FRG: Collaborative Research: New Challenges in the Derivation and Dynamics of Quantum Systems

FRG：协作研究：量子系统推导和动力学的新挑战

• 批准号: 2052740
• 负责人: Andrea Nahmod
• 金额: $ 39万
• 依托单位: University of Massachusetts Amherst
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-07-01 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2052740&HistoricalAwards=false
• 关键词: FRG; Collaborative; Research; New; Challenges

The main scientific goal of this project is to study the interplay between certain nonlinear evolution partial differential equations (PDE) and the natural progenitor particle systems from which these equations are derived. The equations considered are fundamental models for wave propagation phenomena ranging from the microscopic (Bose Einstein Condensate) to the macroscopic (rogue waves in deep sea), and for the dynamics of gases (Vlasov equation). To address important challenges in studying these equations, the PIs adopt an innovative approach combining deterministic and probabilistic perspectives. Informed by the qualitative properties of these PDE, the principal objective of this project is to identify the correct analogues of such properties at the many particle level, and to demonstrate that these correspond to the known properties at the PDE level. The award will foster collaborations among US based researchers at various stages of their careers and provide research opportunities and support for students and postdoctoral scholars. Additional activities include three annual research workshops aimed at training, dissemination, and stimulate further research. In their analysis, the PIs consider two different but intimately related research directions at the forefront of mathematical physics, nonlinear PDE and probability. The first direction concerns the derivation of the Hamiltonian structure for nonlinear evolution equations, including kinetic equations such as the Vlasov equation, as well as a novel viewpoint on such derivations guided by Ebin-Marsden's seminal program in the context of hydrodynamics. The second direction is rooted on the integrability of the 1D cubic nonlinear Schrodinger (NLS) equation and pursues two lines of inquiry. One of these questions focuses on exploring the origins of integrability of the 1D cubic NLS through a series of projects aimed at unveiling correct analogues of integrability at the many particle level, and then at demonstrating that these correspond to the known properties at the NLS level. A second line of inquiry stems from the work of Lebowitz, Rose and Speer who posited that the grand canonical ensemble description of equilibrium behavior is expected to be false for integrable PDE. The PIs plan to settle this major open problem by constructing a suitable substitute.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.

该项目的主要科学目标是研究某些非线性进化部分微分方程（PDE）与得出这些方程的天然祖细胞粒子系统之间的相互作用。所考虑的方程是波传播现象的基本模型，从微观（玻色爱因斯坦冷凝物）到宏观（深海中的流氓波）以及气体动力学（vlasov方程）。为了应对研究这些方程式的重要挑战，PI采用了一种创新的方法，结合了确定性和概率的观点。在这些PDE的定性特性中，该项目的主要目标是确定许多粒子级别的此类属性的正确类似物，并证明这些属性与PDE级别的已知属性相对应。该奖项将在其职业生涯的各个阶段促进美国研究人员之间的合作，并为学生和博士后学者提供研究机会和支持。其他活动包括三个旨在培训，传播和刺激进一步研究的年度研究研讨会。 在他们的分析中，PI在数学物理学，非线性PDE和概率的最前沿考虑了两个不同但密切相关的研究方向。第一个方向涉及非线性演化方程的哈密顿结构的推导，包括诸如弗拉索夫方程等动力学方程，以及在流体动力学的背景下以埃比·玛尔斯（Ebin-Marsden）的精神分裂程序指导的这种派生的新观点。第二个方向植根于1D立方非线性Schrodinger（NLS）方程的集成性，并追求两行查询。这些问题之一的重点是通过一系列旨在在许多粒子级别揭示可合准性的类似物的一系列项目来探索1D立方NLS的整合性的起源，然后证明这些项目与NLS级别的已知属性相对应。第二行调查源于Lebowitz，Rose and Speer的工作，他们认为，对均值PDE的宏伟规范合奏的描述预计将是错误的。 PIS计划通过构建合适的替代品来解决这一重大开放问题。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛影响的评论标准来评估值得支持的。