Stochastic Dynamics on Energy Landscapes with Applications in Physics and Biology

物理学和生物学中的大量系统在时间上是随机演化的，人们使用特殊的数学工具来理解这些涨落的功能。小规模材料的模型，如纳米磁体或设计的超材料，会有空间相关性的波动，这会影响它们的性能。这些系统需要开发工具来考虑多个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运输，以及可能有助于或对系统造成灾难性影响的不太可能的事件。这个项目的数学进步包括利用这些工具找到潜在的数学机制，这些机制可以与实验观察一起使用，从而增加我们对物理世界的理解。该项目包含物理、生物和数学的自然交叉学科融合，以培养当代应用数学学生，因为应用数学的研究领域日益多样化。研究人员将继续努力增加对数学的参与。学生们有机会参加STEM的高中女生当地夏令营，建立各级的指导网络，创造包容的和积极的情感空间，并使用出色的可视化来促进数学。该项目的应用为发展分析大型和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纳入朗之万系统的能源格局框架，并进一步利用这个框架来解释系统的行为。具体地说，这个项目回答了关于随机粗粒化和大偏差的公开问题。研究人员提供了既保留确定性项又保留波动项的方法，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，而不是通过标准平均来消除噪声。从粗粒度方程出发，该项目提供了渐近近似亚稳态之间的转变时间的方法，并在不同限制上显示了不同技术之间的等价性。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.