Inverse problems in classical and quantum physics

The subject of this thesis is in the area of Applied Mathematics known as Inverse Problems. Inverse problems are those where a set of measured data is analysed in order to get as much information as possible on a model which is assumed to represent a system in the real world. We study two inverse problems in the fields of classical and quantum physics: QCD condensates from tau-decay data and the inverse conductivity problem. Despite a concentrated effort by physicists extending over many years, an understanding of QCD from first principles continues to be elusive. Fortunately, data continues to appear which provide a rather direct probe of the inner workings of the strong interactions. We use a functional method which allows us to extract within rather general assumptions phenomenological parameters of QCD (the condensates) from a comparison of the time-like experimental data with asymptotic space-like results from theory. The price to be paid for the generality of assumptions is relatively large errors in the values of the extracted parameters. Although we do not claim that our method is superior to other approaches, we hope that our results lend additional confidence to the numerical results obtained with the help of methods based on QCD sum rules. EIT is a technology developed to image the electrical conductivity distribution of a conductive medium. The technique works by performing simultaneous measurements of direct or alternating electric currents and voltages on the boundary of an object. These are the data used by an image reconstruction algorithm to determine the electrical conductivity distribution within the object. In this thesis, two approaches of EIT image reconstruction are proposed. The first is based on reformulating the inverse problem in terms of integral equations. This method uses only a single set of measurements for the reconstruction. The second approach is an algorithm based on linearisation which uses more then one set of measurements. A promising result is that one can qualitatively reconstruct the conductivity inside the cross-section of a human chest. Even though the human volunteer is neither two-dimensional nor circular, such reconstructions can be useful in medical applications: monitoring for lung problems such as accumulating fluid or a collapsed lung and noninvasive monitoring of heart function and blood flow.

Coarse-graining and quantum-classical adaptive coupling in soft matter

Die vorliegende Arbeit untersucht den Zusammenhang zwischen Skalen in Systemen weicher Materie, der für Multiskalen-Simulationen eine wichtige Rolle spielt. Zu diesem Zweck wurde eine Methode entwickelt, die die Approximation der Separierbarkeit von Variablen für die Molekulardynamik und ähnliche Anwendungen bewertet. Der zweite und größere Teil dieser Arbeit beschäftigt sich mit der konzeptionellen und technischen Erweiterung des Adaptive Resolution Scheme'' (AdResS), einer Methode zur gleichzeitigen Simulation von Systemen mit mehreren Auflösungsebenen. Diese Methode wurde auf Systeme erweitert, in denen klassische und quantenmechanische Effekte eine Rolle spielen.rnrnDie oben genannte erste Methode benötigt nur die analytische Form der Potentiale, wie sie die meisten Molekulardynamik-Programme zur Verfügung stellen. Die Anwendung der Methode auf ein spezielles Problem gibt bei erfolgreichem Ausgang einen numerischen Hinweis auf die Gültigkeit der Variablenseparation. Bei nicht erfolgreichem Ausgang garantiert sie, dass keine Separation der Variablen möglich ist. Die Methode wird exemplarisch auf ein zweiatomiges Molekül auf einer Oberfläche und für die zweidimensionale Version des Rotational Isomer State (RIS) Modells einer Polymerkette angewandt.rnrnDer zweite Teil der Arbeit behandelt die Entwicklung eines Algorithmus zur adaptiven Simulation von Systemen, in denen Quanteneffekte berücksichtigt werden. Die Quantennatur von Atomen wird dabei in der Pfadintegral-Methode durch einen klassischen Polymerring repräsentiert. Die adaptive Pfadintegral-Methode wird zunächst für einatomige Flüssigkeiten und tetraedrische Moleküle unter normalen thermodynamischen Bedingungen getestet. Schließlich wird die Stabilität der Methode durch ihre Anwendung auf flüssigen para-Wasserstoff bei niedrigen Temperaturen geprüft.

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.

Quantum effects and dynamics in hydrogen-bonded systems: a first-principles approach to spectroscopic experiments

Computer simulations have become an important tool in physics. Especially systems in the solid state have been investigated extensively with the help of modern computational methods. This thesis focuses on the simulation of hydrogen-bonded systems, using quantum chemical methods combined with molecular dynamics (MD) simulations. MD simulations are carried out for investigating the energetics and structure of a system under conditions that include physical parameters such as temperature and pressure. Ab initio quantum chemical methods have proven to be capable of predicting spectroscopic quantities. The combination of these two features still represents a methodological challenge. Furthermore, conventional MD simulations consider the nuclei as classical particles. Not only motional effects, but also the quantum nature of the nuclei are expected to influence the properties of a molecular system. This work aims at a more realistic description of properties that are accessible via NMR experiments. With the help of the path integral formalism the quantum nature of the nuclei has been incorporated and its influence on the NMR parameters explored. The effect on both the NMR chemical shift and the Nuclear Quadrupole Coupling Constants (NQCC) is presented for intra- and intermolecular hydrogen bonds. The second part of this thesis presents the computation of electric field gradients within the Gaussian and Augmented Plane Waves (GAPW) framework, that allows for all-electron calculations in periodic systems. This recent development improves the accuracy of many calculations compared to the pseudopotential approximation, which treats the core electrons as part of an effective potential. In combination with MD simulations of water, the NMR longitudinal relaxation times for 17O and 2H have been obtained. The results show a considerable agreement with the experiment. Finally, an implementation of the calculation of the stress tensor into the quantum chemical program suite CP2K is presented. This enables MD simulations under constant pressure conditions, which is demonstrated with a series of liquid water simulations, that sheds light on the influence of the exchange-correlation functional used on the density of the simulated liquid.

Quantum gravity effects in rotating black hole spacetimes

The aim of this work is to explore, within the framework of the presumably asymptotically safe Quantum Einstein Gravity, quantum corrections to black hole spacetimes, in particular in the case of rotating black holes. We have analysed this problem by exploiting the scale dependent Newton s constant implied by the renormalization group equation for the effective average action, and introducing an appropriate "cutoff identification" which relates the renormalization scale to the geometry of the spacetime manifold. We used these two ingredients in order to "renormalization group improve" the classical Kerr metric that describes the spacetime generated by a rotating black hole. We have focused our investigation on four basic subjects of black hole physics. The main results related to these topics can be summarized as follows. Concerning the critical surfaces, i.e. horizons and static limit surfaces, the improvement leads to a smooth deformation of the classical critical surfaces. Their number remains unchanged. In relation to the Penrose process for energy extraction from black holes, we have found that there exists a non-trivial correlation between regions of negative energy states in the phase space of rotating test particles and configurations of critical surfaces of the black hole. As for the vacuum energy-momentum tensor and the energy conditions we have shown that no model with "normal" matter, in the sense of matter fulfilling the usual energy conditions, can simulate the quantum fluctuations described by the improved Kerr spacetime that we have derived. Finally, in the context of black hole thermodynamics, we have performed calculations of the mass and angular momentum of the improved Kerr black hole, applying the standard Komar integrals. The results reflect the antiscreening character of the quantum fluctuations of the gravitational field. Furthermore we calculated approximations to the entropy and the temperature of the improved Kerr black hole to leading order in the angular momentum. More generally we have proven that the temperature can no longer be proportional to the surface gravity if an entropy-like state function is to exist.

Fluorescence close to dielectric interfaces

The aim of the work was to study the correlation between the orientation and excited-state lifetimes of organic dyes close to dielectric interfaces. For this purpose, an experimental setup was designed and built, guiding the light through a prism in total internal reflection geometry. Fluorescence intensities and lifetimes for an ensemble of dye molecules were analyzed as a function of the excitation and detection polarizations. Working close to the total internal reflection angle, the differences between polarization combinations were enhanced. A classical electromagnetic model that assumes a chromophore as a couple of point-like electrical dipoles was developed. A numerical method to calculate the excitation and emission of dye molecules embedded in a multilayer system was implemented, by which full simulation of the time resolved fluorescence experiments was achieved. Free organic dyes and organic dyes covalently bound to polyelectrolyte chains were used. The polymer functionalization process avoided aggregation and provided control over the dyes position, within a few nanometers to the interface. Moreover, by varying the pH, the polymer chains could be deposited on different substrates with different conformations and the resulting fluorescence characteristics analyzed. Initially the fluorescence of organic dyes embedded in a polymer matrix was studied as a function of the distance between the fluorophores and the polymer-air interface. The non-radiative decay rate, vacuum decay rate and the relative angle between the excitation and emission dipoles of the chromophores could be determined. Different free organic dyes were deposited onto different dielectric spacers, as close as possible to the air-dielectric interface. Surprisingly, the fluorescence characteristics of dyes deposited onto polyelectrolyte layer were in good agreement with theoretical predictions of dyes in a polymer matrix, even when the layer was only 2 nm thick. When functionalized chains were deposited at low pH, on top of a polyelectrolyte spacer, the fluorescence had the characteristics of emitters embedded in a polymer matrix as well. Surface deposition at high pH showed an intermediate behaviour between emitters embedded in polymer and on top of the surface, in air. In general, for low pH values, the chains are deposited on a substrate in a train-like conformation. For high pH values, the chains are deposited in a loop-like conformation. As a consequence at low pH the functionalized polymer strongly interdigitates with the polyelectrolyte chains of the spacer, bringing most of the dyes inside the polymer. Thus, the fluorophores may experience the polymer as surrounding environment. On the other hand, for high pH values the dye-loaded chains adsorbed have a conformational arrangement of dense loops that extend away from the surface. Therefore many fluorophores experience the air as surrounding environment. Changing the spacer from polyelectrolyte to negatively charged silane produced contradictory results for lifetimes and intensities. The fluorescence intensities indicated the behaviour of emitters embedded in a polymer matrix, regardless of the pH value. On the other hand, for low pH values, the excited-state lifetimes showed that the emitters behaved as in air. For higher pH values, an intermediate behaviour between fluorophores located within and above of a dielectric film was observed. The poor agreement between theoretical and experimental data may be due to the simplified model utilized, by which the dipoles are assumed either in one side or in the other with respect to a geometrical air-dielectric interface. In the case when the dielectric film is constituted by the functionalized polymer chains themselves, reality is more complex and a different model may apply. Nevertheless, possible applications of the technique arise from a qualitative analysis.

Structure and dynamics of ionic liquid,electrode interfaces

In the early 20th century, Gouy, Chapman, and Stern developed a theory to describe the capacitance and the spatial ion distribution of diluted electrolytes near an electrode. After a century of research, considerable progress has been made in the understanding of the electrolyte/electrode interface. However, its molecular-scale structure and its variation with an applied potential is still under debate. In particular for room-temperature ionic liquids, a new class of solventless electrolytes, the classical theories for the electrical double layer are not applicable. Recently, molecular dynamics simulations and phenomenological theories have attempted to explain the capacitance of the ionic liquid/electrode interface with the molecular-scale structure and dynamics of the ionic liquid near the electrode. rnHowever, experimental evidence is very limited. rnrnIn the presented study, the ion distribution of an ionic liquid near an electrode and its response to applied potentials was examined with sub-molecular resolution. For this purpose, a new sample chamber was constructed, allowing in situ high energy X-ray reflectivity experiments under potential control, as well as impedance spectroscopy measurements. The combination of structural information and electrochmical data provided a comprehensive picture of the electric double layer in ionic liquids. Oscillatory charge density profiles were found, consisting of alternating anion- and cation-enriched layers at both, cathodic and anodic, potentials. This structure was shown to arise from the same ion-ion correlations dominating the liquid bulk structure that were observed as a distinct X-ray diffraction peak. Therefore, existing physically motivated models were refined and verified by comparison with independent measurements. rnrnThe relaxation dynamics of the interfacial structure upon potential variation were studied by time resolved X-ray reflectivity experiments with sub-millisecond resolution. The observed relaxation times during charging/discharging are consistent with the impedance spectroscopy data revealing three processes of vastly different characteristic time-scales. Initially, the ion transport normal to the interface happens on a millisecond-scale. Another 100-millisecond-scale process is associated with molecular reorientation of electrode-adsorbed cations. Further, a minute-scale relaxation was observed, which is tentatively assigned to lateral ordering within the first layer.