Variational Problems in Analysis and Physics

数学物理的目的是在我们周围的世界和物理定律之间提供一种合理的联系。我们理解为什么物质是在非相对论量子力学的基础上扩展的。我们能够解释为什么某些物质在温度降低时会改变它们的质量，例如，当水变成冰的时候。然而，有许多现象并没有得到严格的理解。物理系统趋于平衡是一种日常体验：热咖啡通过向环境释放能量来冷却，直到温度保持不变。对称性可以被打破；例如，在高温下看起来是均匀的物质往往会失去其均质性，也就是说，当冷却时它会形成团块。物理系统试图达到能量最低的状态。核心问题是如何描述这种状态，以及系统如何进行过渡。热咖啡如何通过冷却来达到与环境的平衡？物质如何通过转变成不那么均匀的形状来降低其能量？另一个可能不那么明显的例子是一根导热棒，它的一端保持在高温下，另一端保持在低温下。这个系统不是处于平衡状态，而是处于稳定状态；热量不断地从热到冷流动。虽然这是一个非常古老的问题，但对于这一观察结果，没有令人满意的数学严谨的微观解释。这项建议的目的是在特定的数学和物理模型中研究这些问题。其中一些是具有许多相互作用的代理的大型物理系统。其他的，从表面上看，是非常简单的，比如磁铁中的一个带电粒子。要找到这些问题的答案，需要有新的数学见解。该项目的一个重要特点是利用物理洞察力和数学技术的相互作用。该项目交织了数学物理的几个分支：通过经典和量子力学主方程的非平衡统计力学，涉及磁场和更一般矢量场的变分问题，以及与经典辐射场相互作用的带电系统。这项调查有一个共同的主题--变量不平等。其中一项努力是延续最近在解决经典和量子力学领域中其他问题的平衡方法方面的强劲进展。同样，用于现实模型的Kac型主方程的差距现在是触手可及的。Kac主方程非常适合于用有限的蓄水池来近似恒温器。PI将研究两个恒温器系统在不同温度下的非平衡稳态(NESS)的性质。下一步是研究有限水库对这类系统的逼近，特别是确定这些逼近所适用的各种时间尺度。一个广阔的领域是涉及矢量场的变分演算。这个项目将研究一类共形不变不等式，其中包括尖锐常数的计算。目标是利用这些洞察力来阐明具有磁场的系统，其中波函数是复杂的。另一个不同但密切相关的问题是超临界电荷的Maxwell-Pauli-Coulomb方程的分析。此次调查的焦点将集中在是否存在解决方案爆炸的问题上。这一裁决反映了NSF的法定使命，并已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。

The aim of mathematical physics is to provide a well reasoned connection between the world around us and the laws of physics. We understand why matter is extended on the basis of non-relativistic quantum mechanics. We are able to explain why certain materials change their quality when the temperature is lowered, as for example, when water turns to ice. There are, however, many phenomena that are not understood in a rigorous way. It is an everyday experience that physical systems tend towards equilibrium: Hot coffee cools by giving up energy to the environment until the temperatures are the same. Symmetry can be broken; for instance matter that appears to be homogeneous at high temperature tends to loose its homogeneity, that is, it forms clumps when cooled down. Physical systems try to achieve a state of lowest energy. Central questions are how to describe this state and how the system makes the transition. How does hot coffee approach an equilibrium with its environment by cooling down? How can matter lower its energy by transitioning to a less homogeneous shape? Another example, maybe less obvious, is a heat-conducting rod with one end held at a high temperature and the other at low temperature. This system is not in equilibrium but in a steady state; heat keeps flowing from hot to cold. Although a very old problem, there is no satisfactory mathematically-rigorous microscopic explanation for this observation. The aim of this proposal is to study these questions in specific mathematical and physical models. Some of these are large physical systems with many interacting agents. Others are, from a superficial perspective, quite simple, such as a single charged particle in a magnet. Finding answers to these questions requires new mathematical insights. An important feature of the project is to exploit the interaction of physical insight and mathematical techniques. This interdisciplinary quality makes it an ideal training ground for students at all levels.The project interweaves several strands of mathematical physics: non-equilibrium statistical mechanics through classical and quantum mechanical master equations, variational problems involving magnetic fields and more general vector fields as well as charged systems interacting with a classical radiation field. The investigation has variational inequalities as a common theme. One endeavor is to carry over recent robust advances concerning approach to equilibrium to other problems, both in the classical and quantum mechanical realm. Likewise, the gap for Kac type master equations for realistic models is now within reach. The Kac master equation is ideal for approximating thermostats by finite reservoirs. The PI will investigate the properties of the non-equilibrium steady state (NESS) for a system of two thermostats at different temperatures. The next step is to study the approximation of such systems by finite reservoirs, in particular determining the various time scales over which these approximations hold. A wide open area is the calculus of variations involving vector fields. This project will investigate a class of conformally invariant inequalities that includes the computation of the sharp constants. The goal is to use these insights to shed some light on systems with magnetic fields where the wave function is complex. A different but closely related problem is the analysis of the Maxwell-Pauli-Coulomb equations for supercritical charges. The focus of this inquiry will be on the question of whether there is blow up of solutions.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.