Particle-particle interactions are of paramount importance in every multi-body system as they determine the collective behaviour and coupling strength. Many well-known interactions like electro-static, van der Waals or screened Coulomb, decay exponentially or with negative powers of the particle spacing r. Similarly, hydrodynamic interactions between particles undergoing Brownian motion decay as 1/r in bulk, and are assumed to decay in small channels. Such interactions are ubiquitous in biological and technological systems. Here I confine multiple particles undergoing Brownian motion in narrow, microfluidic channels and study their coupling through hydrodynamic interactions. Our experiments show that the hydrodynamic particle-particle interactions are distance-independent in these channels. We also show that these interactions affect actively propelled particles via electrophoresis or gravity, resulting in non-linear transport phenomena. These findings are of fundamental importance for understanding transport of dense mixtures of particles or molecules through finite length, water-filled channels or pore networks.

This work is a study of the relationship between Brownian motion and elementary, linear partial differential equations. In the text, I have shown that Brownian motion is a Markov process, and that Brownian motion itself, and certain Stochastic processes involving Brownian motion are also martingales. In particular, Dynkin's formula for Brownian motion was shown. Using Dynkin's formula and Brownian motion, I then constructed solutions for the classical Dirichlet problem and the heat equation, given by Δu=0 and ut= 1/2Δu+g, respectively. I have shown that the bounded solution is unique if Brownian motion will always exit the domain of the function once it has started at a point in the domain. The heat equation also has a unique bounded solution.

"November, 2001" / Includes bibliographical references (leaves 229-246) / xxiv, 254 leaves : ill. (col.) ; 30 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / Thesis (Ph.D.)--University of Adelaide, Dept. of Electrical and Electronic Engineering, 2002

Heterophase polymerization is a technique widely used for the synthesis of high performance polymeric materials with applications including paints, inks, adhesives, synthetic rubber, biomedical applications and many others. Due to the heterogeneous nature of the process, many different relevant length and time scales can be identified. Each of these scales has a direct influence on the kinetics of polymerization and on the physicochemical and performance properties of the final product. Therefore, from the point of view of product and process design and optimization, the understanding of each of these relevant scales and their integration into one single model is a very promising route for reducing the time-to-market in the development of new products, for increasing the productivity and profitability of existing processes, and for designing products with improved performance or cost/performance ratio. The process considered is the synthesis of structured or composite polymer particles by multi-stage seeded emulsion polymerization. This type of process is used for the preparation of high performance materials where a synergistic behavior of two or more different types of polymers is obtained. Some examples include the synthesis of core-shell or multilayered particles for improved impact strength materials and for high resistance coatings and adhesives. The kinetics of the most relevant events taking place in an emulsion polymerization process has been investigated using suitable numerical simulation techniques at their corresponding time and length scales. These methods, which include Molecular Dynamics (MD) simulation, Brownian Dynamics (BD) simulation and kinetic Monte Carlo (kMC) simulation, have been found to be very powerful and highly useful for gaining a deeper insight and achieving a better understanding and a more accurate description of all phenomena involved in emulsion polymerization processes, and can be potentially extended to investigate any type of heterogeneous process. The novel approach of using these kinetic-based numerical simulation methods can be regarded as a complement to the traditional thermodynamic-based macroscopic description of emulsion polymerization. The particular events investigated include molecular diffusion, diffusion-controlled polymerization reactions, particle formation, absorption/desorption of radicals and monomer, and the colloidal aggregation of polymer particles. Using BD simulation it was possible to precisely determine the kinetics of absorption/desorption of molecular species by polymer particles, and to simulate the colloidal aggregation of polymer particles. For diluted systems, a very good agreement between BD simulation and the classical theory developed by Smoluchowski was obtained. However, for concentrated systems, significant deviations from the ideal behavior predicted by Smoluchowski were evidenced. BD simulation was found to be a very valuable tool for the investigation of emulsion polymerization processes especially when the spatial and geometrical complexity of the system cannot be neglected, as is the case of concentrated dispersions, non-spherical particles, structured polymer particles, particles with non-uniform monomer concentration, and so on. In addition, BD simulation was used to describe non-equilibrium monomer swelling kinetics, which is not possible using the traditional thermodynamic approach because it is only valid for systems at equilibrium. The description of diffusion-controlled polymerization reactions was successfully achieved using a new stochastic algorithm for the kMC simulation of imperfectly mixed systems (SSA-IM). In contrast to the traditional stochastic simulation algorithm (SSA) and the deterministic rate of reaction equations, instead of assuming perfect mixing in the whole reactor, the new SSA-IM determines the volume perfectly mixed between two consecutive reactions as a function of the diffusion coefficient of the reacting species. Using this approach it was possible to describe, using a single set of kinetic parameters, typical mass transfer limitations effects during a free radical batch polymerization such as the cage effect, the gel effect and the glass effect. Using multiscale integration it was possible to investigate the formation of secondary particles during the seeded emulsion polymerization of vinyl acetate over a polystyrene seed. Three different cases of radical generation were considered: generation of radicals by thermal decomposition of water-soluble initiating compounds, generation of radicals by a redox reaction at the surface of the particles, and generation of radicals by thermal decomposition of surface-active initiators "inisurfs" attached to the surface of the particles. The simulation results demonstrated the satisfactory reduction in secondary particles formation achieved when the locus of radical generation is controlled close to the particles surface. / Eine der industriell am meisten verwendeten Methoden zur Herstellung von Hochleistungspolymeren ist die Heterophasenpolymerisation. Industriell von besonderer Bedeutung ist die sogenannte Saatemulsionspolymerisation bei der kleine Saatteilchen durch die sequentielle Zugabe von weiteren Monomeren gezielt modifiziert werden, um Kompositpolymerteilchen mit den gewünschten mechanischen und chemischen Gebrauchseigenschaften herzustellen. Ein häufig auftretendes Problem während dieser Art der Heterophasenpolymerisation ist die Bildung von neuen, kleinen Teilchen im Polymerisationsverlauf. Diese sogenannte sekundäre Teilchenbildung muss vermieden werden, da sie die Herstellung der gewünschten Teilchen mit den angestrebten Eigenschaften verhindert. Ein spezieller Fall der Saatemulsionspolymerisation ist die Kombination von Vinylacetat als Monomer, das auf Saatteilchen aus Polystyrol polymerisieren soll. Die Unterdrückung der Teilchenneubildung ist in diesem Beispiel besonders schwierig, da Vinylacetat eine sehr hohe Wasserlöslichkeit besitzt. In der vorliegenden Arbeit wurden zur Lösung der Aufgabenstellung verschiedene numerische Simulierungsalgorithmen verwendet, die entsprechend den charakteristischen Längen- und Zeitskalen der im Verlauf der Polymerisation ablaufenden Prozesse ausgewählt wurden, um die passenden Bedingungen für die Unterdrückung der sekundären Teilchenbildung zu finden. Die verwendeten numerischen Methoden umfassen Molekulare Dynamik Simulationen, die benutzt werden, um molekulare Bewegungen zu berechnen; Brownsche Dynamik Simulationen, die benutzt werden, um die zufälligen Bewegungen der kolloidalen Teilchen und der molekularen Spezies zu beschreiben, und kinetische Monte Carlo Simulationen, die das zufällige Auftreten von individuellen physikalischen oder chemischen Ereignissen modellieren. Durch die Kombination dieser Methoden ist es möglich, alle für die Beschreibung der Polymerisation relevanten Phänomene zu berücksichtigen. Damit können nicht nur die Reaktionsgeschwindigkeit und die Produktivität des Prozesses simuliert werden sondern auch Aussagen bezüglich der physikalischen und chemischen Eigenschaften des Produktes sowie den Applikationseigenschaften getroffen werden. In dieser Arbeit wurden zum ersten Mal Modelle für die unterschiedlichen Längen- und Zeitskalen bei Heterophasenpolymerisationen entwickelt und erfolgreich zur Modellierung des Prozesses angewendet. Die Ergebnisse führten zu bedeutenden Verbesserungen der Theorie von Emulsionspolymerisationen insbesondere für die Beschreibung des Massenaustausches zwischen den Phasen (bspw. Radikaleintritt in und Radikalaustritt aus die Polymerteilchen), der Bildung von neuen Teilchen, und der Polymerisationskinetik unter den heterogenen Reaktionsbedingungen mit uneinheitlicher Durchmischung.

The first part of this thesis studies tail probabilities forelliptical distributions and probabilities of extreme eventsfor multivariate stochastic processes. It is assumed that thetails of the probability distributions satisfy a regularvariation condition. This means, roughly speaking, that thereis a non-negligible probability for very large or extremeoutcomes to occur. Such models are useful in applicationsincluding insurance, finance and telecommunications networks.It is shown how regular variation of the marginals, or theincrements, of a stochastic process implies regular variationof functionals of the process. Moreover, the associated tailbehavior in terms of a limit measure is derived. The second part of the thesis studies problems related toparameter estimation in stochastic models with long memory.Emphasis is on the estimation of the drift parameter in somestochastic differential equations driven by the fractionalBrownian motion or more generally Volterra-type processes.Observing the process continuously, the maximum likelihoodestimator is derived using a Girsanov transformation. In thecase of discrete observations the study is carried out for theparticular case of the fractional Ornstein-Uhlenbeck process.For this model Whittle’s approach is applied to derive anestimator for all unknown parameters.

In this thesis we study the standard probabilistic model for hashing with linear probing. The main purpose is to determine the asymptotic distribution for the maximum displacement. Depending on the ratio between the number of items and the number of cells, there are several cases to consider. Paper I solves the problem for the special case of almost full hash tables. That is, hash tables where every cell but one is occupied. Paper II completes the analysis by solving the problem for all remaining cases. That is, for every case where the number of items divided by the number of cells lies in the interval [0,1]. The last two papers treat quite different topics. Paper III studies the area covered by the supremum process of Brownian motion. One of the main theorems in Paper I is expressed in terms of the Laplace transform of this area. Paper IV provides a new sufficient condition for a collection of independent random variables to be negatively associated when conditioned on their total sum. The condition applies to a collection of independent Borel-distributed random variables, which made it possible to prove a Poisson approximation that where essential for the completion of Paper II.

This is an introduction to Wiener measure and the Feynman-Kac formula on general Riemannian manifolds for Riemannian geometers with little or no background in stochastics. We explain the construction of Wiener measure based on the heat kernel in full detail and we prove the Feynman-Kac formula for Schrödinger operators with bounded potentials. We also consider normal Riemannian coverings and show that projecting and lifting of paths are inverse operations which respect the Wiener measure.

The first passage time (FPT) problem for Brownian motion has been extensively studied in the literature. In particular, many incarnations of integral equations which link the density of the hitting time to the equation for the boundary itself have appeared. Most interestingly, Peskir (2002b) demonstrates that a master integral equation can be used to generate a countable number of new integrals via its differentiation or integration. In this thesis, we generalize Peskir's results and provide a more powerful unifying framework for generating integral equations through a new class of martingales. We obtain a continuum of new Volterra type equations and prove uniqueness for a subclass. The uniqueness result is then employed to demonstrate how certain functional transforms of the boundary affect the density function. Furthermore, we generalize a class of Fredholm integral equations and show its fundamental connection to the new class of Volterra equations. The Fredholm equations are then shown to provide a unified approach for computing the FPT distribution for linear, square root and quadratic boundaries. In addition, through the Fredholm equations, we analyze a polynomial expansion of the FPT density and employ a regularization method to solve for the coefficients. Moreover, the Volterra and Fredholm equations help us to examine a modification of the classical FPT under which we randomize, independently, the starting point of the Brownian motion. This randomized problem seeks the distribution of the starting point and takes the boundary and the (unconditional) FPT distribution as inputs. We show the existence and uniqueness of this random variable and solve the problem analytically for the linear boundary. The randomization technique is then drawn on to provide a structural framework for modeling mortality. We motivate the model and its natural inducement of 'risk-neutral' measures to price mortality linked financial products. Finally, we address the inverse FPT problem and show that in the case of the scale family of distributions, it is reducible to nding a single, base boundary. This result was applied to the exponential and uniform distributions to obtain analytical approximations of their corresponding base boundaries and, through the scaling property, for a general boundary.

This thesis is a report of the work I have done over the last five years to study thermodynamic and kinetic aspects of single-molecule behavior in the condensed phase. It is concerned specifically with the development of analytically tractable models of various phenomena that have been observed in experiments on such single-molecule systems as colloids, double-stranded DNA, multi-unit proteins, and enzymes. In fluid environments, the energetics, spatial conformations, and chemical reactivity of these systems undergo fluctuations that can be characterized experimentally in terms of time correlation functions, survival probabilities, mean first passage times, and related statistical parameters. The thesis shows how many of these quantities can be calculated in closed form from a model based on simple Brownian motion, or generalizations of it involving fractional calculus. The theoretical results obtained here have been shown to agree qualitatively or quantitatively with a range of experimental data. The thesis therefore demonstrates the effectiveness of Brownian motion concepts as a paradigm of stochasticity in biological processes.

We use the Garman-Kohlhagen model to compute the hedge and price of a participating forward contract on the US dollar that is written by a Turkish Bank. The algorithm is computed using actual market data and a weekly updated hedge is computed. We note that despite a weekly update and many assumptions made on the volatility and the interest rates the model gives a very reasonable hedge.