Stochastic Dynamics on Energy Landscapes with Applications in Physics and Biology

物理学和生物学中的大量系统随时间随机进化，并且使用特殊的数学工具来理解这些波动的功能。纳米磁铁或设计的超材料等小尺度材料的模型具有空间依赖性波动，影响其性能。这些系统需要开发工具来解释多种噪声源和空间相关噪声源的相互作用，以预测可能的结构和传输，以及可能对系统有益或灾难性的不太可能发生的事件。这个项目的数学进步包括利用这些工具来找到可以与实验观察一起使用的潜在数学机制，从而增加我们对物理世界的理解。随着应用数学研究领域的多样化，该项目包含了物理、生物和数学的自然跨学科融合，以培养当代应用数学学生。研究人员将继续致力于提高学生对数学的参与度。学生们有机会参加当地的STEM高中女生营地，建立各级指导网络，创造包容和情感积极的空间，并利用出色的可视化来促进数学。本课题的应用为提高分析大、无限维随机系统的数学工具提供了一个舞台。这些工具的核心是将一个系统引入朗之万系统的能源景观框架，并进一步利用这个框架来解释系统的行为。具体来说，这个项目回答了关于随机粗粒度和大偏差的开放性问题。研究人员提供了保留确定性项和波动项的方法，以解释多个噪声源的相互作用，并推导出全局变量（能量）的有效方程，而不是用标准平均法来去除噪声。从粗粒度方程出发，给出了渐近逼近亚稳态过渡时间的方法，并证明了不同技术在不同极限上的等价性。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

A large number of systems in physics and biology evolve randomly in time, and special mathematical tools are used to understand the function of these fluctuations. Models of small scale materials like nano magnets or designed meta-materials have fluctuations with spatial dependence that affect their performance. These systems require the development of tools to account for the interaction of multiple sources of noise and spatially-correlated noise to predict both likely structure and transport, as well as unlikely events that could either help or be catastrophic to the system. The mathematical advancement of this project includes utilizing these tools to find underlying mathematical mechanisms that can be used along with experimental observations, thus increasing our understanding of the physical world. This project contains a natural interdisciplinary blend of physics, biology and mathematics to train the current generation of applied mathematics students as the research field of applied mathematics grows in diversity. The investigators will continue working to increase participation in mathematics. Students have the opportunity to participate in local camps for high school girls in STEM, build mentoring networks at all levels, create inclusive and emotionally positive spaces, and use outstanding visualizations to promote mathematics.The applications in this project provide an arena for advancing the mathematical tools for analyzing large and infinite dimensional stochastic systems. These tools center around bringing a system into the energy landscape framework of Langevin systems and further utilizing this framework to explain the system’s behavior. Specifically, this project answers open problems regarding stochastic coarse-graining and large deviations. Rather than standard averaging to remove the noise, investigators provide methods to retain both a deterministic term and a fluctuating term to account for the interaction of multiple sources of noise, as well as derive effective equations for a global variable (energy). From the coarse-grained equations, the project offers methods to asymptotically approximate transition times between metastable states and show equivalence between different techniques on different limits.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.