Variational problems in physics

物理学中的变分问题

• 批准号: 1301555
• 负责人: Michael Loss
• 金额: $ 17.1万
• 依托单位: Georgia Tech Research Corporation
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-08-01 至 2017-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1301555&HistoricalAwards=false
• 关键词: Variational; problems; physics

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.

该提案是关于不同的物理和数学应用程序，有变分法作为一个共同的主题。研究的一个领域是非平衡统计力学，其目标是理解简单模型中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。第二个主题是有关对称性破缺或不存在一类非线性变分问题。 我们的计划是找到一个演化方程，让任何函数演化成一个优化函数。这是一个已经在许多有趣的案例中起作用的概念。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，其目标是将随机位移模型的最新结果扩展到更大的一类几何。 第四个主题，将调查，是几何之间的联系曲线和某些特殊的Lieb-Thirring型不等式。 虽然这些主题听起来很不一样，但它们都可以被视为优化问题，并可能导致对变分法的新的和意想不到的数学见解。数学是重要的，因为它的普遍性，无论是作为一种语言，还是作为解决科学和工程问题的工具箱。 这个建议的目的是了解从物理科学中产生的一系列问题。 一个核心问题是如何描述相互作用的主体的大型系统的演化，无论是碰撞的粒子，成群的鸟类还是在经济中交换商品的个人。 这样的系统通常处于平衡状态，但会对外部干扰作出反应。重要的是要了解这样一个系统是否会恢复平衡，如果是的话，有多快。 这项资助的一个重点是在碰撞粒子的大系统的背景下理解这些问题。研究的第二个领域涉及随机性对量子力学粒子运动的影响。我们的目标是比目前所知的更详细地了解随机移动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建模绝缘体的方法。 这一研究领域在数学上的迷人之处在于，大多数问题都可以被表述为优化问题，从而利用了一个大型的数学工具包，同时又增加了它。这一研究领域也是学生的一个很好的训练基地，因为他们被迫超越问题的纯数学方面。