Exact solvability of the Kardar-Parisi-Zhang stochastic partial differential equation

Kardar-Parisi-Zhang 随机偏微分方程的精确可解性

• 批准号: 1438867
• 负责人: Ivan Corwin
• 金额: $ 10.57万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-02-20 至 2016-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1438867&HistoricalAwards=false
• 关键词: Exact; solvability; Kardar; Parisi; Zhang

The Kardar-Parisi-Zhang (KPZ) equation is a non-linear stochastic partial differential equation whose statistical properties are believed to describe a large class of mathematical models including interacting particle systems, random growth models, directed polymers, and branching diffusion processes. The purpose of this project is to make significant progress towards developing a theory of the exact solvability of the KPZ equation. In particular, this theory should lead to exact formulas for the probability distributions of the solution to the KPZ equation when started with various important types of initial data. This project will involve a number of fields of mathematics and this direction has already produced results of independent interest to these fields, which include: Macdonald symmetric function theory, tropical combinatorics and Whittaker functions, and certain quantum integrable systems. Since its discovery two hundred years ago the Gaussian distribution (bell curve) has come to represent one of mathematics greatest societal and scientific contributions ? a robust theory explaining and analyzing much of the randomness inherent in the world. Physical and mathematical systems accurately described in terms of Gaussian statistics are said to be in the Gaussian universality class. This class, however, is not all encompassing. For example, classical extreme value statistics or Poisson statistics better capture the randomness and severity of events ranging from natural disasters to emergency room visits. More recently, significant research efforts have been focused on understanding systems which are not well-described in terms of any of the classically developed statistics. The failure of these systems to conform to classical descriptions is generally due to a non-linear relationship between natural observables and underlying sources of random inputs and noise. A variety of models for complex systems such as growth processes, polymer chains, mass transport, traffic flow, queueing theory, driven gases, and turbulence have been actively studied for over forty years in mathematics, physics, material science, chemistry and biology. All of these systems fail to conform with classical Gaussian statistics, as has been observed through experimental evidence involving turbulent liquid crystals, crystal growth on a thin film, facet boundaries, bacteria colony growth, paper wetting, crack formation, and burning fronts. Surprisingly, despite their differences, all of the systems fall into a new statistical universality class whose properties are described in terms of a single model called the Kardar-Parisi-Zhang (KPZ) equation. The purpose of this project is to develop a statistical understand of the KPZ equation and its universality class.

Kardar-Parisi-Zhang（KPZ）方程是一种非线性随机偏微分方程，其统计特性被认为描述了大量数学模型，包括相互作用的粒子系统，随机生长模型，定向聚合物和分支扩散过程。该项目的目的是在发展KPZ方程的确切可溶性理论方面取得重大进展。特别地，该理论应导致确切的公式，以便从各种重要类型的初始数据开始时，将解决方案的概率分布到KPZ方程。该项目将涉及许多数学领域，并且该方向已经产生了这些领域的独立兴趣结果，其中包括：MacDonald对称功能理论，热带组合和Whittaker功能以及某些量子集成系统。自从两百年前的发现以来，高斯分布（贝尔曲线）已经代表了数学最大的社会和科学贡献之一？一种强大的理论，解释和分析了世界上固有的许多随机性。从高斯统计数据中精确描述的物理和数学系统被认为是高斯普遍性类别。但是，这个课程并非全部包含。例如，经典的极值统计或泊松统计量更好地捕获了从自然灾害到急诊室就诊的事件的随机性和严重性。最近，重大的研究工作集中在理解系统上，这些系统在任何经典开发的统计数据方面都没有很好地描述。这些系统无法符合经典描述的未能是由于自然可观察物与随机输入和噪声的潜在来源之间的非线性关系。在数学，物理学，材料科学，化学和生物学领域，已经积极研究了多种复杂系统的模型，例如生长过程，聚合物链，大众传输，交通流量，排队理论和湍流。所有这些系统都无法符合经典的高斯统计数据，这是通过涉及湍流液晶，薄膜上的晶体生长，刻面边界，细菌菌落生长，纸张润湿，裂纹形成和燃烧前沿的实验证据观察到的。令人惊讶的是，尽管它们的差异差异，但所有系统都属于一个新的统计普遍性类别，其属性是用一个称为Kardar-Parisi-Zhang（KPZ）方程的单个模型来描述的。该项目的目的是建立对KPZ方程及其普遍性类别的统计理解。