Critical Phenomena and Disorder Effects

关键现象和紊乱效应

• 批准号: 1613296
• 负责人: Michael Aizenman
• 金额: $ 24万
• 依托单位: Princeton University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-08-01 至 2020-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1613296&HistoricalAwards=false
• 关键词: Critical; Phenomena; Disorder; Effects

This award will fund research on the use of mathematical tools from probability theory to address long-standing questions in the physics of phase transitions, in an area known as statistical physics. Mathematical versions of the concepts of phase transitions, threshold behavior, critical phenomena, and scaling limits have value which is now well recognized in areas which at first sight might have seemed far from the statistical physics where these concepts originated. Mathematical studies of such topics have led to fundamental developments in modern probability theory. In turn, rigorous analysis has provided useful feedback on our understanding of the relevant physics. Examples of the latter are found in the disorder effects on phase transitions in two and three dimensional classical and quantum systems systems (such as the so-called Imry-Ma phenomenon), and in the spectral and dynamical effects of disorder in the context of random quantum operators. The PI was instrumental in past works of this type, including recently, in which he contributed key results which helped uncover new classical and quantum physical phenomena thanks to studies in probability theory. The projected research will both continue and redirect the PI's efforts, and will necessarily continue be interdisciplinary in nature. Beyond its use of probability theory and mathematical analysis on physics questions, the results of the research could potentially bring broader impacts to other areas where modern probability theory is helping elucidate new phenomena, such as computer science and engineering, and data science. Of particular note with this project's other broader impacts is the potential for outcomes of the highest caliber in training the next generation of US scientists. Based on the PI's past and recent supervision of extraordinarily talented graduate students and postdocs under NSF support, the project will certainly allow the PI to offer research experience of the highest level for such future trainees at Princeton University. Some of the research will be carried out in collaboration with top researchers from other institutions, providing valuable networking opportunities for the trainees. The PI's attention will be redirected towards critical phenomena in a number of instructive models below their upper critical dimension, a concept whose understanding was firmly advanced by PI's previous work on percolation, Ising spin systems and phi^4 field theory. It was recently noted that the techniques by which the PI has previously established the Gaussian (bosonic) nature of the Ising model's scaling limits in dimensions greater than four yield also simple proofs of the fermionic nature of certain correlation functions in the planar case. Some of these relations have been known through exact solution of the two dimensional model, but the new argument suggests a path towards explanation of "universal" emergent planarity in a class of critical non-planar and non-solvable two dimensional models. Related questions concerning critical phenomena will be explored for the three-dimensional case, which is neither trivial nor solvable yet of obvious interest. Work will also continue on classical and quantum effects of disorder on the spectral and dynamical properties of random operators, and on the structure of Gibbs equilibrium states of systems with quenched disorder.

该奖项将资助有关使用概率理论的数学工具使用的研究，以解决相变的物理学中长期存在的问题，该领域被称为统计物理学。相变，阈值行为，批判现象和缩放限制的数学版本具有价值，现在在乍看之下似乎与这些概念起源的统计物理学相去甚远。 此类主题的数学研究导致了现代概率理论的基本发展。 反过来，严格的分析为我们对相关物理学的理解提供了有用的反馈。 在疾病的影响中发现了后者的示例，对两个维度的经典和量子系统系统（例如所谓的IMRY-MA现象）以及在随机量子算子的背景下的频谱和动力学效应中发现了对相变的影响。 PI在包括最近在内的过去作品中发挥了重要作用，在其中，他取得了关键的结果，这有助于发现新的经典和量子物理现象，这要归功于概率理论的研究。 预计的研究将继续并重定向PI的努力，并且必然会继续是跨学科的本质。除了使用概率理论和对物理问题的数学分析外，研究的结果可能会带来更广泛的影响，以帮助现代概率理论有助于阐明新现象，例如计算机科学和工程以及数据科学。 特别值得注意的是，该项目的其他更广泛的影响是在培训下一代美国科学家的最高水平结果的潜力。根据PI在NSF支持下对非常有才华的研究生和博士后的过去和最新监督，该项目肯定会允许PI为普林斯顿大学的此类未来学员提供最高水平的研究经验。一些研究将与其他机构的顶级研究人员合作进行，为学员提供宝贵的网络机会。 PI的注意力将在其上方的许多临界维度以下的许多指导性模型中重定向到关键现象，这一概念通过PI先前在渗透，Ising Spin Systems和Phi^4现场理论方面坚定地提出了理解。 最近注意到，PI先前已经建立了Ising模型在尺寸上的缩放限制的高斯（骨）性质的技术，大于四个产率也简单地证明了平面案例中某些相关函数的费尔米金性质。 这些关系中的一些通过两个维度模型的精确解决方案已知，但是新的论点提出了在一类关键的非平面和不可分解的两个维度模型中解释“通用”新兴平面性的道路。 对于三维案例，将探讨有关关键现象的相关问题，该案例既不是微不足道的也不可解决的。 疾病对随机操作员的光谱和动力学特性以及静脉疾病系统的吉布斯平衡状态的结构的经典和量子影响也将继续进行工作。