Pattern formation, energy landscapes, and scaling laws
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Welcome to the website of the research group "Pattern formation, energy landscapes and scaling laws" at the Max-Planck-Institute for Mathematics in the Sciences. On the following pages, we give an overview over our work of the past years, as well as a glance at the ongoing research.
Experiments and, nowadays, numerical simulation are a main tool for gaining insight into nature. However, these approaches share two main disadvantages: On the one hand, there are certain phenomena that are difficult to access via experiments or numerical simulation, for example problems involving several length scales that greatly differ in their order of magnitude. On the other hand, even in the case where a problem can be treated experimentally or numerically, there might still be a lack of understanding - one searches not only for a description, but also for a satisfactory explanation of the mechanisms behind an observation.
This is the point at which a more theoretical analysis comes into play: There are many mathematical models in physics where the interplay of only a very small number of basic effects is believed to create a wealth of phenomena, including ones that are experimentally and numerically largely inaccessible due to their multi-scale nature. But with the help of mathematical analysis, one can often rigorously derive predictions that explain the real-world observations by relating them to the few basic effects that are the foundation of the model - thereby seconding (or disproving) its validity.
The focus of our research group is the analysis of (typically continuum) models that mostly come from materials science and fluid mechanics. We seek to understand specific phenomena, like the formation of certain patterns or the emergence of certain scaling laws, some of which are presented below. Our technical expertise is in the calculus of variations and in partial differential equations. Moreover, we use numerical simulation for quantitative comparison of our analysis to experiments. In recent years, probabilistic elements of the models and techniques play an increasing role in our group.
Our ambition is to give more insight on physically relevant phenomena while at the same time innovating mathematics.
Overview over our work
Below, we very briefly introduce some of the problems that we have investigated. For details, please visit the related links or contact the researchers working on the topic.
The magnetization of a ferromagnetic sample is influenced by external magnetic fields, by the magnetic field it creates itself, material anisotropies and the quantum mechanical spin exchange effect.
Together, these lead to various interesting patterns, for example in magnetic thin films. Amongst other questions, we have investigated the coarsening of the concertina pattern and the transition from symmetric to asymmetric domain walls.
Binary mixtures far from thermodynamic equilibrium tend to separate their two components; droplets spread on a thin liquid substrate tend to merge. Typically, such systems develop a characteristic length scale that reflects the current state of ordering and which grows during the evolution.
Despite the possibly very complicated pattern, the rate at which the coarsening proceeds follows simple growth laws. We have approached some of these coarsening rates rigorously.
Many heterogeneous media are described by partial differential equations with random coefficients varying on small scales. On macroscopic scales such media often show a less complex, deterministic behavior. The corrector problem from stochastic homogenization provides an equation that relates the microscopic random description of the medium to its macroscopic behavior. Since, in practice, the corrector problem has to be approximated, a quantitative analysis is needed to estimate the approximation error.
Motivated by this, we have developed quantitative methods for elliptic equations with random (possibly correlated) coefficients. In particular, we obtained an optimal error estimate for the approximation of the homogenized coefficients by periodization.
The hydrodynamic limit associates to a simple microscopic model, that is exposed to thermal noise, a macroscopic continuous evolution. This macroscopic evolution turns out to be deterministic like in most physical models. In this way, the hydrodynamic limit helps to answer the question how macroscopic phenomena originate from microscopic rules.
A prototype for a real world application would be understanding how the interaction of spins leads to the continuous description of a magnet. Because such a real world application is out of reach yet, we mainly consider toy models from statistical mechanics to gain insight and develop tools needed for more realistic and hence more complicated models.
Viscous Thin Films
We investigate the time evolution of thin viscous films on a flat solid. In particular we are interested in properties of mathematical models for the triple junction between liquid, solid, and the surrounding gas. The choice of proper boundary conditions at the junction for the underlying partial differential equation is decisive for the macroscopic behavior of the thin film.