Variational Questions in Mathematics and Physics

数学和物理中的变分问题

• 批准号: 2154340
• 负责人: Michael Loss
• 金额: $ 23.62万
• 依托单位: Georgia Tech Research Corporation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-07-01 至 2025-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2154340&HistoricalAwards=false
• 关键词: Variational; Questions; Mathematics; Physics

In an overwhelming number of questions in mathematics, physics, engineering and economics and other disciplines one seeks solutions that satisfy certain demands or constraints and importantly, one looks for an optimal one. The mathematical theory that addresses these questions is the calculus of variations. Variational questions are deeply rooted in nature in the sense that systems tend to settle in a state that has energy as low as possible. An example is an excited atom that falls to its lowest energy state while emitting light. Likewise, a hot liquid tends to an equilibrium with the environment by cooling down. The aim of research in this area is always twofold: describe the state to which a system tends and explain how the system reaches that state. In the case of the atom this means finding the lowest energy state and describing how it decays to that state; in the case for the liquid this means describing how heat flows from hot to cold. The research will pursue such questions in relation to new and relevant examples. There has been considerable progress in understanding how systems such as gases of colliding particles tend to their equilibrium. In research the investigator will extend these ideas to quantum mechanical systems and to analyze how they tend to their equilibrium. In quantum mechanics these questions are to a large extent open and their answers will shed some light on various issues in quantum information theory. Another question to be pursued concerns magnetic fields that keep electrons confined to a region. The challenge will be to find fields that are optimal in some precise sense and to describe how the electrons are distributed. It is expected that the fields’ lines have interesting patterns that resemble the ones created in fusion research where one uses magnetic fields to confine hot plasma. Graduate students will be exposed to and contribute to several questions in research level mathematics. The principal investigator is expected to significantly impact the community, as evidenced by their role as secretary of the International Association for Mathematical Physics.The project addresses various questions in the calculus of variations. One question is to study zero modes of the three-dimensional Dirac equation. Zero modes are important in quantum field theory and in the question of proving the stability of matter. It is well known that if the ‘3/2 norm’ of the magnetic field is small then the magnetic field cannot support a zero mode. A similar result holds for the ‘3 norm’ of the vector potential. The aim is to find sharp necessary conditions on these quantities zero modes exists. It is conjectured that the optimal fields have field lines that look like the Hopf fibration. In this connection investigations will be pursued concerning possible blow up of solutions of the coupled Maxwell-Pauli equations where the magnetic moment is larger than 2. A different research direction will be to understand the approach to equilibrium in certain quantum mechanical systems. Such systems can be described by Lindblad equations. In general, not much can be said about the rate of approach to equilibrium but there are interesting Quantum Markov Operators that are analogs of the classical Kac master equation. In these cases, quantitative determination is expected of the rate of approach to equilibrium. One possibility is to do this by computing the gap of the generator. Another attempt is to prove approach to equilibrium in entropy. This will be achieved by establishing analogs of classical entropy inequalities.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.

在数学、物理学、工程学、经济学和其他学科的大量问题中，人们寻求满足某些要求或约束的解决方案，重要的是，人们寻找最优的解决方案。 解决这些问题的数学理论是变分法。变分问题在自然界中根深蒂固，因为系统倾向于在能量尽可能低的状态下稳定下来。一个例子是一个受激原子，当它发光时，福尔斯降到它的最低能量状态。 同样地，热的液体通过冷却趋于与环境平衡。这一领域的研究目标通常是双重的：描述系统趋向的状态，并解释系统如何达到该状态。在原子的情况下，这意味着找到最低能量状态并描述它如何衰变到该状态;在液体的情况下，这意味着描述热量如何从热到冷。研究将结合新的相关实例探讨这些问题。在理解诸如气体碰撞粒子等系统如何趋向于平衡方面已经取得了相当大的进展。 在研究中，研究人员将这些想法扩展到量子力学系统，并分析它们如何趋于平衡。在量子力学中，这些问题在很大程度上是开放的，它们的答案将揭示量子信息理论中的各种问题。另一个需要研究的问题是磁场将电子限制在一个区域内。挑战将是找到在某种精确意义上最优的场，并描述电子是如何分布的。预计这些场的线具有有趣的图案，类似于在聚变研究中使用磁场来限制热等离子体所产生的图案。研究生将接触到并有助于研究水平数学的几个问题。首席研究员预计将显着影响社区，作为他们的作用证明了国际数学物理协会的秘书。该项目解决变分法中的各种问题。一个问题是研究三维狄拉克方程的零模。零模在量子场论和证明物质稳定性的问题中很重要。 众所周知，如果磁场的“3/2范数”很小，则磁场不能支持零模式。类似的结果也适用于矢势的“3范数”。目的是找到这些量零模式存在的尖锐的必要条件。它被证明是最佳领域的磁力线看起来像霍普夫纤维化。 在这方面的调查，将追求有关可能的爆炸的解决方案的耦合麦克斯韦-泡利方程的磁矩大于2。 一个不同的研究方向将是理解某些量子力学系统中的平衡方法。这样的系统可以用Lindblad方程来描述。一般来说，关于接近平衡的速率可以说的不多，但是有一些有趣的量子马尔可夫算子，它们是经典Kac主方程的类似物。在这些情况下，需要定量确定接近平衡的速度。一种可能性是通过计算发电机的差距来实现这一点。另一个尝试是证明熵接近平衡。这个奖项反映了NSF的法定使命，并被认为是值得通过使用基金会的知识价值和更广泛的影响审查标准进行评估的支持。