Analysis of complex systems: From stochastic time series to pattern formation in microscopic fluidic films

The current thesis consists of two parts, both of which are dealing with the analysis of complex systems, one with stochastic time series analysis and one with pattern formation in microscopic fluid layers. The first part is about an analysis method which allows to extract stochastic order parameter equations from measured time series data. Employing this method termed Kramers-Moyal analysis, significant problems can occur if the temporal resolution of the measurement data is not sufficient. In the course of this thesis it is analyzed what the term “sufficient” means in this context, and where the limitations of the applicability of the Kramers-Moyal analysis are. Furthermore, the analysis method is extended in order to be able to analyze data with small sampling frequencies more reliably. In this context an estimation of the uncertainties of determined model parameters plays an important role. The latter is done via a Monte Carlo error propagation technique. First, the extended method is tested employing examples of synthetic data. Then, an application to real-world data from an optical tweezers experiment is performed. In this experiment a micrometer-sized particle diffusing in a fluid is trapped in the center of a highly focused laser beam. The particle then performs Brownian motion subject to an external force that is induced by the optical light pressure of the laser beam, whereas the motion of the particle is filmed by a CCD camera. Subsequently, the postions of the particle are extracted from the recorded images. An analysis of the measurement data shows a surprisingly large Markov-Einstein time scale, which can be traced back to hydrodynamic memory effects. Above this time scale the process can be mapped to an Ornstein-Uhlenbeck process via the application of the developed extended method. This is in agreement with the classical theory of overdamped Brownian motion. In the second part of the thesis, an experiment in the field of organic semiconductor research is modelled. The goal of these experiments is to deposit ordered structures of thin layers of small organic semiconducting molecules onto a substrate. This is done via organic molecular beam deposition. Within this technique organic molecules are sublimed at high temperatures in vacuum and subsequently condense onto a cooled substrate. In order to create ordered structures on the substrate, the latter is prestructured with inorganic substances, such that the organic molecules preferrably accumulate at specific sites. One specific type of experiments is regarded, where the prestructure consists of periodic gold stripes on a SiO2 substrate. Depending on the geometry of the prestructure and the amount of deposited material, instabilities are observed that hinder the formation of homogeneous ridges on top of the gold stripes. In order to provide a theoretical description of the experiments, a so-called thin film equation is employed that describes the dynamics of the height profile of the layer of deposited molecules on the substrate. Based on this equation, the linear stability of stationary solutions is analyzed that form ridges centered on the gold stripes. This stability analysis is performed employing numerical continuation. Thereby two instabilities are found which are also observed in the experiments. If the amount of deposited molecules is too small, a ridge breaks up into small droplets on the gold stripe. If too much material is deposited, large bulges form which also partly cover the bare substrate between the gold stipes. For both instabilities direct numerical simulations are performed to analyze the full nonlinear dynamics.