The occurrence of earthquakes is characterized by a high degree of spatiotemporal complexity. Although numerous patterns, e.g. fore- and aftershock sequences, are well-known, the underlying mechanisms are not observable and thus not understood. Because the recurrence times of large earthquakes are usually decades or centuries, the number of such events in corresponding data sets is too small to draw conclusions with reasonable statistical significance. Therefore, the present study combines both, numerical modeling and analysis of real data in order to unveil the relationships between physical mechanisms and observational quantities. The key hypothesis is the validity of the so-called "critical point concept" for earthquakes, which assumes large earthquakes to occur as phase transitions in a spatially extended many-particle system, similar to percolation models. New concepts are developed to detect critical states in simulated and in natural data sets. The results indicate that important features of seismicity like the frequency-size distribution and the temporal clustering of earthquakes depend on frictional and structural fault parameters. In particular, the degree of quenched spatial disorder (the "roughness") of a fault zone determines whether large earthquakes occur quasiperiodically or more clustered. This illustrates the power of numerical models in order to identify regions in parameter space, which are relevant for natural seismicity. The critical point concept is verified for both, synthetic and natural seismicity, in terms of a critical state which precedes a large earthquake: a gradual roughening of the (unobservable) stress field leads to a scale-free (observable) frequency-size distribution. Furthermore, the growth of the spatial correlation length and the acceleration of the seismic energy release prior to large events is found. The predictive power of these precursors is, however, limited. Instead of forecasting time, location, and magnitude of individual events, a contribution to a broad multiparameter approach is encouraging. / Das Auftreten von Erdbeben zeichnet sich durch eine hohe raumzeitliche Komplexität aus. Obwohl zahlreiche Muster, wie Vor- und Nachbeben bekannt sind, weiß man wenig über die zugrundeliegenden Mechanismen, da diese sich direkter Beobachtung entziehen. Die Zeit zwischen zwei starken Erdbeben in einer seismisch aktiven Region beträgt Jahrzehnte bis Jahrhunderte. Folglich ist die Anzahl solcher Ereignisse in einem Datensatz gering und es ist kaum möglich, allein aus Beobachtungsdaten statistisch signifikante Aussagen über deren Eigenschaften abzuleiten. Die vorliegende Arbeit nutzt daher numerische Modellierungen einer Verwerfungszone in Verbindung mit Datenanalyse, um die Beziehung zwischen physikalischen Mechanismen und beobachteter Seismizität zu studieren. Die zentrale Hypothese ist die Gültigkeit des sogenannten "kritischen Punkt Konzeptes" für Seismizität, d.h. starke Erdbeben werden als Phasenübergänge in einem räumlich ausgedehnten Vielteilchensystem betrachtet, ähnlich wie in Modellen aus der statistischen Physik (z.B. Perkolationsmodelle). Es werden praktische Konzepte entwickelt, die es ermöglichen, kritische Zustände in simulierten und in beobachteten Daten sichtbar zu machen. Die Resultate zeigen, dass wesentliche Eigenschaften von Seismizität, etwa die Magnitudenverteilung und das raumzeitliche Clustern von Erdbeben, durch Reibungs- und Bruchparameter bestimmt werden. Insbesondere der Grad räumlicher Unordnung (die "Rauhheit") einer Verwerfungszone hat Einfluss darauf, ob starke Erdbeben quasiperiodisch oder eher zufällig auftreten. Dieser Befund zeigt auf, wie numerische Modelle genutzt werden können, um den Parameterraum für reale Verwerfungen einzugrenzen. Das kritische Punkt Konzept kann in synthetischer und in beobachteter Seismizität verifiziert werden. Dies artikuliert sich auch in Vorläuferphänomenen vor großen Erdbeben: Die Aufrauhung des (unbeobachtbaren) Spannungsfeldes führt zu einer Skalenfreiheit der (beobachtbaren) Größenverteilung; die räumliche Korrelationslänge wächst und die seismische Energiefreisetzung wird beschleunigt. Ein starkes Erdbeben kann in einem zusammenhängenden Bruch oder in einem unterbrochenen Bruch (Vorbeben und Hauptbeben) stattfinden. Die beobachtbaren Vorläufer besitzen eine begrenzte Prognosekraft für die Auftretenswahrscheinlichkeit starker Erdbeben - eine präzise Vorhersage von Ort, Zeit, und Stärke eines nahenden Erdbebens ist allerdings nicht möglich. Die genannten Parameter erscheinen eher vielversprechend als Beitrag zu einem umfassenden Multiparameteransatz für eine verbesserte zeitabhängige Gefährdungsabschätzung.

Plants and some unicellular algae store carbon in the form of transitory starch on a diurnal basis. The turnover of this glucose polymer is tightly regulated and timely synthesis as well as mobilization is essential to provide energy for heterotrophic growth. Especially for starch degradation, novel enzymes and mechanisms have been proposed recently. However, the catalytic properties of these enzymes and their coordination with metabolic regulation are still to be discovered. This thesis develops theoretical methods in order to interpret and analyze enzymes and their role in starch degradation. In the first part, a novel description of interfacial enzyme catalysis is proposed. Since the initial steps of starch degradation involve reactions at the starch-stroma interface it is necessary to have a framework which allows the derivation of interfacial enzyme rate laws. A cornerstone of the method is the introduction of the available area function - a concept from surface physics - to describe the adsorption step in the catalytic cycle. The method is applied to derive rate laws for two hydrolases, the Beta-amylase (BAM3) and the Isoamylase (DBE/ISA3), as well as to the Glucan, water dikinase (GWD) and a Phosphoglucan phosphatase (DSP/SEX4). The second part uses the interfacial rate laws to formulate a kinetic model of starch degradation. It aims at reproducing the stimulatory effect of reversible phosphorylation by GWD and DSP on the breakdown of the granule. The model can describe the dynamics of interfacial properties during degradation and suggests that interfacial amylopectin side-chains undergo spontaneous helix-coil transitions. Reversible phosphorylation has a synergistic effect on glucan release especially in the early phase dropping off during degradation. Based on the model, the hypothesis is formulated that interfacial phosphorylation is important for the rapid switch from starch synthesis to starch degradation. The third part takes a broader perspective on carbohydrate-active enzymes (CAZymes) but is motivated by the organization of the downstream pathway of starch breakdown. This comprises Alpha-1,4-glucanotransferases (DPE1 and DPE2) and Alpha-glucan-phosphorylases (Pho or PHS) both in the stroma and in the cytosol. CAZymes accept many different substrates and catalyze numerous reactions and therefore cannot be characterized in classical enzymological terms. A concise characterization is provided by conceptually linking statistical thermodynamics and polymer biochemistry. Each reactant is interpreted as an energy level, transitions between which are constrained by the enzymatic mechanisms. Combinations of in vitro assays of polymer-active CAZymes essential for carbon metabolism in plants confirmed the dominance of entropic gradients. The principle of entropy maximization provides a generalization of the equilibrium constant. Stochastic simulations confirm the results and suggest that randomization of metabolites in the cytosolic pool of soluble heteroglycans (SHG) may contribute to a robust integration of fluctuating carbon fluxes coming from chloroplasts. / Stärke hat eine herausragende Bedeutung für die menschliche Ernährung. Sie ist ein komplexes, wasserunlösliches Glucosepolymer und dient - als eine der wichtigsten Speicherformen von Kohlenhydraten in Pflanzen - der Aufrechterhaltung des Energiestoffwechsels. Unterschiedliche Organe enthalten Stärke. In Knollen und Samen wird die sogenannte Speicherstärke über lange Zeiträume auf- und abgebaut. Die im Allgemeinen weniger bekannte transitorische Stärke in Blättern und einigen einzelligen Algen wird in einem täglichen Rhythmus umgesetzt: Sie wird während der Photosynthese aufgebaut und in der Nacht abgebaut. Experimentelle Studien haben nachgewiesen, dass die Fähigkeit der Pflanze, den Abbau transitorischer Stärke zu regeln, essentiell ist, um während der Nacht das Wachstum der Pflanze zu gewährleisten. Da die Geschwindigkeit von biochemischen Reaktionen über Enzyme reguliert wird, ist die Aufklärung ihrer Funktion im Stoffwechsel eine notwendige Voraussetzung, um den komplexen Prozess des Wachstums zu erklären. Die vorliegende Arbeit stellt einen Versuch dar, die Funktion von Enzymen beim Stärkeabbau anhand von mathematischen Modellen und Computersimulationen besser zu verstehen. Dieser Ansatz erlaubt es, Eigenschaften des Systems durch Abstraktion anhand eines idealisierten Abbildes herzuleiten. Die mathematisch notwendigen Folgerungen dienen der Aufstellung von Hypothesen, die wiederum mit experimentellen Resultaten konfrontiert werden können. Stoffwechselsysteme sind komplexe Untersuchungsobjekte, bei denen eine rein qualitative Argumentation schnell an Grenzen gerät, wo mathematische Methoden die Möglichkeit von Aussagen noch zulassen. Der erste Teil der Arbeit entwickelt einen theoretischen Rahmen, um Gleichungen für die Geschwindigkeit oberflächenaktiver Enzyme herzuleiten. Dies ist notwendig, da die ersten Reaktionen, die dem Stärkeabbau zugeordnet werden, an ihrer Oberfläche stattfinden. Die Methode wird auf vier essentielle Enzyme angewandt: zwei abbauende Enzyme (Beta-Amylase und Isoamylase) und zwei den Abbau unterstützende Enzyme (Alpha-Glucan,Wasser-Dikinase und Phosphoglucan Phosphatase). Der zweite Teil entwickelt ein kinetisches Modell des Stärkeabbaus unter Verwendung der hergeleiteten Ratengleichungen. Das Modell bildet die Dynamik des Systems realistisch ab und legt nahe, dass ein spontaner Phasenübergang an der Oberfläche von geordneten zu weniger geordneten Zuständen stattfindet. Ferner wird die Hypothese aufgestellt, dass die reversible Modifikation der Oberfläche durch Enzyme besonders in der Anfangsphase des Abbaus einen synergetischen Effekt hat, d.h. den Abbau enorm beschleunigt. Dies könnte beim schnellen Umschalten von Stärkeaufbau zu Stärkeabbau regulatorisch relevant sein. Im letzten Teil werden kohlenhydrataktive Enzyme betrachtet, die in der löslichen Phase die Produkte des Stärkeabbaus weiterverarbeiten. Da diese sogenannten Transferasen auch in vielen anderen Organismen und Stoffwechselwegen vorkommen, wird ein allgemeiner Standpunkt eingenommen. Anhand von Methoden aus der statistischen Physik wird theoretisch wie experimentell nachgewiesen, dass diese Enzyme spontan die Entropie innerhalb des Stoffwechselsystems erhöhen. Diese Neigung, "Unordnung" zu schaffen, wird vom Organismus aber paradoxerweise ausgenutzt, um die Weiterverarbeitung von Kohlenhydraten im Stärkestoffwechsel zu stabilisieren. Dieser Mechanismus eröffnet einen neuen Blick auf energie- und entropiegetriebene Prozesse in Zellen.

A computational model is developed to investigate fundamental flow physics and transport phenomena of evaporating wavy-laminar falling liquid films of water and black liquor. The computational model is formulated from first principles based on the conservation laws for mass, momentum, energy and species in addition to a phase transport equation for capturing interface deformation and evolution. Free surface waves are generated by monochromatic perturbation of velocity. Continuum models for interfacial evaporation define source terms for liquid vaporization and species enrichment in the conservation laws. A phenomenological crystallization model is derived to account for species depletion due to salt precipitation during black liquor falling film evaporation. Using highly resolved numerical grids on parallel computers, the computational model is implemented to analyze the dynamics of capillary separation eddies in low Reynolds number falling films, investigate the dominant mechanisms of heat transfer enhancement in falling films at moderately high Reynolds numbers and study the fundamental wave structures and wave induced transport in black liquor falling films on flat and cylindrical walls. From simulation results, a theory based on the dynamics of wavefront streamwise pressure gradient is proposed to explain interfacial waves interaction that give rise to multiple backflow regions in films dominated by solitary-capillary waves. The study shows that the mechanism of heat transfer enhancement in moderately high Reynolds number films follows from relatively lower conduction thermal resistance and higher crosswise convective transport at newly formed intermediate wavefronts. Interfacial phenomena such as wave-breaking and vapor entrainment observed in black liquor falling films is explained in terms of a mechanistic theory based on evolution of secondary instabilities and large amplitude wave force imbalances.

Complex networks are employed as a mathematical description of complex systems in many different fields, ranging from biology to sociology, economy and ecology. Dynamical processes in these systems often display phase transitions, where the dynamics of the system changes qualitatively. In combination with these phase transitions certain components of the system might irretrievably go extinct. In this case, we talk about absorbing transitions. Developing mathematical tools, which allow for an analysis and prediction of the observed phase transitions is crucial for the investigation of complex networks. In this thesis, we investigate absorbing transitions in dynamical networks, where a certain amount of diversity is lost. In some real-world examples, e.g. in the evolution of human societies or of ecological systems, it is desirable to maintain a high degree of diversity, whereas in others, e.g. in epidemic spreading, the diversity of diseases is worthwhile to confine. An understanding of the underlying mechanisms for emergence and persistence of diversity in complex systems is therefore essential. Within the scope of two different network models, we develop an analytical approach, which can be used to estimate the prerequisites for diversity. In the first part, we study a model for opinion formation in human societies. In this model, regimes of low diversity and regimes of high diversity are separated by a fragmentation transition, where the network breaks into disconnected components, corresponding to different opinions. We propose an approach for the estimation of the fragmentation point. The approach is based on a linear stability analysis of the fragmented state close to the phase transition and yields much more accurate results compared to conventional methods. In the second part, we study a model for the formation of complex food webs. We calculate and analyze coexistence conditions for several types of species in ecological communities. To this aim, we employ an approach which involves an iterative stability analysis of the equilibrium with respect to the arrival of a new species. The proposed formalism allows for a direct calculation of coexistence ranges and thus facilitates a systematic analysis of persistence conditions for food webs. In summary, we present a general mathematical framework for the calculation of absorbing phase transitions in complex networks, which is based on concepts from percolation theory. While the specific implementation of the formalism differs from model to model, the basic principle remains applicable to a wide range of different models.

Complex systems can often be modelled as networks, in which their basic units are represented by abstract nodes and the interactions among them by abstract links. This network of interactions is the key to understanding emergent collective phenomena in such systems. In most cases, it is an adaptive network, which is defined by a feedback loop between the local dynamics of the individual units and the dynamical changes of the network structure itself. This feedback loop gives rise to many novel phenomena. Adaptive networks are a promising concept for the investigation of collective phenomena in different systems. However, they also present a challenge to existing modelling approaches and analytical descriptions due to the tight coupling between local and topological degrees of freedom. In this thesis, I present a simple rule-based framework for the investigation of adaptive networks, using which a wide range of collective phenomena can be modelled and analysed from a common perspective. In this framework, a microscopic model is defined by the local interaction rules of small network motifs, which can be implemented in stochastic simulations straightforwardly. Moreover, an approximate emergent-level description in terms of macroscopic variables can be derived from the microscopic rules, which we use to analyse the system\'s collective and long-term behaviour by applying tools from dynamical systems theory. We discuss three adaptive-network models for different collective phenomena within our common framework. First, we propose a novel approach to collective motion in insect swarms, in which we consider the insects\' adaptive interaction network instead of explicitly tracking their positions and velocities. We capture the experimentally observed onset of collective motion qualitatively in terms of a bifurcation in this non-spatial model. We find that three-body interactions are an essential ingredient for collective motion to emerge. Moreover, we show what minimal microscopic interaction rules determine whether the transition to collective motion is continuous or discontinuous. Second, we consider a model of opinion formation in groups of individuals, where we focus on the effect of directed links in adaptive networks. Extending the adaptive voter model to directed networks, we find a novel fragmentation mechanism, by which the network breaks into distinct components of opposing agents. This fragmentation is mediated by the formation of self-stabilizing structures in the network, which do not occur in the undirected case. We find that they are related to degree correlations stemming from the interplay of link directionality and adaptive topological change. Third, we discuss a model for the evolution of cooperation among self-interested agents, in which the adaptive nature of their interaction network gives rise to a novel dynamical mechanism promoting cooperation. We show that even full cooperation can be achieved asymptotically if the networks\' adaptive response to the agents\' dynamics is sufficiently fast.

Untersuchungen zum statistischen Verhalten von Polymerketten auf anziehenden Oberflächen stellen ein spannendes Forschungsgebiet dar aufgrund des Wechselspiels zwischen dem Entropiegewinn bei Ablösung von der einschränkenden Oberfläche und dem Energiegewinn bei der Bildung von Oberflächenkontakten. Für gute und Theta-Lösungen und lange Ketten ist dieses Gebiet recht alt und gut verstanden, doch gibt es immer noch eine Reihe von offenen Fragen, insbesondere zu endlich langen Polymeren, die gerade im Zeitalter zunehmender Miniaturisierung und experimenteller Auflösung Klärung bedürfen, aber nicht zuletzt auch von prinzipiellem Interesse sind. Die vorliegende Arbeit beschäftigt sich mit dem Gleichgewichtsverhalten einer endlich langen Polymerkette in Lösung in der Nähe einer anziehenden Oberfläche. Die Anziehungsstärke wird dabei systematisch variiert und der Einfluss auf die Konformation des Homopolymers studiert. Dies geschieht im kanonischen und im mikrokanonischen Ensemble, die im betrachteten endlichen System nicht identisch sind. Da die Lösungsmittelstärke des selbstwechselwirkenden Polymers durch die Temperatur variiert werden kann, gelang so eine systematische Studie einer Reihe von Konformationsübergängen. Ob das Polymer an einem Ende irreversibel mit der Oberfläche verbunden ist oder sich zu einem gewissen Grad von ihr entfernen kann, spielt für insbesondere den Adsorptionsübergang eine Rolle, die untersucht wird. Anschließend wurde der Einfluss nicht homogener Oberflächenanziehung in Form von attraktiven Streifenpotentialen auf der Oberfläche auf die zuvor beschriebenen Konformationsübergänge studiert. Die Natur der so forcierten Mustererkennung konnte unter anderem abhängig von Streifenbreite und -stärke detailliert beleuchtet und mit dem Verhalten an homogenen Oberflächen in Bezug gesetzt werden. Sämtliche Daten wurden mit Monte-Carlo-Computersimulationen in generalisierten Ensemblen und einem Polymermodell, das atomare Details vernachlässigt, gewonnen.

Fundamental aspects of turbulence and turbulent mixing are investigated using direct numerical simulations (DNS) of stationary isotropic turbulence, with Taylor-scale Reynolds numbers ranging from 8 to 650 and Schmidt numbers from 1/8 to 1024. The primary emphasis is on important scaling issues that arise in the study of intermittency, mixing and turbulence under solid-body rotation. Simulations up to 2048^3 in size have been performed using large resource allocations on Terascale computers at leading supercomputing centers. Substantial efforts in algorithmic development have also been undertaken and resulted in a new code based on a two-dimensional domain decomposition which allows the use of very large number of processors.Benchmark tests indicate very good parallel performance for resolutions up to 4096^3 on up to 32768 processors. Investigation of intermittency through the statistics of dissipation and enstrophy in a series of simulations at the same Reynolds number but different resolution indicate that accurate results in high-order moments require a higher degree of fine-scale resolution than commonly practiced. At the highest Reynolds number in our simulations (400 and 650) dissipation and enstrophy exhibit extreme fluctuations of O(1000) the mean which have not been studied in the literature before and suggest a universal scaling of small scales. Simulations at Reynolds number of 650 on 2048^3 grids with scalars at Sc=1/8 and 1 have allowed us to obtain the clearest evidence of attainment of inertial-convective scaling in the scalar spectrum in numerical simulations to date whereas results at high Sc support k^{-1} viscous-convective scaling. Intermittency for scalars as measured by the tail of the PDF of scalar dissipation and moments of scalar gradient fluctuations is found to saturate at high Sc. Persistent departures from isotropy are observed as the Reynolds number increases. However, results suggest a return to isotropy at high Schmidt numbers, a tendency that appears to be stronger at high Reynolds numbers. The effects of the Coriolis force on turbulence under solid-body rotation are investigated using simulations on enlarged solution domains which reduce the effects of periodic boundary conditions.

Shock compaction experiments were performed on soft magnetic phases Fe₄N and Fe₁₆N₂, and hard magnetic phases Nd₂Fe₁₄B and Sm₂Fe₁₇N₃ in order to determine their thermo-mechanical stability during shock loading and explore the possibility of fabricating a textured nanocomposite magnet. Gas gun experiments performed on powders pressed in a three capsule fixture showed phase transformations occurring in Fe₄N, Fe₁₆N₂, and Nd₂Fe₁₄B, while Sm₂Fe₁₇N₃ was observed to be relatively stable. Shock compaction of FCC Fe₄N resulted in a partial transformation to HCP Fe₃N, consistent with previous reports of the transition occurring at a static pressure of ~3 GPa. Shock compaction of Fe₁₆N₂ produced decomposition products alpha-Fe, Fe₄N, and FeN due to a combination of thermal effects associated with dynamic void collapse and plastic deformation. Decomposition of Nd-Fe-B, producing alpha-Fe and amorphous Nd-Fe-B, was observed in several shock consolidated samples and is attributed to deformation associated with shock compaction, similar to decomposition reported in ball milled Nd-Fe-B. No decomposition was observed in shock compacted samples of Sm-Fe-N, which is consistent with literature reports showing decomposition occurring only in samples compacted at a pressure above ~15 GPa. Nd-Fe-B and Sm-Fe-N were shown to accommodate deformation primarily by grain size reduction, especially in large grained materials. Hard/Soft composite magnetic materials were formed by mixing single crystal particles of Nd-Fe-B with iron nanoparticles, and the alignment-by-magnetic-field technique was able to introduce significant texture into green compacts of this mixture. While problems with decomposition of the Nd₂Fe₁₄B phase prevented fabricating bulk magnets from the aligned green compacts, retention of the nanoscale morphology of the alpha-Fe particles and the high alignment of the green compacts shows promise for future development of textured nanocomposite magnets through shock compaction.

One of the most important design issues for phase change materials is to engineer the phase transition process. The challenge of accurately predicting a phase transition is estimating the true value of transition rate, which is determined by the saddle point with the minimum energy barrier between stable states on the potential energy surface (PES). In this thesis, a new algorithm for searching the minimum energy path (MEP) is presented. The new algorithm is able to locate both the saddle point and local minima simultaneously. Therefore no prior knowledge of the precise positions for the reactant and product on the PES is needed. Unlike existing pathway search methods, the algorithm is able to search multiple transition paths on the PES simultaneously, which gives us a more comprehensive view of the energy landscape than searching individual ones. In this method, a Bézier curve is used to represent each transition path. During the searching process, the reactant and product states are located by minimizing the two end control points of the curve, while the shape of the transition pathway is refined by moving the intermediate control points of the curve in the conjugate directions. A curve subdivision scheme is developed so that multiple transitions paths can be located. The algorithm is demonstrated by examples of LEPS potential, LEPS plus harmonic oscillator potential, and PESs defined by Rastrigin function and Schwefel function.

Many seemingly simple systems can display extraordinarily complex dynamics which has been studied and uncovered through nonlinear dynamical theory. The leitmotif of this thesis is changing phase-space structures and their (linear or nonlinear) stabilities by adding control functions (which act on the system as external perturbations) to the relevant Hamiltonians. These phase-space structures may be periodic orbits, invariant tori or their stable and unstable manifolds. One-electron systems and diatomic molecules are fundamental and important staging ground for new discoveries in nonlinear dynamics. In past years, increasing emphasis and effort has been put on the control or manipulation of these systems. Recent developments of nonlinear dynamical tools can provide efficient ways of doing so. In the first subtopic of the thesis, we are adding a control function to restore tori at prescribed locations in phase space. In the remainder of the thesis, a control function with parameters is used to change the linear stability of the periodic orbits which govern the processes in question. In this thesis, we report our theoretical analyses on multiphoton ionization of Rydberg atoms exposed to strong microwave fields and the dissociation of diatomic molecules exposed to bichromatic lasers using nonlinear dynamical tools. This thesis is composed of three subtopics. In the first subtopic, we employ local control theory to reduce the stochastic ionization of hydrogen atom in a strong microwave field by adding a relatively small control term to the original Hamiltonian. In the second subtopic, we perform periodic orbit analysis to investigate multiphoton ionization driven by a bichromatic microwave field. Our results show quantitative and qualitative agreement with previous studies, and hence identify the mechanism through which short periodic orbits organize the dynamics in multiphoton ionization. In addition, we achieve substantial time savings with this approach. In the third subtopic we extend our periodic orbit analysis to the dissociation of diatomic molecules driven by a bichromatic laser. In this problem, our results based on periodic orbit analysis again show good agreement with previous work, and hence promise more potential applications of this approach in molecular physics.