Stochastic Dynamics on Energy Landscapes with Applications in Physics and Biology

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.