii of (7,10)^Be and the one neutron halo nucleus 11^Be at the COLLAPS experiment at ISOLDE/ CERN. In the course of the work reported in this thesis it was possible to measure the absolute transition frequencies and the isotope shifts in the D1 line for the Be isotopes mentioned above with an accuracy of better than 2 MHz. Combination with the most recent calculations of the mass effect allowed the extraction of the nuclear charge radii of (7,10,11)^Be with an relative accuracy better than 1%. The nuclear charge radius decreases from 7^Be continuously to 10^Be and increases again for 11^Be. This result is compared with predictions of ab-initio nuclear models which reproduce the observed trend. Particularly the "Greens Function Monte Carlo" and the "Fermionic Molecular Dynamic" model show very good agreement.

Global simulated soil biogenic nitric oxide (NO) emissions: Impact, improvement and innovation

rnNitric oxide (NO) is important for several chemical processes in the atmosphere. Together with nitrogen dioxide (NO2 ) it is better known as nitrogen oxide (NOx ). NOx is crucial for the production and destruction of ozone. In several reactions it catalyzes the oxidation of methane and volatile organic compounds (VOCs) and in this context it is involved in the cycling of the hydroxyl radical (OH). OH is a reactive radical, capable of oxidizing most organic species. Therefore, OH is also called the “detergent” of the atmosphere. Nitric oxide originates from several sources: fossil fuel combustion, biomass burning, lightning and soils. Fossil fuel combustion is the largest source. The others are, depending on the reviewed literature, generally comparable to each other. The individual sources show a different temporal and spatial pattern in their magnitude of emission. Fossil fuel combustion is important in densely populated places, where NO from other sources is less important. In contrast NO emissions from soils (hereafter SNOx) or biomass burning are the dominant source of NOx in remote regions.rnBy applying an atmospheric chemistry global climate model (AC-GCM) I demonstrate that SNOx is responsible for a significant part of NOx in the atmosphere. Furthermore, it increases the O3 and OH mixing ratio substantially, leading to a ∼10% increase in the oxidizing efficiency of the atmosphere. Interestingly, through reduced O3 and OH mixing ratios in simulations without SNOx, the lifetime of NOx increases in regions with other dominating sources of NOx