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

• Main Author: Dalinger, Alexander
• Format: Others
• Language: en
• Published: 2019
• Online Access: https://tuprints.ulb.tu-darmstadt.de/9194/1/dalinger2019.pdf; Dalinger, Alexander <http://tuprints.ulb.tu-darmstadt.de/view/person/Dalinger=3AAlexander=3A=3A.html> : On the hydrodynamic behaviour of a particle system with nearest neighbour interactions. Technische Universität, Darmstadt [Ph.D. Thesis], (2019)

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.