Applications of Number Theory and Probability in problems in Mathematical Physics

数论和概率在数学物理问题中的应用

• 批准号: EP/J004529/2
• 负责人: Igor Wigman
• 金额: $ 9.51万
• 依托单位: King's College London
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2013
• 资助国家: 英国
• 起止时间: 2013 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FJ004529%2F2
• 关键词: Applications; Number; Theory; Probability; problems

First observed by the physicist and musician Ernst Chladni in the 18th century, the nodal structures (also referred to as the Chladni Plates or Chladni Modes) appear in many problems in engineering, physics and natural sciences. The nodal patterns describe the sets that remain stationary during membrane vibrations, hence their importance in such diverse areas as musical instruments industry, mechanical structures, earthquake study and other fields. For example it is believed that the symmetries of musical instruments' nodal lines reflect or infer the beauty or quality of their sound. In addition, empirical observations show that the maximal destruction or damage inflicted by earthquakes is along nodal patterns, and hence they are important in a city planning considerations and related issues. They also arise in the study of wave propagation, cosmology, and astrophysics; this is a very active and rapidly developing research area.So far, the nodal structures have been mainly addressed in the physicists' literature. In his seminal paper, Michael Berry (1977) suggested that the behaviour of the nodal patterns corresponding to the high frequency vibration on chaotic-shaped membranes (meaning that the trajectory of free particles is equidistributed on that membrane; this, for example, excludes the sphere and the torus) is universal, and corresponds to the Random Wave, studied earlier by Longuet-Higgins as a model for ocean waves. Extensive numerical experiments confirm his predictions. Later, Blum, Gnutzmann and Smilansky (2002) studied some aspects of nodal structures numerically; in particular, they distinguish between the chaotic case and the completely integrable one (such as the torus or the sphere; here the dynamics of the free motion is well understood). Following this work, Bogomolny and Schmit (2002) introduced the elegant percolation-like model, that explains some of the aspects of nodal patterns discovered by Blum-Gnutzmann-Smilanky. Despite the fact that, thanks to those efforts, a lot is understood about the nodal structures, only few rigorous statements are known.In this research I propose to investigate the nodal structures with mathematical rigour. In particular I would like to find some evidence that would either support or contradict the physicists' predictions and empirical observations.

由物理学家和音乐家Ernst Chladni于18世纪首次观察到的节点结构(也称为Chladni板或Chladni模)出现在工程、物理和自然科学的许多问题中。节点模式描述了在薄膜振动过程中保持静止的集合，因此它们在乐器工业、机械结构、地震研究等领域中具有重要意义。例如，人们认为乐器节线的对称性反映或推断了它们的声音的美或质量。此外，经验观察表明，地震造成的最大破坏或破坏是沿着节点模式的，因此它们在城市规划考虑和相关问题中是重要的。它们也出现在波传播、宇宙学和天体物理学的研究中，这是一个非常活跃和迅速发展的研究领域。到目前为止，节点结构主要是在物理学家的文献中讨论的。Michael Berry(1977)在他的开创性论文中提出，与混沌形状的膜上的高频振动相对应的节块图案的行为是普遍的(这意味着自由粒子的轨迹在该膜上均匀分布；例如，这不包括球体和环面)，并且对应于Longuet-Higgins早先作为海浪模型研究的随机波。大量的数值实验证实了他的预测。后来，Blum，Gnutzmann和Smilansky(2002)用数值方法研究了节点结构的某些方面；特别是，他们区分了混沌情况和完全可积情况(如环面或球体；这里自由运动的动力学是很好理解的)。在这项工作之后，Bogomolny和Schmit(2002)引入了优雅的类渗流模型，该模型解释了Blum-Gnutzmann-Smilanky发现的节点模式的一些方面。尽管由于这些努力，人们对节点结构有了很多了解，但对节点结构的严格描述却知之甚少。在这项研究中，我建议用严格的数学方法来研究节点结构。特别是，我希望找到一些证据来支持或反驳物理学家的预测和经验观察。