Ziel dieser Arbeit ist es die Fick-Jacobs Näherung, welche eine genaue Beschreibung zahlreiche Transporteigenschaften von Brownschen Teilchen in räumlich beschränkten Geometrien liefern kann, auf experimentell vorherrschende Gegebenheiten, z.B., sich stark ändernde Geometrien, komplizierte Kraftfelder, Teilchenausdehnung und endliche viskose Reibung, zu erweitern. Dazu wird zuerst die exakte Lösung für die stationäre Wahrscheinlichkeitsdichte mittels Entwicklung in einem geometrischen Parameter, der die Kanalmodulation misst, berechnet. Die höheren Entwicklungsterme ermöglichen die Berechnung von Korrekturen zu den Transportkoeffizienten für sich stark ändernde Geometrien. Ferner kann die Fick-Jacobs Näherung mittels der Entwicklungsmethode auf beliebige Kraftfelder verallgemeinert werden. Am Beispiel des mikrofluidischen Kanals zeigen wir, dass das Zusammenspiel von externen Kräften (skalare Potentiale) und Strömungen (Vektorpotentiale) zur effizienten Trennung von Objekten, mittels des Effektes des hydrodynamisch induzierten entropischen Einsperrens, genutzt werden kann. Da das effiziente Sortieren nach Größe eine der wichtigsten Ziele in der Grundlagenforschung ist, zeigen wir wie die Teilchenausdehnung in die Fick-Jacobs Näherung integriert werden kann. Abschließend wird der Einfluss der Mediumsviskosität auf den Teilchentransport untersucht. Wenn die Zeitskalen separieren, führt adiabatische Eliminierung auch für endliche Reibung zu einer Fick-Jacobs ähnlichen Beschreibung. Diese ist unweigerlich mit Energiegleichverteilung und mit verschwindender Geschwindigkeitskorrelation verbunden. Numerische Simulationen zeigen, dass diese Beschreibung für moderate bis starke Dämpfung und schwache externe Kräfte akkurat ist. Für starke Kräfte wird die angenommene Energiegleichverteilung infolge von Teilchen-Wand Kollisionen verletzt. Dies führt zu einer nichtlinearen Abhängigkeit der Teilchengeschwindigkeit und des effektiven Diffusionskoeffizienten von der Kraftstärke. / This work intends to show how experimentally relevant issues such as strong channel corrugation, sophisticated external force fields, particle size, and the solvent''s viscosity can be incorporated into the commonly used Fick-Jacobs approach which provides a powerful tool to capture many properties of Brownian particles'' transport in confined geometries. First, we derive exact solutions of the stationary probability distribution in terms of an expansion parameter specifying the channel corrugation. Thereby, the leading order is equivalent to the Fick-Jacobs approach. By means of higher expansion orders, which become significant for strong channel corrugation, we obtain corrections to the key particle transport quantities. Going one step further, we generalize the Fick-Jacobs approach to the most general forces. As an exemplary application, we consider microfluidic devices in which the interplay of conservative forces and pressure-driven flows (vector potentials) offers a unique opportunity to efficiently separate Brownian particles of the same size using the newly discovered effect of hydrodynamically enforced entropic trapping. Since separation and sorting by size is a main challenge in basic research, we demonstrate that within certain limits the analytic expressions for the key transport quantities, derived for point-like particles, can be applied to extended objects, too. Lastly, we study the impact of the solvent''s viscosity on particle transport. If the time scales separate, adiabatic elimination results in an effective description even for finite damping. The possibility of such description is intimately connected with equipartition and vanishing velocity correlation. Numerical simulations show that this approach is accurate for moderate to strong damping and for weak forces. For strong external forces, equipartition may break down due to reflections at the boundaries. This leads to a non-monotonic dependence of the particle mobility on the force strength.

Made available in DSpace on 2014-12-17T15:14:51Z (GMT). No. of bitstreams: 1 JoaoMS.pdf: 634960 bytes, checksum: 579213d21eb24a6a8d342d2c3b15bc3d (MD5) Previous issue date: 2003-06-20 / Coordena??o de Aperfei?oamento de Pessoal de N?vel Superior / In this work we investigate the stochastic behavior of a large class of systems with variable damping which are described by a time-dependent Lagrangian. Our stochastic approach is based on the Langevin treatment describing the motion of a classical Brownian particle of mass m. Two situations of physical interest are considered. In the first one, we discuss in detail an application of the standard Langevin treatment (white noise) for the variable damping system. In the second one, a more general viewpoint is adopted by assuming a given expression to the so-called collored noise. For both cases, the basic diffententiaql equations are analytically solved and al the quantities physically relevant are explicitly determined. The results depend on an arbitrary q parameter measuring how the behavior of the system departs from the standard brownian particle with constant viscosity. Several types of sthocastic behavior (superdiffusive and subdiffusive) are obteinded when the free pamameter varies continuosly. However, all the results of the conventional Langevin approach with constant damping are recovered in the limit q = 1 / Neste trabalho n?s investigamos o comportamento estoc?stico de uma grande classe de sistemas com amortecimento vari?vel descritos por uma lagraniana dependente do tempo. Nossa abordagem estoc?stica ? baseada no formalismo de Langevin descrevendo o comportamento de uma part?cula browniana cl?ssica de massa m. Duas situa??es de interesse f?sico s?o consideradas. Inicialmente, uma aplica??o do tratamento padr?o de Langevin (ru?do branco) para viscosidade vari?vel ? discutido em detalhe. Na segunda abordagem, um ponto de vista mais geral ? adotado supondo uma dada express?o para o chamado ru?do branco. Em ambos os casos as equa??es diferenciais b?sicas s?o analiticamente resolvidas, e todas as quantidades fisicamente relevantes s?o explicitamente determinadas. Os resultados dependem de um par?metro arbitr?rio (q) medindo como o comportamento din?mico do sistema se afasta daquele apresentado pela part?cula browniana com viscosidade constante. V?rios tipos de comportamentos estoc?sticos (subsifusivos and superdifusivos) s?o obtidos quando o par?metro livre q varia continuamente. Contudo, no limite q -> 1, todos os resultados da abordagem de Langevin convencional s?o recuperados