CAREER: Transition Pathways in Complex Systems. Theory and Numerical Methods.

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Abstract of NSF Proposal 0239625PI: Eric Vanden-Eijnden, Courant Institute of Mathematical SciencesTitle: Transition pathways in complex systemsThe description of complex system driven by rare events, orconformations, is a major challenge in applied mathematics andcomputational sciences. Well-known examples of conformations includenucleation events during phase transition, conformational changes inmolecules, thermal activated switching of magnetic materials, andchemical reactions. The disparity between the execution time of theconformation event and the waiting time between these events istypically so large that it impossible to simulate the dynamics ofthese processes by solving directly the underlying dynamic equations.This project is a combined theorical and numerical effort to describeand simulate these rare conformation changes. The theoreticaldescription of transition pathways in complex systems leads tochallenging problem in probability and stochastic processes theorywhich require to go beyond the standard tools of large deviationtheory and extensions thereof. These new theoretical tools will bedeveloped so as to naturally lead to efficient algorithms, well-suitedfor the numerical investigation of the transition pathways required inrealistic complex systems.Applied mathematics has much to contribute, and much to gain inunderstanding via a coordinate program of analysis, simulation, andeducation the mechanisms by which rare conformation changes arise incomplex systems. Topics to be addressed include: (a) the developmentof appropriate theoretical tools for the description of the transitionpathways in complex systems; (b) the development of efficientnumerical algorithms for the identification of these pathways; and,(c) the development of courses at undergraduate and graduate levels inapplied stochastic methods which will provide the students with thenecessary background to do research on these topics at the interfacebetween mathematics, computer science, and the natural sciences.Successful pursuit of these issues requires a multidisciplinaryeffort, drawing expertise from computational sciences, appliedmathematics, in the multiple fields in the natural sciences whichinvolve complex dynamical systems. The students and postdoctoralscientists participating this effort will gain flexibility andperspective through exposure to these many disciplines.