B08 – Screening and long-range interactions in problems in materials science

During the first funding period of this project we studied the effective properties of interacting particle systems in which so-called screening effects play a crucial role. We considered the two examples of solid particles settling under gravity in a Stokes flow and coarsening systems with strong local correlations. For sedimentation we derived and analyzed kinetic limit problems for the particle densities and proved alocal version of Einstein’s law in a static setting. We also studied coarsening systems with local correlations. For a toy model of particles on a line which interact by a set of infinitely many ordinary differential equations we proved well-posedness, obtained upper bounds for coarsening rates and provided examples of initial data for which the system evolves according to the expected coarsening rate. We want to extend this line of research by constructing invariant self-similar probability measures for this type of models and by studying variants in higher space dimensions. In the context of sedimentation we plan to study the validity of Einstein’s law for the effective viscosity in a dynamical setting, we will investigate the derivation of macroscopic models for randomly distributed particles and analyze the limit models for initial data with noise. Similar methods will allow us to derive Darcy’s law in a stochastic setting under minimal assumptions on the particle distribution. The limit models for sedimenting particles provide a motivation to study two-component viscous flows. Here we plan to investigate the dynamics of the flow if one of the fluids is distributed in a thin layer. We will also extend our study to understand certain localization phenomena, for example in topological insulators in which the electric currents are localized to the edges of the material. The goal is to determine conditions that yield the existence of so-called edge states, i.e. eigenvalues localized at boundaries and interfaces, for the magnetic Laplacian and perturbations of it.

 

 

Name	Institute	Location	Phone	Email
Niethammer, Barbara	IAM	En60/2.039	2216	niethammer@iam.uni-bonn.de
Velázquez, Juan José	IAM	En60/2.023	62378	velazquez@iam.uni-bonn.de