Variational Problems in Analysis and Physics

物理系统趋于平衡。机械系统通过向环境释放能量来达到能量最低的状态。同样，热力学系统中的局部扰动，如房间内的温度波动，往往会趋于平衡；即，系统会松弛到热平衡。然而，在某些情况下，由于外部影响，系统不会趋向于平衡。一个著名的例子是一块金属，它的一端加热，另一端冷却。在这种情况下，当热量从热端流向冷端时，系统不会松弛到平衡状态，而是保持稳定状态。这个建议的目标是用各种数学模型来描述这些状态，范围从基本的优化问题到相互作用的大型代理系统。我们将考虑在数学中很重要或在描述物理现象时普遍存在的最优化问题，例如超导。我们的目标是回答这样的问题：一个大系统达到平衡或趋于稳定状态的速度有多快？这些均衡的性质是什么？如果一个系统有一些潜在的对称性，那么平衡点也有相同的对称性吗？这一研究领域最吸引人的方面是，问题的范围从可以使用标准方法解决的问题到需要相当巧妙的解决方案的问题。这些特点也使它成为研究生的理想训练基地。系统失衡的分析仍然是一个广泛的开放领域；例如，物理系统与热水浴的相互作用还没有被很好地理解。这样的系统并不趋向于平衡，而是趋向于非平衡稳态(NESS)。PI建议在主方程的框架内研究这种现象。研究目标之一是找到李亚普诺夫函数，以控制通向NESS的方法。对这类系统的研究往往会导致关于能隙和熵产生的变分问题，而它们的解需要泛函不等式。重要的是要以其尖锐的形式证明这些不平等，并知道所有对其存在相等的函数。研究的一个方向是使用流来研究这种不平等，这是Lyapunov函数的逆思想。这种方法在一些非常重要的情况下取得了成功，预计它不仅会带来尖锐的不平等，而且还会带来额外的修正项。描述电子与晶体振动相互作用的极化子模型也符合这个框架。这是量子场论最简单的例子之一。对能量的强耦合极限已有了很好的认识，并建议研究更微妙的问题，即电子密度的强耦合极限。

Physical systems tend towards equilibrium. Mechanical systems achieve a state of lowest energy by giving off energy to the environment. Likewise, local disturbances in a thermodynamic system, such as temperature fluctuations in a room, tend to even out; i.e., the system relaxes towards a thermal equilibrium. However, there are instances where due to external influences, systems do not tend towards an equilibrium. A famous example is a piece of metal that is heated on one end and cooled at the other. In this case, the system does not relax into equilibrium but remains in a steady state as heat flows from the hot to the cold end. The objective of this proposal is to describe these states in terms of various mathematical models ranging from basic optimization problems to large systems of interacting agents. We will consider optimization problems that are important in mathematics or ubiquitous in descriptions of physical phenomena, such as superconductivity. Our goal is to answer questions such as: How fast does a large system equilibrate or tend towards a steady state? What are the properties of these equilibria? If a system has some underlying symmetry, do equilibria have the same symmetry? The fascinating aspect of this research area is that problems run the gamut from ones that can be solved using standard methods to those that need considerable ingenuity for their solution. These characteristics also make it an ideal training ground for graduate students.The analysis of systems out of equilibrium is still a wide open field; e.g., the interaction of a physical system with heat baths is not well understood. Such systems do not tend towards equilibrium but instead towards a non-equilibrium steady state (NESS). The PI proposes to examine such phenomena within the frame work of master equations. One of the research goals is to find Lyapunov functions that allow to control the approach towards an NESS. The investigation of such systems often leads to variational problems for the gap as well as for the entropy production, and their solution requires functional inequalities. It is important to prove these inequalities in their sharp form and to know all functions for which there is equality. One direction of research is to investigate such inequalities using flows which is the idea of Lyapunov functions in reverse. This method has been successful in some highly non-trivial cases and it is expected that it will deliver not only sharp inequalities but also additional correction terms. The Polaron model which describes an electron interacting with the vibrations of a crystal, fits into this framework as well. This is one of the simplest examples of a quantum field theory. The strong coupling limit of the energy is well understood and it is proposed to study the more subtle problem, namely the strong coupling limit of the electron density.