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

New York University, New York NY

Investigators

Abstract

Abstract of NSF Proposal 0239625 PI: Eric Vanden-Eijnden, Courant Institute of Mathematical Sciences Title: Transition pathways in complex systems The description of complex system driven by rare events, or conformations, is a major challenge in applied mathematics and computational sciences. Well-known examples of conformations include nucleation events during phase transition, conformational changes in molecules, thermal activated switching of magnetic materials, and chemical reactions. The disparity between the execution time of the conformation event and the waiting time between these events is typically so large that it impossible to simulate the dynamics of these processes by solving directly the underlying dynamic equations. This project is a combined theorical and numerical effort to describe and simulate these rare conformation changes. The theoretical description of transition pathways in complex systems leads to challenging problem in probability and stochastic processes theory which require to go beyond the standard tools of large deviation theory and extensions thereof. These new theoretical tools will be developed so as to naturally lead to efficient algorithms, well-suited for the numerical investigation of the transition pathways required in realistic complex systems. Applied mathematics has much to contribute, and much to gain in understanding via a coordinate program of analysis, simulation, and education the mechanisms by which rare conformation changes arise in complex systems. Topics to be addressed include: (a) the development of appropriate theoretical tools for the description of the transition pathways in complex systems; (b) the development of efficient numerical algorithms for the identification of these pathways; and, (c) the development of courses at undergraduate and graduate levels in applied stochastic methods which will provide the students with the necessary background to do research on these topics at the interface between mathematics, computer science, and the natural sciences. Successful pursuit of these issues requires a multidisciplinary effort, drawing expertise from computational sciences, applied mathematics, in the multiple fields in the natural sciences which involve complex dynamical systems. The students and postdoctoral scientists participating this effort will gain flexibility and perspective through exposure to these many disciplines.