Applications of Number Theory and Probability in problems in Mathematical Physics

物理学家和音乐家Ernst Chladni在18世纪世纪首次观察到，节点结构（也称为Chladni板块或Chladni模式）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、地震研究和其他领域中具有重要意义。例如，人们认为乐器的节线的对称性反映或推断了它们声音的美或质量。此外，经验观察表明，地震造成的最大破坏或损害是沿沿着节点模式，因此，他们是重要的城市规划考虑和相关问题。波节结构也出现在波传播、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。迈克尔·贝里（Michael Berry，1977）在他的开创性论文中提出，对应于混沌形膜上高频振动的波节模式的行为（意味着自由粒子的轨迹在该膜上均匀分布;例如，这排除了球体和环面）是普遍的，并且对应于朗格特-希金斯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 and 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的情况（如环面或球面;这里自由运动的动力学很好理解）。在这项工作之后，Bogomolny和Schmit（2002）引入了优雅的类双折射模型，解释了Blum-Gnutzmann-Smilanky发现的节点模式的一些方面。尽管事实上，由于这些努力，很多是了解的节点结构，只有少数严格的语句是已知的。在这项研究中，我建议调查的节点结构与数学的严谨性。特别是我想找到一些证据来支持或反驳物理学家的预测和经验观察。

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.