Distances in random structures

随机结构中的距离

• 批准号: 341845-2012
• 负责人: AddarioBerry, Louigi
• 金额: $ 1.82万
• 依托单位: McGill University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=572383
• 关键词: Distances; random; structures

Phase transitions are ubiquitous in the natural world -- they are so commonplace that their strangeness is almost invisible to us. Why should water turn suddenly to ice at 0 degrees Celsius, rather than becoming gradually thicker as the temperature drops? Why will iron magnetize at 770 degrees but not at 769? Developing a mathematical theory for such phenomena is one of the most difficult and exciting mathematical challenges of the age. One of the classical mathematical models of phase transitions is known as percolation, which models defects in an ordered material. For example, if microscopic faults develop within a crystal, whose molecular bonds form a lattice structure, one may ask whether the material will remain structurally sound or whether macroscopic cracks will develop. It turns out that there is a critical density of faults, just below which the material remains sound, and just above which the crystal will disintegrate. The percolation phase transition -- and indeed, phase transitions in many other systems -- exhibits the intriguing phenomenon of universality, in which the precise macroscopic structure of the model does not seem to affect the qualitative macroscopic behaviour of the phase transition. Universality has been rigorously established in very few cases, and the mathematical treatment of universality is one of the major driving problems of the subject. One of my primary research interests, and a key focus of this proposal, is studying percolation phase transitions in high dimensional systems, with an ultimate aim of establishing universality. High dimensional systems appear throughout the physical and biological sciences: in this context the dimension of a system should be thought of as its number of degrees of freedom, or in other words the number of parameters that must be specified to fully describe the state of the system. For example, in classical Euclidean mechanics, to describe the state of a particle requires six parameters -- three to describe spatial position and three to specify velocity -- so one particle already forms a six-dimensional system.

相变在自然界中无处不在--它们是如此常见，以至于我们几乎看不见它们的奇怪之处。为什么水在0摄氏度时会突然结冰，而不是随着温度的下降而逐渐变厚？为什么铁在770度时会磁化，而在769度时不会磁化？为这种现象开发数学理论是这个时代最困难和最令人兴奋的数学挑战之一。 相变的经典数学模型之一被称为渗流，它模拟有序材料中的缺陷。例如，如果晶体内部出现微观缺陷，其分子键形成晶格结构，人们可能会问，材料是否会保持结构完好，或者宏观裂缝是否会发展。事实证明，存在一个断层的临界密度，低于这个密度，材料就保持完好，超过这个密度，晶体就会解体。渗流相变--实际上，许多其他系统中的相变--表现出有趣的普适性现象，在这种现象中，模型的精确宏观结构似乎不影响相变的定性宏观行为。普适性在极少数情况下得到了严格的确立，普适性的数学处理是这一课题的主要驱动问题之一。 我的主要研究兴趣之一，也是这个提议的一个关键焦点，是研究高维系统中的渗流相变，最终目的是建立普适性。高维系统出现在整个物理和生物科学中：在这种情况下，系统的维度应该被认为是其自由度的数量，或者换句话说，必须指定的参数的数量才能完全描述系统的状态。例如，在经典的欧几里德力学中，描述一个粒子的状态需要六个参数--三个描述空间位置，三个指定速度--所以一个粒子已经形成了一个六维系统。