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Anomalous diffusion or, more generally, anomalous transport, with nonlinear dependence of the mean-squared displacement on the measurement time, is ubiquitous in nature. It has been observed in processes ranging from microscopic movement of molecules to macroscopic, large-scale paths of migrating birds. Using data from multiple empirical systems, spanning 12 orders of magnitude in length and 8 orders of magnitude in time, we employ a method to detect the individual underlying origins of anomalous diffusion and transport in the data. This method decomposes anomalous transport into three primary effects: long-range correlations (“Joseph effect”), fat-tailed probability density of increments (“Noah effect”), and nonstationarity (“Moses effect”). We show that such a decomposition of real-life data allows us to infer nontrivial behavioral predictions and to resolve open questions in the fields of single-particle tracking in living cells and movement ecology.

Classical option pricing schemes assume that the value of a financial asset follows a geometric Brownian motion (GBM). However, a growing body of studies suggest that a simple GBM trajectory is not an adequate representation for asset dynamics, due to irregularities found when comparing its properties with empirical distributions. As a solution, we investigate a generalisation of GBM where the introduction of a memory kernel critically determines the behaviour of the stochastic process. We find the general expressions for the moments, log-moments, and the expectation of the periodic log returns, and then obtain the corresponding probability density functions using the subordination approach. Particularly, we consider subdiffusive GBM (sGBM), tempered sGBM, a mix of GBM and sGBM, and a mix of sGBMs. We utilise the resulting generalised GBM (gGBM) in order to examine the empirical performance of a selected group of kernels in the pricing of European call options. Our results indicate that the performance of a kernel ultimately depends on the maturity of the option and its moneyness.

Numerous examples for a priori unexpected non-Gaussian behaviour for normal and anomalous diffusion have recently been reported in single-particle tracking experiments. Here, we address the case of non-Gaussian anomalous diffusion in terms of a random-diffusivity mechanism in the presence of power-law correlated fractional Gaussian noise. We study the ergodic properties of this model via examining the ensemble- and time-averaged mean-squared displacements as well as the ergodicity breaking parameter EB quantifying the trajectory-to-trajectory fluctuations of the latter. For long measurement times, interesting crossover behaviour is found as function of the correlation time tau characterising the diffusivity dynamics. We unveil that at short lag times the EB parameter reaches a universal plateau. The corresponding residual value of EB is shown to depend only on tau and the trajectory length. The EB parameter at long lag times, however, follows the same power-law scaling as for fractional Brownian motion. We also determine a corresponding plateau at short lag times for the discrete representation of fractional Brownian motion, absent in the continuous-time formulation. These analytical predictions are in excellent agreement with results of computer simulations of the underlying stochastic processes. Our findings can help distinguishing and categorising certain nonergodic and non-Gaussian features of particle displacements, as observed in recent single-particle tracking experiments.

We consider the emerging dynamics of a separable continuous time random walk (CTRW) in the case when the random walker is biased by a velocity field in a uniformly growing domain. Concrete examples for such domains include growing biological cells or lipid vesicles, biofilms and tissues, but also macroscopic systems such as expanding aquifers during rainy periods, or the expanding Universe. The CTRW in this study can be subdiffusive, normal diffusive or superdiffusive, including the particular case of a Lévy flight. We first consider the case when the velocity field is absent. In the subdiffusive case, we reveal an interesting time dependence of the kurtosis of the particle probability density function. In particular, for a suitable parameter choice, we find that the propagator, which is fat tailed at short times, may cross over to a Gaussian-like propagator. We subsequently incorporate the effect of the velocity field and derive a bi-fractional diffusion-advection equation encoding the time evolution of the particle distribution. We apply this equation to study the mixing kinetics of two diffusing pulses, whose peaks move towards each other under the action of velocity fields acting in opposite directions. This deterministic motion of the peaks, together with the diffusive spreading of each pulse, tends to increase particle mixing, thereby counteracting the peak separation induced by the domain growth. As a result of this competition, different regimes of mixing arise. In the case of Lévy flights, apart from the non-mixing regime, one has two different mixing regimes in the long-time limit, depending on the exact parameter choice: in one of these regimes, mixing is mainly driven by diffusive spreading, while in the other mixing is controlled by the velocity fields acting on each pulse. Possible implications for encounter–controlled reactions in real systems are discussed.

We consider the first-passage problem for N identical independent particles that are initially released uniformly in a finite domain Ω and then diffuse toward a reactive area Γ, which can be part of the outer boundary of Ω or a reaction centre in the interior of Ω. For both cases of perfect and partial reactions, we obtain the explicit formulas for the first two moments of the fastest first-passage time (fFPT), i.e., the time when the first out of the N particles reacts with Γ. Moreover, we investigate the full probability density of the fFPT. We discuss a significant role of the initial condition in the scaling of the average fFPT with the particle number N, namely, a much stronger dependence (1/N and 1/N² for partially and perfectly reactive targets, respectively), in contrast to the well known inverse-logarithmic behaviour found when all particles are released from the same fixed point. We combine analytic solutions with scaling arguments and stochastic simulations to rationalise our results, which open new perspectives for studying the relevance of multiple searchers in various situations of molecular reactions, in particular, in living cells.

Employing extensive Monte Carlo computer simulations, we investigate in detail the properties of multichain adsorption of charged flexible polyelectrolytes (PEs) onto oppositely charged spherical nanoparticles (SNPs). We quantify the conditions of critical adsorption-the phase-separation curve between the adsorbed and desorbed states of the PEs-as a function of the SNP surface-charge density and the concentration of added salt. We study the degree of fluctuations of the PE-SNP electrostatic binding energy, which we use to quantify the emergence of the phase subtransitions, including a series of partially adsorbed PE configurations. We demonstrate how the phase-separation adsorption-desorption boundary shifts and splits into multiple subtransitions at low-salt conditions, thereby generalizing and extending the results for critical adsorption of a single PE onto the SNP. The current findings are relevant for finite concentrations of PEs around the attracting SNP, such as the conditions for PE adsorption onto globular proteins carrying opposite electric charges.

Brownian motion and viscoelastic anomalous diffusion in homogeneous environments are intrinsically Gaussian processes. In a growing number of systems, however, non-Gaussian displacement distributions of these processes are being reported. The physical cause of the non-Gaussianity is typically seen in different forms of disorder. These include, for instance, imperfect "ensembles" of tracer particles, the presence of local variations of the tracer mobility in heteroegenous environments, or cases in which the speed or persistence of moving nematodes or cells are distributed. From a theoretical point of view stochastic descriptions based on distributed ("superstatistical") transport coefficients as well as time-dependent generalisations based on stochastic transport parameters with built-in finite correlation time are invoked. After a brief review of the history of Brownian motion and the famed Gaussian displacement distribution, we here provide a brief introduction to the phenomenon of non-Gaussianity and the stochastic modelling in terms of superstatistical and diffusing-diffusivity approaches.

The Cattaneo or telegrapher's equation describes the crossover from initial ballistic to normal diffusion. Here we study and survey time-fractional generalisations of this equation that are shown to produce the crossover of the mean squared displacement from superdiffusion to subdiffusion. Conditional solutions are derived in terms of Fox H-functions and the dth-order moments as well as the diffusive flux of the different models are derived. Moreover, the concept of the distribution-like is proposed as an alternative to the probability density function.

Quorum-sensing bacteria in a growing colony of cells send out signalling molecules (so-called “autoinducers”) and themselves sense the autoinducer concentration in their vicinity. Once—due to increased local cell density inside a “cluster” of the growing colony—the concentration of autoinducers exceeds a threshold value, cells in this clusters get “induced” into a communal, multi-cell biofilm-forming mode in a cluster-wide burst event. We analyse quantitatively the influence of spatial disorder, the local heterogeneity of the spatial distribution of cells in the colony, and additional physical parameters such as the autoinducer signal range on the induction dynamics of the cell colony. Spatial inhomogeneity with higher local cell concentrations in clusters leads to earlier but more localised induction events, while homogeneous distributions lead to comparatively delayed but more concerted induction of the cell colony, and, thus, a behaviour close to the mean-field dynamics. We quantify the induction dynamics with quantifiers such as the time series of induction events and burst sizes, the grouping into induction families, and the mean autoinducer concentration levels. Consequences for different scenarios of biofilm growth are discussed, providing possible cues for biofilm control in both health care and biotechnology.

Wealth and income distributions are known to feature country-specific Pareto exponents for their long power-law tails. To propose a rationale for this, we introduce an agent-based dynamic model and use Monte Carlo simulations to unveil the wealth distributions in closed and open economical systems. The standard money-exchange scenario is supplemented with the position-exchange agent dynamics that vitally affects the Pareto law. Specifically, in closed systems with position-exchange dynamics the power law changes to an exponential shape, while for open systems with traps the Pareto law remains valid.

Fractional Brownian motion (FBM) is a Gaussian stochastic process with stationary, long-time correlated increments and is frequently used to model anomalous diffusion processes. We study numerically FBM confined to a finite interval with reflecting boundary conditions. The probability density function of this reflected FBM at long times converges to a stationary distribution showing distinct deviations from the fully flat distribution of amplitude 1/L in an interval of length L found for reflected normal Brownian motion. While for superdiffusion, corresponding to a mean squared displacement (MSD) 〈X² (t)〉 ⋍ tᵅ with 1 < α < 2, the probability density function is lowered in the centre of the interval and rises towards the boundaries, for subdiffusion (0 < α < 1) this behaviour is reversed and the particle density is depleted close to the boundaries. The MSD in these cases at long times converges to a stationary value, which is, remarkably, monotonically increasing with the anomalous diffusion exponent α. Our a priori surprising results may have interesting consequences for the application of FBM for processes such as molecule or tracer diffusion in the confines of living biological cells or organelles, or other viscoelastic environments such as dense liquids in microfluidic chambers.

The power spectral density (PSD) of any time-dependent stochastic processX (t) is ameaningful feature of its spectral content. In its text-book definition, the PSD is the Fourier transform of the covariance function of X-t over an infinitely large observation timeT, that is, it is defined as an ensemble-averaged property taken in the limitT -> infinity. Alegitimate question is what information on the PSD can be reliably obtained from single-trajectory experiments, if one goes beyond the standard definition and analyzes the PSD of a single trajectory recorded for a finite observation timeT. In quest for this answer, for a d-dimensional Brownian motion (BM) we calculate the probability density function of a single-trajectory PSD for arbitrary frequency f, finite observation time T and arbitrary number k of projections of the trajectory on different axes. We show analytically that the scaling exponent for the frequency-dependence of the PSD specific to an ensemble of BM trajectories can be already obtained from a single trajectory, while the numerical amplitude in the relation between the ensemble-averaged and single-trajectory PSDs is afluctuating property which varies from realization to realization. The distribution of this amplitude is calculated exactly and is discussed in detail. Our results are confirmed by numerical simulations and single-particle tracking experiments, with remarkably good agreement. In addition we consider a truncated Wiener representation of BM, and the case of a discrete-time lattice random walk. We highlight some differences in the behavior of a single-trajectory PSD for BM and for the two latter situations. The framework developed herein will allow for meaningful physical analysis of experimental stochastic trajectories.

The emerging diffusive dynamics in many complex systems show a characteristic crossover behaviour from anomalous to normal diffusion which is otherwise fitted by two independent power-laws. A prominent example for a subdiffusive–diffusive crossover are viscoelastic systems such as lipid bilayer membranes, while superdiffusive–diffusive crossovers occur in systems of actively moving biological cells. We here consider the general dynamics of a stochastic particle driven by so-called tempered fractional Gaussian noise, that is noise with Gaussian amplitude and power-law correlations, which are cut off at some mesoscopic time scale. Concretely we consider such noise with built-in exponential or power-law tempering, driving an overdamped Langevin equation (fractional Brownian motion) and fractional Langevin equation motion. We derive explicit expressions for the mean squared displacement and correlation functions, including different shapes of the crossover behaviour depending on the concrete tempering, and discuss the physical meaning of the tempering. In the case of power-law tempering we also find a crossover behaviour from faster to slower superdiffusion and slower to faster subdiffusion. As a direct application of our model we demonstrate that the obtained dynamics quantitatively describes the subdiffusion–diffusion and subdiffusion–subdiffusion crossover in lipid bilayer systems. We also show that a model of tempered fractional Brownian motion recently proposed by Sabzikar and Meerschaert leads to physically very different behaviour with a seemingly paradoxical ballistic long time scaling.

We develop a mathematical model considering a random walker with long-range hops on arbitrary graphs. The random multi-hopper can jump to any node of the graph from an initial position, with a probability that decays as a function of the shortest-path distance between the two nodes in the graph. We consider here two decaying functions in the form of Laplace and Mellin transforms of the shortest-path distances. We prove that when the parameters of these transforms approach zero asymptotically, the hitting time in the multi-hopper approaches the minimum possible value for a normal random walker. We show by computational experiments that the multi-hopper explores a graph with clusters or skewed degree distributions more efficiently than a normal random walker. We provide computational evidences of the advantages of the random multi-hopper model with respect to the normal random walk by studying deterministic, random and real-world networks.

How ergodic is diffusion under harmonic confinements? How strongly do ensemble- and time-averaged displacements differ for a thermally-agitated particle performing confined motion for different initial conditions? We here study these questions for the generic Ornstein-Uhlenbeck (OU) process and derive the analytical expressions for the second and fourth moment. These quantifiers are particularly relevant for the increasing number of single-particle tracking experiments using optical traps. For a fixed starting position, we discuss the definitions underlying the ensemble averages. We also quantify effects of equilibrium and nonequilibrium initial particle distributions onto the relaxation properties and emerging nonequivalence of the ensemble- and time-averaged displacements (even in the limit of long trajectories). We derive analytical expressions for the ergodicity breaking parameter quantifying the amplitude scatter of individual time-averaged trajectories, both for equilibrium and outof-equilibrium initial particle positions, in the entire range of lag times. Our analytical predictions are in excellent agreement with results of computer simulations of the Langevin equation in a parabolic potential. We also examine the validity of the Einstein relation for the ensemble- and time-averaged moments of the OU-particle. Some physical systems, in which the relaxation and nonergodic features we unveiled may be observable, are discussed.

A considerable number of systems have recently been reported in which Brownian yet non-Gaussian dynamics was observed. These are processes characterised by a linear growth in time of the mean squared displacement, yet the probability density function of the particle displacement is distinctly non-Gaussian, and often of exponential(Laplace) shape. This apparently ubiquitous behaviour observed in very different physical systems has been interpreted as resulting from diffusion in inhomogeneous environments and mathematically represented through a variable, stochastic diffusion coefficient. Indeed different models describing a fluctuating diffusivity have been studied. Here we present a new view of the stochastic basis describing time dependent random diffusivities within a broad spectrum of distributions. Concretely, our study is based on the very generic class of the generalised Gamma distribution. Two models for the particle spreading in such random diffusivity settings are studied. The first belongs to the class of generalised grey Brownian motion while the second follows from the idea of diffusing diffusivities. The two processes exhibit significant characteristics which reproduce experimental results from different biological and physical systems. We promote these two physical models for the description of stochastic particle motion in complex environments.

We study by Monte Carlo simulations a kinetic exchange trading model for both fixed and distributed saving propensities of the agents and rationalize the person and wealth distributions. We show that the newly introduced wealth distribution – that may be more amenable in certain situations – features a different power-law exponent, particularly for distributed saving propensities of the agents. For open agent-based systems, we analyze the person and wealth distributions and find that the presence of trap agents alters their amplitude, leaving however the scaling exponents nearly unaffected. For an open system, we show that the total wealth – for different trap agent densities and saving propensities of the agents – decreases in time according to the classical Kohlrausch–Williams–Watts stretched exponential law. Interestingly, this decay does not depend on the trap agent density, but rather on saving propensities. The system relaxation for fixed and distributed saving schemes are found to be different.

The biomolecule is among the most important building blocks of biological systems, and a full understanding of its function forms the scaffold for describing the mechanisms of higher order structures as organelles and cells. Force is a fundamental regulatory mechanism of biomolecular interactions driving many cellular processes. The forces on a molecular scale are exactly in the range that can be manipulated and probed with single molecule force spectroscopy. The natural environment of a biomolecule is inside a living cell, hence, this is the most relevant environment for probing their function. In vivo studies are, however, challenged by the complexity of the cell. In this review, we start with presenting relevant theoretical tools for analyzing single molecule data obtained in intracellular environments followed by a description of state-of-the art visualization techniques. The most commonly used force spectroscopy techniques, namely optical tweezers, magnetic tweezers, and atomic force microscopy, are described in detail, and their strength and limitations related to in vivo experiments are discussed. Finally, recent exciting discoveries within the field of in vivo manipulation and dynamics of single molecule and organelles are reviewed.

We investigate both analytically and by computer simulations the ensemble- and time-averaged, nonergodic, and aging properties of massive particles diffusing in a medium with a time dependent diffusivity. We call this stochastic diffusion process the (aging) underdamped scaled Brownian motion (UDSBM). We demonstrate how the mean squared displacement (MSD) and the time-averaged MSD of UDSBM are affected by the inertial term in the Langevin equation, both at short, intermediate, and even long diffusion times. In particular, we quantify the ballistic regime for the MSD and the time-averaged MSD as well as the spread of individual time-averaged MSD trajectories. One of the main effects we observe is that, both for the MSD and the time-averaged MSD, for superdiffusive UDSBM the ballistic regime is much shorter than for ordinary Brownian motion. In contrast, for subdiffusive UDSBM, the ballistic region extends to much longer diffusion times. Therefore, particular care needs to be taken under what conditions the overdamped limit indeed provides a correct description, even in the long time limit. We also analyze to what extent ergodicity in the Boltzmann-Khinchin sense in this nonstationary system is broken, both for subdiffusive and superdiffusive UDSBM. Finally, the limiting case of ultraslow UDSBM is considered, with a mixed logarithmic and power-law dependence of the ensemble-and time-averaged MSDs of the particles. In the limit of strong aging, remarkably, the ordinary UDSBM and the ultraslow UDSBM behave similarly in the short time ballistic limit. The approaches developed here open ways for considering other stochastic processes under physically important conditions when a finite particle mass and aging in the system cannot be neglected.

We investigate the ergodic properties of a random walker performing (anomalous) diffusion on a random fractal geometry. Extensive Monte Carlo simulations of the motion of tracer particles on an ensemble of realisations of percolation clusters are performed for a wide range of percolation densities. Single trajectories of the tracer motion are analysed to quantify the time averaged mean squared displacement (MSD) and to compare this with the ensemble averaged MSD of the particle motion. Other complementary physical observables associated with ergodicity are studied, as well. It turns out that the time averaged MSD of individual realisations exhibits non-vanishing fluctuations even in the limit of very long observation times as the percolation density approaches the critical value. This apparent non-ergodic behaviour concurs with the ergodic behaviour on the ensemble averaged level. We demonstrate how the non-vanishing fluctuations in single particle trajectories are analytically expressed in terms of the fractal dimension and the cluster size distribution of the random geometry, thus being of purely geometrical origin. Moreover, we reveal that the convergence scaling law to ergodicity, which is known to be inversely proportional to the observation time T for ergodic diffusion processes, follows a power-law BT� h with h o 1 due to the fractal structure of the accessible space. These results provide useful measures for differentiating the subdiffusion on random fractals from an otherwise closely related process, namely, fractional Brownian motion. Implications of our results on the analysis of single particle tracking experiments are provided.

First-passage properties in general, and the mean first-passage time (MFPT) in particular, are widely used in the context of diffusion-limited processes. Real processes are not always purely Brownian: in the last few years, non-Brownian behaviors have been observed in an increasing number of systems. Especially single particle experiments in living cells provide striking examples for systems in which non-Brownian behavior of subdiffusive kind has been repeatedly observed experimentally. Here we present a method based on first-passage properties to gain more detailed insight into the actual physical processes underlying the anomalous diffusion behavior, and to probe the environment in which this diffusion process evolves. This method allows us to discriminate between three prominent models of subdiffusion: continuous time random walks, diffusion on fractals, and fractional Brownian motion. We also investigate the search efficiency of random walks on discrete networks for a specific target. We show how to compute first-passage properties on those networks in order to optimize the search process, as well as general bounds on the global mean first-passage time (GMFPT). Using those results, we estimate the impact on the search efficiency of several parameters, namely the target connectivity, the target motion, or the network topology. Eigenschaften der First Passage, dem erstmaligen Überschreiten eines festgelegten Grenzwertes, und die Mean First Passage Time (MFPT) im besonderen sind häufig benutzte Methoden zur Charakterisierung von stochastischen Prozessen. Dabei sind reale Prozesse nicht immer Brownscher Natur: in den letzten Jahren wurde nicht-Brownsches Verhalten in einer zunehmenden Anzahl von Systemen beobachtet. Insbesondere Single particle tracking Experimente in lebenden Zellen sind ein frappierende Beispiel, wo nicht-Brownsches Verhalten in der Form von Subdiffusion vielfach beobachtet wird. Wir schlagen hier eine Methode vor, die auf First Passage Eigenschaften beruht, um weitergehende physikalische Informationen über den zugrunde liegenden stochastischen Prozess sowie die physische Umgebung, in der der Diffusionsprozeß abläuft, zu gewinnen. Diese Methode erlaubt es uns, zwischen den drei prominenten Modellen der Subdiffusion zu unterscheiden: Continuous Time Random Walks, Diffusion auf Fraktalen und fractional Brownian motion. Wir untersuchen außerdem Random Walks auf diskreten Netzwerken. Wir zeigen, wie via First Passage Eigenschaften auf diesen Netzwerken die diffusive Suche nach einem bestimmten Ziel optimiert werden kann. Wir erhalten allgemeine Schranken für die globale MFPT. Mit diesen Ergebnissen schätzen wir die Effizienz des Suchprozesses ab, wenn verschiedene Parameter wie die Ziel-Konnektivität, Bewegung des Ziels oder Veränderungen der Netzwerktopologie stattfinden.

In this thesis we present a case study of EcoRV restriction enzymes involved in vital cellular defence. We show that due to its so far elusive propensity to an inactive state the enzyme avoids non-specific binding and remains well-distributed in the bulk cytoplasm of the cell. Despite the reduced volume exploration of subdiffusion processes, the low activity of the enzyme surprisingly guarantees a high efficiency of the enzyme. Analysing different stochastic processes for subdiffusion we demonstrate that reduced non-specific binding are beneficial for efficient subdiffusive enzyme activity even in relatively small bacteria cells. Diese Dissertation stellt eine Fallstudie des Restriktionsenzyms EcoRV, welches in der Zellverteidigung eine wichtige Rolle spielt, vor. Es wird gezeigt, dass die Eigenschaft von EcoRV, bevorzugt eine inaktiven Konformation anzunehmen und daduch nichtspezifisches Anbinden an die zelluläre DNA zu vermeiden, eine gleichmäßigere Verbreitung in der Zelle bewirkt. Trotz der reduzierten Volumenausbreitung subdiffusiver Prozesse garantiert diese Eigenschaft eine unerwartet hohe Effizienz des Enzyms. Die Analyse unterschiedlicher stochastischer Prozesse zeigt, dass reduziertes nichtspezifisches Binden die Effizienz subdiffundierender Enzyme auch in relativ kleinen Bakterienzellen erhöht.

With the recent advances in nanotribology and single molecule experiments, it has been revealed that at nanoscopic scales, conventional laws do not apply due to adhesion and thermal fluctuations. This work focuses on the friction dynamics of nano-soft matter. In the first part of this work, the friction of single peptides on hydrophobic and hydrophilic surfaces and in peptide bundles were put under investigation by using extended Molecular Dynamics simulations and stochastic Fokker-Planck equation. We show that the friction force is product of the velocity, hydrogen bond number and a hydrogen bond friction coefficient, which itself depends on the collective nature of bond ruptures. Secondly, in a related project, the transient compressible Navier-Stokes equation was solved analytically to obtain frequency-dependent friction response functions for spherical, cylindrical and planar geometries in unbounded fluid, with finite slip length. Neue Entwicklungen in der Nanotribologie und Einzelmolekülexperimente zeigen, dass herkömmliche Gesetze auf der Nano-Skala aufgrund von Haftung und thermischen Fluktuationen nicht mehr gelten. Der Schwerpunkt dieser Arbeit ist die Reibungsdynamik nanoskopischer, weicher Materie. Im 1. Teil der Arbeit wird die Reibung einzelner Peptide auf hydrophoben und hydrophilen Oberflächen und in Peptid-Bündeln untersucht. Hierfür werden Molekulardynamik-Simulationen und eine stochastische Analyse basierend auf der Fokker-Planck-Gleichung verwendet. Es wird gezeigt, dass die Reibungskraft proportional zur Geschwindigkeit, Wasserstoffbrückenanzahl und dem Wasserstoffbrücken-Reibungskoeffizienten ist, der vom kollektiven Charakter der Bindungsbrüche abhängt. Im 2. Teil der Arbeit wird die kompressible Navier-Stokes-Gleichung analytisch gelöst und die frequenzabhängige Reibungs-Antwortfunktion von Kugeln, Zylindern und Ebenen mit Slip-Länge berechnet.

In this work a fiber optical interferometric magnetometer with a piezo based 4-axis nano-positioning- system was designed and build up. The system was designed to work in a 3He cryostat and under high magnetic fields. Angle and temperature dependent magnetization measurements were performed on electrons forming a pseudo-spin-system in an AlAs/GaAs quantum well. We found convincing evidence for strong electron-electron-interaction effects. Additionally, nanomechanical bridges of ferromagnet /semiconductor hybrid structures were produced. These showed mechanical oscillations with high eigenfrequency of 0.5 GHz. By performing power dependent microwave spectroscopy mechanical and magnetic coupling phenomena were investigated. Im Rahmen dieser Arbeit wurde ein faseroptisches interferometrisches Magnetometer mit einem piezobasierten 4-achsigen Nanopositioniersystem konzipiert und aufgebaut. Das System ist für den Betrieb in einem 3He-Kryostaten und unter hohen Magnetfeldern konzipiert. Es wurden winkel- und temperaturabhängige Magnetisierungsmessungen an Elektronen in AlAs/GaAs Quantentrögen durchgeführt, die ein Pseudo-Spin-System bilden. Dabei wurden deutliche Hinweise auf starke Elektron-Elektronwechselwirkungseffekte gefunden. Zusätzlich wurden Nanobrücken aus Ferromagnet/Halbleiterhybridstrukturen hergestellt. Diese zeigen mechanische Schwingungsanregungen mit hohen Eigenfrequenzen von 0.5 GHz. Durch leistungsabhängige Mikrowellenspektroskopie wurden die mechanischen und magnetischen Kopplungsphänomene untersucht.

This research work focuses on the structural changes in lamellar poly(styrene-b-butadiene) thin films during solvent vapor treatment. The slightly polybutadiene-selective cyclohexane and slightly polystyrene-selective cyclohexanone are used as solvent vapors. In situ grazing-incidence small angle X-ray scattering (GISAXS) is carried out and the fast structural changes are well followed. The lamellae and film swelling, the improvement of the long-range order and the phase transition are observed. We attribute the observations to the change of the glass transition temperature, the Flory-Huggins segment-segment interaction parameter and/or the volume fraction of one block. Diese Arbeit behandelt die strukturellen Änderungen in dünnen lamellaren Filmen aus Poly(Styrol-b-Butadien) während der Lösungsmitteldampfbehandlung. Zur Dampfbehandlung wurden das für Polybutadien schwach selektive Cyclohexan und das für Polystyrol schwach selektive Cyclohexanon verwendet. Mit in situ Kleinwinkel-Röntgenstreuung unter streifendem Einfall (grazing-incidence small-angle x-ray scattering, GISAXS) konnten schnell ablaufende strukturelle Änderungen verfolgt werden. Das Aufquellen der Lamellen und des Filmes, die Erhöhung der weitreichenden Ordnung und der Phasenübergang wurden festgestellt. Wir ordnen diese Beobachtungen der Änderung der Glasübergangstemperatur, dem Flory-Huggins Segment-Segment Wechselwirkungsparameter und/oder dem Volumenanteil eines Blockes zu.

Als halbmetallischer Ferrimagnet mit einer Curie-Temperatur von 860K ist Magnetit interessant für die Spinelektronik. Mit Lasermolekularstrahlepitaxie wurden Magnetitfilme unterschiedlicher Orientierung hergestellt. Wie bei vielen metallischen Systemen skaliert deren anomale Hall- und Längsleitfähigkeit mit einem Exponenten von 5/3, obwohl ein anderer Transportmechanismus vorliegt. Durch Magnetotransportmessungen an Magnetitfilmen sowie an auf diesen Filmen basierenden Tunnelkontakten für verschiedene Magnetfeldorientierungen wurden magnetische Anisotropiekonstanten bestimmt. Diese Methode erlaubt im Vergleich zur volumensensitiven ferromagnetischen Resonanz Zugang zu den Anisotropieeigenschaften von ferromagnetischen Mikro- und Nanostrukturen. Mit dem Ziel einer Spininjektion in einen Halbleiter, wurde Magnetit erstmals epitaktisch auf ZnO gewachsen und Experimente zum Nachweis einer Spininjektion von Magnetit in ZnO vorgestellt. The halfmetallic ferrimagnet magnetite with a high Curie temperature of 860K is interesting for spintronics. Magnetite films with different orientations were grown by laser molecular beam epitaxy. Like many metallic systems they show a scaling between Hall conductivity and longitudinal conductivity with an exponent of 5/3, though a different transport mechanism takes place. By magnetotransort measurements performed on magnetite films as well as on magnetite based tunnel junctions with different magnetic field orientations magnetic anisotropy constants were determined. This method offers in comparison to the volume sensitive ferromagnetic resonance access to anisotropy properties of ferromagnetic micro and nano structures. With the objective of a spin injection into a semiconductor magnetite has been grown epitaxially on ZnO for the first time and experiments for a detection of spin injection from magnetite into ZnO are presented.

The interfacial dynamics of polymers and water is theoretically examined. The coupling between thermal fluctuations and the hydrodynamics at a liquid/solid boundary leads to a repulsion of polymers from the interface. Starting from a Fokker-Planck equation, we give scaling relations for this repulsion which are confirmed by computer simulations. The dynamics of end grafted DNA molecules in electric fields is studied and possible applications for biosensing purposes are discussed. In atomistic simulations, the dynamics and structure of water at hydrophilic and hydrophobic substrates and the hydrodynamic boundary condition at these surfaces is analyzed. Die Dynamik von Polymeren und Wasser an Grenzflächen wird theoretisch untersucht. Die Kopplung zwischen thermischen Fluktuationen und der Hydrodynamik an einer fest/flüssig Grenzfläche führt zu einer Abstoßung von Polymeren von der Grenzfläche. Ausgehend von einer Fokker-Planck Gleichung werden Skalenvorhersagen für diese Repulsion aufgestellt und durch Computersimulationen verifiziert. Die Dynamik von an Oberflächen verankerten DNA Molekülen in elektrischen Feldern wird betrachtet und mögliche Anwendungen als Biosensor diskutiert. In atomistischen Simulationen werden die Dynamik und Struktur von Wasser and hydrophilen und hydrophoben Substraten, sowie die hydrodynamische Randbedingung an diesen Oberflächen untersucht.

The dynamics of single polymers at solid-liquid interfaces and their response to external fields are discussed using a wide arsenal of theoretical and numerical tools. Polymer friction on solid surfaces is analyzed with coarse-grained Brownian dynamics polymer models and atomistic molecular dynamics simulations; a model of how to measure such friction with an atomic force microscope is proposed. The simulations show pronounced differences between hydrophobic and hydrophilic surfaces. The adsorption of polymers far from equilibrium is studied using Brownian dynamics simulations. External flow fields generated by a lateral pulling force or by shear flow tend to enhance adsorption. On the other hand, surface corrugations reduce adsorption. Existing polymer elasticity models are extended by incorporating microscopic stretching, bending, and torsional moduli as obtained from ab initio quantum chemistry. Die Dynamik einzelner Polymere an Fest-Flüssig-Grenzflächen und ihr Antwortverhalten auf externe Felder werden mit theoretischen und numerischen Methoden untersucht. Die Reibung von Polymeren an festen Oberflächen wird durch grobgekörnte Brownsche Dynamik- und atomistisch aufgelöste Molekulardynamiksimulationen analysiert; ein Modell zur Messung solcher Reibung mit Hilfe eines Atomkraftmikroskops wird vorgeschlagen. Die Simulationen zeigen ausgeprägte Unterschiede zwischen hydrophoben und hydrophilen Oberflächen auf. Die Nichtgleichgewichtsadsorption von Polymeren wird anhand Brownscher Dynamiksimulationen studiert. Externe Strömungsfelder, wie sie durch eine seitliche Ziehkraft oder Scherfluss auftreten, verstärken Adsorption. Andererseits wird Adsorption durch Oberflächenkorrugation verringert. Elastizitätsmodelle für Polymere werden durch quantenchemische Berechnung der mikroskopischen Streck-, Biege- und Torsionsmoduln verbessert.