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

NSF提案摘要0239625PI：Eric Vanden-Eijnden，库兰特数学科学研究所标题：复杂系统中的转变路径描述由罕见事件或构象驱动的复杂系统，是应用数学和计算科学中的一个主要挑战。众所周知，构象的例子包括相变过程中的成核事件、分子的构象变化、磁性材料的热激活转换和化学反应。信息事件的执行时间和这些事件之间的等待时间之间的差距通常是如此之大，以至于不可能通过直接求解潜在的动态方程来模拟这些过程的动力学。复杂系统中跃迁路径的理论描述给概率和随机过程理论带来了挑战，需要超越大偏差理论及其扩展的标准工具。这些新的理论工具将被开发出来，以自然地产生高效的算法，非常适合于对不现实的复杂系统所需的转变路径进行数值研究。应用数学将通过一个分析、模拟和减少复杂系统中罕见构象变化的协调程序来做出许多贡献，并获得许多理解。要解决的问题包括：(A)开发描述复杂系统中过渡路径的适当理论工具；(B)为识别这些路径开发有效的数值算法；(C)在本科生和研究生水平上开发应用随机方法的课程，这将为学生提供必要的背景，以便在数学、计算机科学和自然科学之间的界面上研究这些主题。要成功地解决这些问题，需要在涉及复杂动力系统的自然科学的多个领域中，利用计算科学的专业知识，应用数学。参与这项工作的学生和博士后科学家将通过接触这些学科获得灵活性和视角。

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.