Topics in Wave Propagation and Quantum Systems

波传播和量子系统主题

• 批准号: 2006416
• 负责人: Olivier Pinaud
• 金额: $ 25.48万
• 依托单位: Colorado State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-11-01 至 2024-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2006416&HistoricalAwards=false
• 关键词: Topics; Wave; Propagation; Quantum; Systems

Wave phenomena are ubiquitous in nature, and are virtually in every aspect of every day life. They are responsible for the sound we hear, the light we see, and the data we receive on our personal devices. Understanding how to mathematically model these waves, how to control them, and how to extract valuable information from them is therefore an important question at the intersection of various fields. The underlying physics principles behind wave propagation are extremely complex, and mathematics has proven to be a crucial tool in their investigation. At the core of this project are the theoretical analysis and the simulation of wave phenomena, with an emphasis on three main problems: how to focus energy with a minimal experimental apparatus; how to infer physical properties of a complex medium using wave propagation; and how to model waves in novel quantum devices. This project integrates research with the training of the next generation of scientists in the field of applied mathematics.The main theme of research of the project is wave phenomena, including both classical and quantum waves. There are three main components to the proposed research. The first one originates from the recent groundbreaking experiments on instantaneous time mirrors. The latter form a novel avenue for time reversal and the control of waves, and is an exciting new topic for mathematicians to explore. The second component concerns inverse problems based on stochastic correctors, which are considered physical systems where forward data are solutions to PDEs with small parameters and random coefficients. In these systems, the data are asymptotically modeled by random correctors to a leading term that is not accessible to measurements. The objectives are first to characterize these stochastic correctors, and second to develop inversion strategies based on these corrections to extract information on the system. A typical example is the “sea ice problem”, which involves random fluctuations around a homogenized solution. The last component of this project concerns quantum systems. The first problem is to develop the mathematical foundations of quantum hydrodynamical models based on the entropy principle, and the second the derivation of efficient numerical schemes for periodic Dirac operators, with applications to the study of exotic phases of matter and the design of novel quantum devices.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.

波动现象在自然界中无处不在，几乎存在于日常生活的各个方面。它们负责我们听到的声音、看到的光以及我们在个人设备上接收的数据。因此，了解如何对这些波进行数学建模、如何控制它们以及如何从中提取有价值的信息是各个领域交叉点的一个重要问题。波传播背后的基本物理原理极其复杂，数学已被证明是其研究的关键工具。该项目的核心是波浪现象的理论分析和模拟，重点解决三个主要问题：如何用最小的实验装置聚集能量；如何利用波传播推断复杂介质的物理性质；以及如何对新型量子设备中的波进行建模。该项目将应用数学领域的研究与下一代科学家的培训结合起来。该项目研究的主题是波现象，包括经典波和量子波。拟议研究由三个主要组成部分组成。第一个源于最近关于瞬时时间镜的突破性实验。后者形成了时间反转和波控制的新途径，是数学家探索的一个令人兴奋的新课题。第二个组成部分涉及基于随机校正器的逆问题，随机校正器被认为是物理系统，其中正向数据是具有小参数和随机系数的偏微分方程的解。在这些系统中，数据由随机校正器渐近建模为测量无法访问的主导项。目标首先是表征这些随机校正器，其次是根据这些校正开发反演策略以提取系统信息。一个典型的例子是“海冰问题”，它涉及均质解决方案周围的随机波动。该项目的最后一个组成部分涉及量子系统。第一个问题是开发基于熵原理的量子流体动力学模型的数学基础，第二个问题是推导周期性狄拉克算子的有效数值方案，并将其应用于物质的奇异相研究和新型量子器件的设计。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进行评估，被认为值得支持 标准。

项目成果

On local quantum Gibbs states

关于局域量子吉布斯态

• DOI: 10.1063/5.0058574
• 发表时间: 2022
• 期刊: Journal of Mathematical Physics
• 影响因子: 1.3
• 作者: Duboscq, Romain;Pinaud, Olivier
• 通讯作者: Pinaud, Olivier

Instantaneous Time Mirrors and Wave Equations with Time-Singular Coefficients

瞬时时间镜和具有时间奇异系数的波动方程

• DOI: 10.1137/20m1384361
• 发表时间: 2021
• 期刊: SIAM Journal on Mathematical Analysis
• 影响因子: 2
• 作者: Pinaud, Olivier

Entropy Minimization for Many-Body Quantum Systems

多体量子系统的熵最小化

• DOI: 10.1007/s10955-021-02824-z
• 发表时间: 2021
• 期刊: Journal of Statistical Physics
• 影响因子: 1.6
• 作者: Duboscq, Romain;Pinaud, Olivier
• 通讯作者: Pinaud, Olivier

A simple real-space scheme for periodic Dirac operators

周期性狄拉克算子的简单实空间方案

• DOI: 10.4310/cms.2021.v19.n6.a12
• 发表时间: 2021
• 期刊: Communications in Mathematical Sciences
• 影响因子: 1
• 作者: Chen, Hua;Pinaud, Olivier;Tahir, Muhammad
• 通讯作者: Tahir, Muhammad

Time reversal of surface plasmons

表面等离子体激元的时间反转

• DOI: 10.3934/dcdsb.2022106
• 发表时间: 2022
• 期刊: Discrete and Continuous Dynamical Systems - B
• 作者: Pinaud, Olivier