This thesis deals with a newly-developed model for the spatiotemporal calcium dynamics within presynaptic terminals. The model is based on single-protein kinetics and has been used to successfully describe different neuron types such as pyramidal neurons in the rat neocortex and the Calyx of Held of neurons from the rat brainstem. A limited number of parameters had to be adjusted to fluorescence measurements of the calcium concentration. These values can be interpreted as a prediction of the model, and in particular the protein densities can be compared to independent experiments. The contribution of single proteins to the total calcium dynamics has been analysed in detail for voltage-dependent calcium channel, plasma-membrane calcium ATPase, sodium-calcium exchanger, and endogenous as well as exogenous buffer proteins. The model can be used to reconstruct the unperturbed calcium dynamics from measurements using fluorescence indicators. The calcium response to different stimuli has been investigated in view of its relevance for synaptic plasticity. This work provides a first step towards a description of the complete synaptic transmission using single-protein data.

The field of geosciences, including geophysics, plays a crucial role in addressing society's pressing concerns related to energy demand, climate change, resource preservation, and environmental protection. Reactive systems encountered in this context are characterized by intricate interactions among various phases, environmental conditions, physical and chemical processes. Achieving a comprehensive understanding of these processes and quantitatively evaluating reactive systems necessitates a holistic scientific approach. This approach encompasses efficient categorization of reactive systems, the development of appropriate experimental and computational tools, and the collection and dissemination of relevant data. In this context, this thesis contributes to geophysics and petrophysics with a focus on reactive systems. It presents and interprets laboratory datasets that address various complex aspects of rock behavior, including the presence of graphite, resulting anisotropy, and the challenging petrophysical characteristics of carbonate rocks. This compilation of research results provides a multifaceted perspective on the complex nature of rocks, including their mineralogical, physical, and chemical properties. It thus contributes to a deeper comprehension of electrical rock properties and their practical utility. Upon examining carbonate rocks and the response of graphitic schist to CO$_\mathrm{2}$ under reservoir conditions, it becomes clear that the impact of increased reactivity in a system on geophysical parameters varies depending on the specific characteristics of the rocks and systems under investigation. Consequently, geophysical methods aiming at a quantitative assessment of reactive systems must exhibit robustness and efficiency in order to be effectively applied in a site- and system-specific manner. Expanding on this foundation, computational methods have been developed to aid in the quantitative analysis of reactive processes in laboratory experiments. These methods also serve as tools for gaining insights into the origin of rock properties and the impact of microstructure variation. Furthermore, inversion techniques are introduced in conjunction with custom-designed experiments within the field of petrophysics. The resultant software tool is made publicly accessible. The research further delves into the exploration of how physical properties of rocks are influenced by their microstructure, as well as how the stochastic nature of pore space geometry can introduce variability and uncertainty in rock physics data. This investigation was carried out through microstructure modeling and finite element simulations. Leveraging these tailored computational techniques allowed for a comprehensive understanding of laboratory data, facilitating robust generalizations and contextualization for field applications and site-specific integrated interpretation. To illustrate the application in a complex natural reactive system, a field study focusing on coastal fumarolic vents in volcanic terrain was carried out and is presented. The challenges, prospects and visualization strategies for integrating simulation or inversion results from different methods are examined. Effective evaluation of complex sites requires open access to existing knowledge, including laboratory datasets. Consequently, this work documents and provides openly accessible examples of complex multi-method laboratory datasets to facilitate better understanding, re-evaluation and application in the field. Finally, the handling of multi-reactive systems in field applications is discussed. It involves the integration of three-dimensional subsurface models with petrophysical insights related to multi-reactive systems. These models are calibrated using additional complementary data from surface or borehole sources. This integrated approach enables a quantitative assessment of site-specific multi-reactive systems.

Diese Arbeit beschäftigt sich mit Coarse-Graining (dt. ``Vergröberung", ``Zusammenfassung von Zuständen") für Gradientensysteme und Markov-Prozesse. Coarse-Graining ist ein etabliertes Verfahren in der Mathematik und in den Naturwissenschaften und hat das Ziel, die Komplexität eines physikalischen Systems zu reduzieren und effektive Modelle herzuleiten. Die mathematischen Probleme in dieser Arbeit stammen aus der Theorie der Systeme interagierender Teilchen. Hierbei werden zwei Ziele verfolgt: Erstens, Coarse-Graining mathematisch rigoros zu beweisen, zweitens, mathematisch äquivalente Beschreibungen für die effektiven Modelle zu formulieren. Die ersten drei Teile der Arbeit befassen sich mit dem Grenzwert schneller Reaktionen für Reaktionssysteme und Reaktions-Diffusions-Systeme. Um effektive Modelle herzuleiten, werden nicht nur die zugehörigen Reaktionsratengleichungen betrachtet, sondern auch die zugrunde liegende Gradientenstruktur. Für Gradientensysteme wurde in den letzten Jahren eine strukturelle Konvergenz, die sogenannte ``EDP-Konvergenz", entwickelt. Dieses Coarse-Graining-Verfahren wird in der vorliegenden Arbeit auf folgende Systeme mit langsamen und schnellen Reaktionen angewandt: lineare Reaktionssysteme (bzw. Markov-Prozesse auf endlichem Zustandsraum), nichtlineare Reaktionssysteme, die das Massenwirkungsgesetz erfüllen, und lineare Reaktions-Diffusions-Systeme. Für den Grenzwert schneller Reaktionen wird eine mathematisch rigorose und strukturerhaltende Vergröberung auf dem Level des Gradientensystems inform von EDP-Konvergenz bewiesen. Im vierten Teil wird der Zusammenhang zwischen Gleichungen mit Gedächtnis und Markov-Prozessen untersucht. Für Gleichungen mit Gedächtnisintegralen wird explizit ein größer Markov-Prozess konstruiert, der die Gleichung mit Gedächtnis als Teilsystem enthält. Der letzte Teil beschäftigt sich mit verschieden Diskretisierungen für den Fokker-Planck-Operator. Dazu werden numerische und analytische Eigenschaften untersucht. / This thesis deals with coarse-graining for gradient systems and Markov processes. Coarse-graining is a well-established tool in mathematical and natural sciences for reducing the complexity of a physical system and for deriving effective models. The mathematical problems in this work originate from interacting particle systems. The aim is twofold: first, providing mathematically rigorous results for physical coarse-graining, and secondly, formulating mathematically equivalent descriptions for the effective models. The first three parts of the thesis deal with fast-reaction limits for reaction systems and reaction-diffusion systems. Instead of deriving effective models by solely investigating the associated reaction-rate equation, we derive effective models using the underlying gradient structure of the evolution equation. For gradient systems a structural convergence, the so-called ``EDP-convergence", has been derived in recent years. In this thesis, this coarse-graining procedure has been applied to the following systems with slow and fast reactions: linear reaction systems (or Markov process on finite state space), nonlinear reaction systems of mass-action type, and linear reaction-diffusion systems. For the fast-reaction limit, we perform rigorous and structural coarse-graining on the level of the gradient system by proving EDP-convergence. In the fourth part, the connection between memory equations and Markov processes is investigated. Considering linear memory equations, which can be motivated from spatial homogenization, we explicitly construct a larger Markov process that includes the memory equation as a subsystem. The last part deals with different discretization schemes for the Fokker–Planck operator and investigates their analytical and numerical properties.