ISBN 978-3-8439-2826-7

978-3-8439-2826-7, Reihe Mathematik

Anne Meurer
Interacting Particle Models with their Limit Equations

182 Seiten, Dissertation Technische Universität Kaiserslautern (2016), Softcover, A5

Zusammenfassung / Abstract

Interacting particle models have many important applications including daily life phenomena such as car traffic or pedestrian flow as well as industrial applications. One industrial example is the movement of bubbles inside a liquid-liquid extraction column which is used to separate liquid compounds, for example for acid recovery.

The movement of such particles can be described by stochastic differential equations.

In this thesis, we consider both mixtures of identical particles and mixtures of different types of particles. In the former situation, the motion of all particles may be described by the same equation, whereas in the latter, the different types of particles have different equations of motion. In addition to a particle description, macroscopic models, where the evolving particle density is modelled, may be derived. Using a mean field limit we first derive the kinetic equations from the particle description. Furthermore, after defining appropriate closure relations, we obtain the hydrodynamic limit equations, which are macroscopic.

A numerical comparison between the particle models and their corresponding macroscopic equations is then performed, for three real life applications; car traffic, pedestrian flow and the movement of bubbles. The car traffic and pedestrian movement can be considered separately as well as in combination, where the cars are influenced by the pedestrians' motion and vice versa. To describe the movement of bubbles we use an ellipsoidal particle model. In this case, the particles have an orientation which also influences the particle interaction.

Our results show that interacting particle models are useful to have a detailed information about the movement of a few single particles. In the case of many particles, the computational costs are quite large and therefore macroscopic models are needed. We additionally show that the macroscopic simulations give qualitatively similar results as the particle simulations for many particles in our considered applications.