Metastability: From Mean Field Models to SPDEs
Kramer’s equation of a diffusion in a double well potential has been the pardigm for a metastable system since 1940. The theme of this note is to partially explain, why and in what sense this is a good model for metastable systems. In the process, I review recent progress in a variety of models, ranging from mean field spin systems to stochastic partial differential equations.
KeywordsGibbs Measure Coarse Graining Dirichlet Form Equilibrium Potential Large Deviation Principle
This notes summarizes thoughts that have come up through extensive work on metastability with numerous people. The entire subject started with an intensive collaboration with Michael Eckhoff, Véronique Gayrard, and Markus Klein. These early works set the stage for the potential theoretic approach. More recently I collaborated with Florent Barret, Alessandra Bianchi, Alessandra Faggionato, Frank den Hollander, Dima Ioffe, Francesco Manzo, Sylvie Méléard, Francesca Nardi, and Cristian Spitoni, on various special issues and models. I thank all of them for sharing their thoughts and insights.
The huge project on metastability was possible also only due to the excellent working conditions I had from 1992 on at the WIAS. This was in in more than one way due to Jürgen Gärtner to whom I am deeply grateful.
One person deserves spatial thanks: Erwin Bolthausen handled our first paper [ 10] on metastability as Editor-in-Chief of PTRF, and through an extensive correspondence, that paper was finally published.
These notes were written while I was holding a Lady Davis Visiting Professorship at the Technion, Haifa. I thank the William Davidson Faculty of Industrial Engineering and Management and in particular Dmitry Ioffe for their kind hospitality. Much of the work reported on here was also supported by a grant from the German-Israeli Foundation (GIF).
Financial support from the German Research Council (DFG) through SFB 611 and the Hausdorff Center for Mathematics is gratefully acknoledged.
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