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On the hydrodynamic behaviour of a particle system with nearest neighbour interactions

Dalinger, Alexander (2019):
On the hydrodynamic behaviour of a particle system with nearest neighbour interactions.
Darmstadt, Technische Universität, [Online-Edition: https://tuprints.ulb.tu-darmstadt.de/9194],
[Ph.D. Thesis]

Abstract

In this thesis we will study a system of Brownian particles on the real line, which are coupled through the nearest neighbours by an attractive potential. This model is related to the Ginzburg-Landau model. We will prove two results. The first result is the hydrodynamic equation for the particle density. More precisely, we show that the empirical measure of the particle positions converges in the hydrodynamic limit to a deterministic and absolutely continuous probability measure, where the density solves a nonlinear heat equation. The crucial idea will be the reduction of the particle model to the height model, in the literature also called Ginzburg-Landau interface model. We will obtain the claimed result by taking the limit in the height model and passing back to the particle model. Further, we will outline how this approach generalises to multiple dimensions. The second result is the characterisation of the equilibrium fluctuations in the case of quadratic potential. We will consider the fluctuation field, which is defined as the square root of the number of particles times the difference of the empirical measure of the particle positions and its expectation. Assuming the initial distribution of the particle system to be stationary, we will show that the fluctuation field converges in the hydrodynamic limit to an infinite-dimensional Ornstein-Uhlenbeck process. The proof will consist of characterising the accumulation points of the distributions of fluctuation fields by means of a martingale problem and showing tightness.

Item Type: Ph.D. Thesis
Erschienen: 2019
Creators: Dalinger, Alexander
Title: On the hydrodynamic behaviour of a particle system with nearest neighbour interactions
Language: English
Abstract:

In this thesis we will study a system of Brownian particles on the real line, which are coupled through the nearest neighbours by an attractive potential. This model is related to the Ginzburg-Landau model. We will prove two results. The first result is the hydrodynamic equation for the particle density. More precisely, we show that the empirical measure of the particle positions converges in the hydrodynamic limit to a deterministic and absolutely continuous probability measure, where the density solves a nonlinear heat equation. The crucial idea will be the reduction of the particle model to the height model, in the literature also called Ginzburg-Landau interface model. We will obtain the claimed result by taking the limit in the height model and passing back to the particle model. Further, we will outline how this approach generalises to multiple dimensions. The second result is the characterisation of the equilibrium fluctuations in the case of quadratic potential. We will consider the fluctuation field, which is defined as the square root of the number of particles times the difference of the empirical measure of the particle positions and its expectation. Assuming the initial distribution of the particle system to be stationary, we will show that the fluctuation field converges in the hydrodynamic limit to an infinite-dimensional Ornstein-Uhlenbeck process. The proof will consist of characterising the accumulation points of the distributions of fluctuation fields by means of a martingale problem and showing tightness.

Place of Publication: Darmstadt
Divisions: 04 Department of Mathematics
04 Department of Mathematics > Stochastik
Date Deposited: 03 Nov 2019 20:56
Official URL: https://tuprints.ulb.tu-darmstadt.de/9194
URN: urn:nbn:de:tuda-tuprints-91942
Referees: Betz, Prof. Dr. Volker and Aurzada, Prof. Dr. Frank
Refereed / Verteidigung / mdl. Prüfung: 17 October 2019
Alternative Abstract:
Alternative abstract Language
In dieser Dissertation wird ein System von Brownschen Teilchen auf den reellen Zahlen studieren, wobei die Teilchen über die nächsten Nachbarn mit einem anziehenden Potential gekoppelt sind. Dieses Modell ist verwandt mit dem Ginzburg-Landau Modell. Wir zeigen zwei Resultate. Das erste Resultat ist die hydrodynamische Gleichung für die Teilchendichte. Genauer gesagt zeigen wir, dass das empirische Maß der Teilchenpositionen im hydrodynamischen Grenzwert gegen ein deterministisches und absolut stetiges Wahrscheinlichkeitsmaß konvergiert, wobei die Dichte eine nichtlineare Wärmeleitungsgleichung löst. Die wesentliche Idee wird es sein, das Teilchenmodell auf das Höhenmodell, in der Literatur Ginzburg-Landau Grenzflächenmodell genannt, zu reduzieren. Indem wir den Grenzwert im Höhenmodell bilden und dann zurück zum Teilchenmodell wechseln, erhalten wir das genannte Resultat. Wir skizzieren außerdem die Verallgemeinerung dieses Ansatzes auf den mehrdimensionalen Fall. Das zweite Resultat ist die Charakterisierung der Gleichgewichtsfluktuationen der Teilchendichte bei quadratischem Potential. Dazu betrachten wir das Fluktuationsfeld, das definiert ist als die Wurzel der Teilchenzahl mal die Differenz von dem empirischen Maß der Teilchenpositionen zu seinem Erwartungswert. Wir nehmen an, dass die Verteilung des Teilchensystem zu Beginn stationär ist. Dann zeigen wir, dass das Fluktuationsfeld im hydrodynamischen Grenzwert gegen einen unendlichdimensionalen Ornstein-Uhlenbeck Prozess konvergiert. Der Beweis wird darin bestehen, die Häufungspunkte der Verteilungen der Fluktuationsfelder durch ein Martingalproblem zu charakterisieren und Straffheit zu zeigen.German
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