Limiting Properties of a Continuous Local Mean-Field Interacting Spin System : Hydrodynamic Limit, Propagation of Chaos, Energy Landscape and Large Deviations

A key interest in the study of interacting spin systems is the rigorous analysis of the macroscopic dynamical behaviour of systems that are described by their microscopic evolution. In this dissertation, we investigate unbounded spin systems where the microscopic evolution is modelled by stochastic differential equations (SDE). To each site of the discrete d-dimensional torus a spin is associated. The spins are distributed on the whole real line and evolve randomly according to the SDEs. The interaction between the spins is of local mean-field type, a long-range spatially variable interaction. The strength of the interaction between two spins depends on the difference of their positions on the torus. We aim to understand rigorously the time evolution of random variables as the size of the system increases. We prove in Chapter I the convergence of (space and spin dependent) empirical processes under proper rescaling to the classical solution of a nonlinear partial differential equation (PDE). This PDE is called hydrodynamic equation. We use the relative entropy method, to show this hydrodynamic limit result. To apply this method, we need to prove the existence of a classical solution of the hydrodynamic equation, which is non-linear and non-elliptic. In Chapter II we prove the propagation of chaos property of the system. We show that finitely many tagged spins are in the limit mutually independent. They evolve in the limit according to stochastic differential equations, without an interaction term. Instead (compared to the original SDEs), there is a term involving the solution of the hydrodynamic equation. In Chapter III we derive large deviation principles for the corresponding equilibrium system. We look at random variables that are distributed according to the invariant measure of the stochastic differential equation. For the empirical measure, defined by these random variables, we derive large deviation principles. We use a generalisation of Varadhan’s lemma that is stated and proven in Appendix C. In Chapter IV we analyse the landscape of the rate function of one of the equilibrium large deviation principles. We interpret this rate function as energy of the system in the limit. This is motivated by the fact that the hydrodynamic equation is the Wasserstein gradient flow of this rate function. We determine minima, critical values, bifurcation properties and lowest paths between minima. Finally in Chapter V we prove a dynamical large deviation principle for the empirical processes and the empirical measures. We derive different representations of the rate functions. By one of these representations it becomes obvious that it is exponentially unlikely that empirical processes deviate from the deterministic flow. In this chapter we allow the system to be more general, e.g. it can contain a random environment and a more general diffusion coefficient. The main distinctive features of the spin system considered in this dissertation are the relevance of the spatially fixed positions of the spins and the possibility of unbounded spins. The spatial positions of the spins affect the interaction and the initial distributions. Therefore new approaches in the proofs are necessary, in particular compared to mean field models. All these results can be used in the future to study long time phenomena like tunnelling and metastability

On the Calculation of Solid-Fluid Contact Angles from Molecular Dynamics

A methodology for the determination of the solid-fluid contact angle, to be employed within molecular dynamics (MD) simulations, is developed and systematically applied. The calculation of the contact angle of a fluid drop on a given surface, averaged over an equilibrated MD trajectory, is divided in three main steps: (i) the determination of the fluid molecules that constitute the interface, (ii) the treatment of the interfacial molecules as a point cloud data set to define a geometric surface, using surface meshing techniques to compute the surface normals from the mesh, (iii) the collection and averaging of the interface normals collected from the post-processing of the MD trajectory. The average vector thus found is used to calculate the Cassie contact angle (i.e., the arccosine of the averaged normal z-component). As an example we explore the effect of the size of a drop of water on the observed solid-fluid contact angle. A single coarse-grained bead representing two water molecules and parameterized using the SAFT-γ Mie equation of state (EoS) is employed, meanwhile the solid surfaces are mimicked using integrated potentials. The contact angle is seen to be a strong function of the system size for small nano-droplets. The thermodynamic limit, corresponding to the infinite size (macroscopic) drop is only truly recovered when using an excess of half a million water coarse-grained beads and/or a drop radius of over 26 nm

Quantitative analysis of backscattered‐electron contrast in scanning electron microscopy

Backscattered-electron scanning electron microscopy (BSE-SEM) imaging is a valuable technique for materials characterisation because it provides information about the homogeneity of the material in the analysed specimen and is therefore an important technique in modern electron microscopy. However, the information contained in BSE-SEM images is up to now rarely quantitatively evaluated. The main challenge of quantitative BSE-SEM imaging is to relate the measured BSE intensity to the backscattering coefficient η and the (average) atomic number Z to derive chemical information from the BSE-SEM image. We propose a quantitative BSE-SEM method, which is based on the comparison of Monte–Carlo (MC) simulated and measured BSE intensities acquired from wedge-shaped electron-transparent specimens with known thickness profile. The new method also includes measures to improve and validate the agreement of the MC simulations with experimental data. Two different challenging samples (ZnS/Zn(O$_x$S$_{1–x}$)/ZnO/Si-multilayer and PTB7/PC$_{71}$BM-multilayer systems) are quantitatively analysed, which demonstrates the validity of the proposed method and emphasises the importance of realistic MC simulations for quantitative BSE-SEM analysis. Moreover, MC simulations can be used to optimise the imaging parameters (electron energy, detection-angle range) in advance to avoid tedious experimental trial and error optimisation. Under optimised imaging conditions pre-determined by MC simulations, the BSE-SEM technique is capable of distinguishing materials with small composition differences

KNEE JOINT LOADING IN GRADED WALKING AS A FUNCTION OF STEP LENGTH AND STEP FREQUENCY

The purpose of this study was to determine the knee joint loading during uphill and downhill walking as a function of step length and step frequency. Twelve subjects were filmed and their ground reaction forces measured during uphill and downhill walking on a ramp at 18" to the horizontal at step lengths of 46, 58 and 69 cm and step frequencies of 1.33, 1.67 and 2.00 Hz, respectively. 20 inverse dynamics were used to calculate knee joint forces, moments and power. In general, knee joint loading increases with both longer steps and higher step frequencies. Most of the differences are statistically significant. The results show that step length and step frequency affects knee joint loading significantly and substantially. Thus knee joint loading can be controlled by regulating these two parameters. This is important when trying to optimise the stimulation of knee joint structures

Quantification of the thickness of TEM samples by low-energy scanning transmission electron microscopy

Precise knowledge of the local sample thickness is often required for quantitative scanning (transmission) electron microscopy (STEM). The local sample thickness can be determined by the comparison of measured intensities from high-angle annular dark-field (HAADF)-STEM at low energies (<30 keV) with Monte-Carlo simulations. However, a suitable choice of the scattering cross-section (CS) used in the simulations is necessary to gain reliable thickness results. In this work, simulations using different CS, including the Screened Rutherford CS and different Mott CSs, were performed. The results were then compared with measurements on samples with known thickness and composition, for which an SEM equipped with a STEM detector was used. In most cases, the Screened Rutherford CS describes the experiment better than other CSs

Electron-beam broadening in electron microscopy by solving the electron transport equation

Scanning transmission electron microscopy (STEM) and scanning electron microscopy (SEM) are prominent techniques for the structural characterization of materials. STEM in particular provides high spatial resolution down to the sub-ångström range. The spatial resolution in STEM and SEM is ultimately limited by the electron-beam diameter provided by the microscope\u27s electron optical system. However, the resolution is frequently degraded by the interaction between electron and matter leading to beam broadening, which depends on the thickness of the analyzed sample. Numerous models are available to calculate beam broadening. However, most of them neglect the energy loss of the electrons and large-angle scattering. These restrictions severely limit the applicability of the approaches for large sample thicknesses in STEM and SEM. In this work, we address beam broadening in a more general way. We numerically solve the electron transport equation without any simplifications, and take into account energy loss along the electron path. For this purpose, we developed the software package CeTE (Computation of electron Transport Equation). We determine beam broadening, energy deposition, and the interaction volume of the scattered electrons in homogeneous matter. The calculated spatial and angular distributions of electrons are not limited to forward scattering and small sample thicknesses. We focus on low electron energies of 30 keV and below, where beam broadening is particularly pronounced. These electron energies are typical for SEM and STEM in scanning electron microscopes

THE ACCURACY OF 3D KINETIC AND KINEMATIC DATA USED FOR JOINT LOADING ANALYSIS IN SKIING AND SNOWBOARDING

Almost 50% of all skiing accidents in men and more then 70% of all skiing accidents in women concerned the lower extremities. In snowboarding about a third of all accidents concerned the lower extremities in both men and women (Burtscher et al., 2003). These high percentages afford systematic research to determine joint loading on the lower extremities in skiing and snowboarding. However, so far only rough estimations of joint loading are reported (van den Bogert et al., 1999; Quinn & Mote, 1992). More precise values would be possible by inverse dynamic analyses. These require representative 3D kinetic and kinematic data which serve as input for the inverse dynamic model to calculate the loading parameters. Therefore, the goal of this presentation is to give an overview and validation of the methodological procedures used in this study to collect and analyse 3D kinetic and kinematic data to determine the loading parameters