"Integrable Systems, Random Matrices and Random Processes"

“可积系统、随机矩阵和随机过程”

• 批准号: 45858-2012
• 负责人: Harnad, John
• 金额: $ 1.46万
• 依托单位: Concordia 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=568929
• 关键词: Integrable; Systems; Random; Matrices; Processes

The range of areas to which the theory of classical and quantum integrable systems is applicable has been rapidly growing since the break-through discovery of the inverse scattering method within the field of soliton physics in the late 1960's. These now include such diverse applications as: the spectral theory of random matrices; integrable random processes, such as "exclusion" processes, crystal and polymer growth, and the remarkable "integrability" properties discovered recently in gauge field theory, governing the fundamental interactions of matter. The problems addressed in this proposal touch upon all these fields. At the core are both new and well-developed methods of analysis, aimed at the solution of concrete problems relating to the underlying "integrable" dynamics common to all these areas. Integrability, whether quantum or classical or probabilistic, implies that, due to special structural properties, the dynamics of complicated, generally nonlinear dynamical systems, having either a finite, or infinite number of "degrees of freedom", can be reduced to the study of systems having only one degree of freedom. This appears in various guises, depending on the domain. In classical dynamical systems and classical field theory, it is related essentially to "separation of variables". In quantum systems, it appears in the form of factorization of the scattering matrix of the interacting system into two-particle amplitudes. In "integrable random processes", it appears in the fact that all correlation functions can be expressed in the form of determinants, having two-particle correlation functions as entries. These results are applicable to a wide variety of physical problems of current physical interest, including: multi-periodic and quasi-periodic solutions to systems describing coherent nonlinear phenomena in optics, fluids, polymers and superconductors; the statistical distributions of the eigenvalues of random matrices and operators; the determination of generating functions for a variety of random processes, such as crystal growth, eigenvalue dynamics of random matrices, mutually-avoiding multiple random-walkers and the computation of scattering amplitudes in superconformally invariant gauge field theory in the "planar" limit.

自 20 世纪 60 年代末在孤子物理领域突破性发现逆散射方法以来，经典和量子可积系统理论的适用范围迅速扩大。这些现在包括多种应用，例如：随机矩阵的谱理论；可积的随机过程，例如“排除”过程、晶体和聚合物生长，以及最近在规范场理论中发现的显着的“可积”特性，控制着物质的基本相互作用。该提案中解决的问题涉及所有这些领域。 其核心是新的和完善的分析方法，旨在解决与所有这些领域共有的潜在“可积”动力学相关的具体问题。 可积性，无论是量子的、经典的还是概率的，都意味着，由于特殊的结构特性，具有有限或无限数量“自由度”的复杂的、通常非线性的动力系统的动力学可以简化为研究只有一个自由度的系统。根据领域的不同，它会以不同的形式出现。在经典动力系统和经典场论中，它本质上与“变量分离”有关。在量子系统中，它以将相互作用系统的散射矩阵分解为两个粒子振幅的形式出现。在“可积随机过程”中，事实表明所有相关函数都可以以行列式的形式表示，以二粒子相关函数作为条目。这些结果适用于当前物理感兴趣的各种物理问题，包括：描述光学、流体、聚合物和超导体中相干非线性现象的系统的多周期和准周期解；随机矩阵和算子的特征值的统计分布；确定各种随机过程的生成函数，例如晶体生长、随机矩阵的特征值动力学、相互避免的多个随机游走以及“平面”极限下超共形不变规范场理论中的散射幅度的计算。