Mathematical theory of phase transitions: modeling.. (PhaTraMAS)
Mathematical theory of phase transitions: modeling, analysis and simulations
Start date: Oct 15, 2008,
End date: Apr 14, 2011
The main topic of this proposal is the study of mathematical models for systems which exhibit phase transitions. We focus on microscopic, many particle systems with complex interactions. Such systems arise in numerous applications such as micromagnetic materials, epitaxial growth, polymers and bio-molecules, to name a few. The goal is two-fold. The first is to consider prototype models which can give a qualitative picture of the complex behavior of these systems, such as domain pattern formation, metastable phenomena and derive rigorous mathematical results. The second is from a modeling and computational perspective and intends to develop systematic mathematical strategies for the speed up of microscopic simulations. We consider this linking of theory and numerics important since numerical experiments serve as an important guide for the theoretical analysis in situations where rigorous results are out of reach, while at the same time theoretical results provide invaluable "benchmarking" for the numerics. Furthermore, the phenomena under investigation take place in several temporal and spatial scales and the analysis of such models requires the use of techniques from several mathematical sub-disciplines and in particular probability theory (Large deviations, Gibbs measures) and analysis (calculus of variations, nonlinear partial differential equations).
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