Fluctuations in Polymers and Random Data Partial Differential Equations

这个项目将研究随机噪声对物理系统理想化模型的影响。随机噪声的加入有时会简化复杂系统的分析，但也可能从根本上改变它们的行为并产生意想不到的现象。这两种情况都适用于本项目中研究的两类系统：所谓的随机聚合物模型，它描述了凝聚态物质中嘈杂的、不断增长的界面，以及用于模拟量子波和经典波的非线性微分方程。该项目还将更广泛地使用数理统计物理作为动机，培训本科生和研究生学习概率理论，概率理论是现代计算机科学和人工智能中的基本工具。更具体地说，该项目的目标是开发强大的工具来估计随机聚合物和相关扩散模型的波动，最终目标是将精确代数方法获得的结果推广到更广泛的模型类别，在这些模型中，精确方法不适用。关于非线性波动和薛定谔方程，该项目将研究给定随机初始数据的解的分布的演变，特别强调极端情况，如小非线性或小分散。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。

This project will investigate the effect of random noise on idealized models of physical systems. The addition of random noise sometimes simplifies the analysis of complex systems, but can also fundamentally modify their behavior and give rise to unexpected phenomena. Both situations apply to the two classes of systems studied in this project: so-called random polymer models, which describe noisy, growing interfaces in condensed matter, and nonlinear differential equations that are used to model quantum and classical waves. The project will also use mathematical statistical physics more generally as a motivation to train undergraduate and graduate students in probability theory, an essential tool in modern computer science and artificial intelligence.More specifically, the aim is to develop robust tools to estimate the fluctuations of random polymers and related diffusion models, with the ultimate goal of extending results obtained by exact algebraic methods to broader classes of models where exact methods do not apply. Concerning nonlinear wave and Schroedinger equations, the project will study the evolution of the distribution of solutions for given random initial data, with special emphasis on extremal situations like small nonlinearity or small dispersion.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.