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随机演化，特殊的数学工具被用来理解这些波动的功能。 像纳米磁体或设计的超材料这样的小尺度材料的模型具有影响其性能的空间依赖性的波动。 这些系统需要开发工具来解释多个噪声源和空间相关噪声的相互作用，以预测可能的结构和运输，以及可能对系统有帮助或造成灾难的不太可能的事件。 这个项目的数学进步包括利用这些工具来寻找潜在的数学机制，这些机制可以沿着实验观察，从而增加我们对物理世界的理解。该项目包含物理学，生物学和数学的自然跨学科融合，以培养当前一代应用数学学生，因为应用数学的研究领域越来越多样化。研究人员将继续努力增加数学的参与。学生有机会参加当地的STEM高中女生夏令营，建立各级指导网络，创造包容和情感积极的空间，并使用出色的可视化来促进数学。该项目中的应用为推进分析大型和无限维随机系统的数学工具提供了一个竞技场。 这些工具围绕着将系统纳入朗之万系统的能量景观框架，并进一步利用这个框架来解释系统的行为。 具体来说，该项目回答了关于随机粗粒化和大偏差的开放问题。 而不是标准的平均来消除噪声，研究人员提供的方法来保留确定性项和波动项，以考虑多个噪声源的相互作用，以及推导出有效的方程的全局变量（能量）。 从粗粒度的方程，该项目提供的方法，渐近近似的过渡时间之间的亚稳态和显示等效的不同技术之间的不同limits.This奖项反映了NSF的法定使命，并已被认为是值得通过评估使用基金会的智力价值和更广泛的影响审查标准的支持。

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.