Variational problems in physics

该提案是关于以变分法为共同主题的不同物理和数学应用。一个研究领域是非平衡统计力学，其目标是在简单模型中理解非平衡稳态的简单方面，例如描述碰撞粒子与恒温器相互作用的kac型模型。第二个主题是关于一类非线性变分问题中的对称性破缺或不对称性破缺。我们的计划是找到能让任何函数演化为优化函数的演化方程。这个概念已经在许多有趣的案例中发挥了作用。第三个主题涉及量子力学中的问题，特别是关于随机薛定谔算子的一些问题，目的是将随机位移模型的最新结果扩展到更大的几何类。要研究的第四个主题是曲线几何与某些特殊的lieb - thirring型不等式之间的联系。虽然这些主题听起来非常不同，但它们都可以被视为优化问题，并且可以导致对变量演算的新的和意想不到的数学见解。数学之所以重要，是因为它具有普适性，既是一种语言，也是解决科学和工程问题的工具箱。这个提议的目的是为了理解物理科学中出现的一系列问题。一个核心问题是如何描述由相互作用的主体组成的大型系统的演化，无论是粒子碰撞、鸟群聚集还是经济体中交换商品的个体。这类系统通常处于平衡状态，但对外部干扰有反应。重要的是要了解这样一个系统是否能恢复平衡，如果能，恢复平衡的速度有多快。这项拨款的一个重点是在大型碰撞粒子系统的背景下理解这些问题。第二个研究领域涉及随机性对量子力学粒子运动的影响。目标是比目前已知的更详细地了解随机位移的障碍物如何影响电子的运动。理解这个问题是很重要的，因为它产生了另一种模拟绝缘体的方法。这条研究路线在数学上的迷人之处在于，这些问题中的大多数都可以被表述为优化问题，因此可以利用一个庞大的数学工具包，同时也可以增加它。这个研究领域对学生来说也是一个很好的训练基地，因为他们被迫看到问题的纯粹数学方面之外的东西。

The proposal is about disparate physical and mathematical applications that have the Calculus of Variations as a common theme. One area of inquiry is non-equilibrium statistical mechanics with the goal to understand simple aspects of non- equilibrium steady states in simple models, such as a Kac-type model describing the interaction of colliding particles with a thermostat. A second topic is related to symmetry breaking or the absence thereof in a class of non-linear variational problems. The plan is to find volution equations that let any function evolve into an optimizing function. This is a concept that has worked in a number of interesting cases already. The third topic concerns problems in quantum mechanics, in particular some problems about random Schroedinger operators with the goal to extend recent results on the Random Displacement Model to a larger class of geometries. The fourth topic that will be investigated, is the connection between the geometry of curves and certain special Lieb-Thirring-type inequalities. While these topics sound quite different, they all can be cast as optimization problems, and can lead to new and unexpected mathematical insights into the calculus of variations.Mathematics is important because of its universal nature, both as a language and as a toolbox for solving problems in science and engineering. The aim of this proposal is to understand a range of questions that arise from the physical sciences. A central problem is how to describe the evolution of large systems of interacting agents, be it colliding particles, flocking birds or individuals that exchange goods in an economy. Such systems are usually in an equilibrium but react to external disturbances. It is important to understand whether such a system returns to equilibrium and if so, how fast. One focus of this grant is on understanding these question in the context of large systems of colliding particles. A second area of inquiry concerns the effect of randomness on the motion of quantum mechanical particles. The goal is to understand in more detail than is presently known how randomly displaced obstacles influence the motion of an electron. An understanding of this question is important since it yields another way of modeling insulators. The mathematically fascinating aspect of this line of research is that most of these problems can be formulated as optimization problems, thus tapping into a large mathematical toolkit while at the same time adding to it. This research area also is an excellent training ground for students because they are forced to see beyond the purely mathematical aspect of a problem.