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

基本信息

批准号：
2052789
负责人：
Natasa Pavlovic
金额：
$ 37.99万
依托单位：
University of Texas at Austin
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=2052789&HistoricalAwards=false
关键词：
FRG Collaborative Research New Challenges

项目摘要

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演化偏微分方程（PDE）与推导出这些方程的自然祖先粒子系统之间的相互作用。所考虑的方程是从微观（玻色爱因斯坦凝聚）到宏观（深海中的异常波）波传播现象以及气体动力学（弗拉索夫方程）的基本模型。为了解决研究这些方程的重要挑战，PI 采用了一种结合确定性和概率观点的创新方法。根据这些偏微分方程的定性性质，该项目的主要目标是在许多粒子水平上识别这些性质的正确类似物，并证明这些与偏微分方程水平上的已知性质相对应。该奖项将促进处于职业生涯各个阶段的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、传播和促进进一步研究的年度研究研讨会。在他们的分析中，PI 考虑了数学物理前沿的两个不同但密切相关的研究方向：非线性偏微分方程和概率。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 Vlasov 方程等动力学方程，以及在流体动力学背景下由 Ebin-Marsden 的开创性计划指导的此类推导的新颖观点。第二个方向植根于一维三次非线性薛定谔 (NLS) 方程的可积性，并追求两条研究路线。其中一个问题的重点是通过一系列项目探索一维立方 NLS 可积性的起源，这些项目旨在揭示多粒子级别的可积性的正确类似物，然后证明这些对应于 NLS 级别的已知属性。第二条探究源于 Lebowitz、Rose 和 Speer 的工作，他们假设均衡行为的宏正则系综描述对于可积偏微分方程预计是错误的。 PI 计划通过构建合适的替代品来解决这一重大的开放问题。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。