Modeling metastability in complex systems

Dynamics in nature often proceed in the form of rare reactive events: The system under study spends very long periods of time at various metastable states and only very rarely transitions from one metastable state to another. Conformation changes of macromolecules, chemical reactions in solution, nucleation events during phase transitions, thermally induced magnetization reversal in micromagnets, etc. are just a few examples of such reactive events. One can often think of the dynamics of these systems as a navigation over a potential or free energy landscape, under the action of small amplitude noise. In the simplest situations the metastable states are then regions around the local minima on this landscape, and transition events between these regions are rare because the noise has to push the system over the barriers separating them. This is the picture underlying classical tools such as transition state theory or Kramers reaction rate theory, and it can be made mathematically precise within the framework of large deviation theory. In complex high dimensional systems, this picture can however be naive because entropic (i.e. volume) effects start to play an important role: Local features of the energy, such as the location of its minima or saddle points, may have much less of an impact on its dynamics than global features such as the width of low lying basins on the landscape: in these situations a more general framework for the description of metastability is required. In this talk, I will discuss tools that have been introduced to that effect, based e.g. on potential theory, and illustrate them on various examples, including the folding of toy models of proteins and the rearrangement of Lennard-Jones clusters.