Random Matrices and Statistical Mechanics of Charged Particle Systems

带电粒子系统的随机矩阵和统计力学

• 批准号: 0103808
• 负责人: Michael Kiessling
• 金额: $ 15.35万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-09-01 至 2005-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0103808&HistoricalAwards=false
• 关键词: Random; Matrices; Statistical; Mechanics; Charged

We work on two outstanding open problems in the statistical mechanics of charged particle systems. The first one is equivalent to the universality conjecture of the local eigenvalue statistics of random matrices, an equilibrium problem. The second one is the construction of relativistic Vlasov kinetic theory from a microscopic model, a nonequilibrium problem. As to the first problem, we use a new, entirely analytic strategy to extend the known universality for the unitary matrices also to the other types of random matrices: real symmetric, complex normal, and quaternionic self-dual. The strategy applies to the bulk and to the edge of the spectrum. We also apply our method to the study of the Laughlin wave function of superconductivity, which is of a related structure. As for the second problem, we use the recently laid microscopic dynamical foundations of relativistic many-particle theory to establish the first derivation of relativistic Vlasov kinetic theory in form of a weak law of large numbers. We also study its fluctuations around the limit in form of a central limit theorem. The universality conjecture is currently one of the top priority problems of random matrix theory, a subfield of probability and mathematical physics. Its applications range from nuclear physics, nanotechnology and superconductivity on the physics and technology side to deep number theoretical implications on the mathematical side - which in turn have applications in cryptography and related fields. The conjecture has been proven so far for the simplest type of matrices, but a proof for more general matrices has so far been elusive. The relativistic Vlasov theory of charged particle systems forms the dynamical basis for a large part of high temperature plasma physics, with applications ranging from thermonuclear fusion research to space plasma research, e.g. about the solar wind and magnetic storms. Its microscopic atomic underpinnings, which have so far eluded researchers, will make it possible for the first time to systematically study the accuracy of Vlasov theory and in particular to compute its leading corrections.

我们致力于带电粒子系统统计力学中两个悬而未决的问题。第一个等价于随机矩阵局部特征值统计量的普适性猜想，即一个平衡问题。第二个是从微观模型，一个非平衡问题，建立相对论性弗拉索夫动力学理论。至于第一个问题，我们使用一个新的，完全解析的策略，以延长已知的普遍性酉矩阵也到其他类型的随机矩阵：真实的对称，复正常，和四元数自对偶。该策略适用于光谱的主体和边缘。我们也将我们的方法应用于超导的Laughlin波函数的研究，这是一个相关的结构。对于第二个问题，我们利用最近建立的相对论多粒子理论的微观动力学基础，建立了相对论Vlasov动力学理论的弱大数定律形式的第一阶导数。我们还研究了它在中心极限定理形式的极限附近的波动。 普适性猜想是概率论和数学物理的一个分支随机矩阵理论的首要问题之一。它的应用范围从物理和技术方面的核物理，纳米技术和超导性到数学方面的深层数论含义-这反过来又在密码学和相关领域中有应用。到目前为止，这个猜想已经在最简单的矩阵类型中得到了证明，但是到目前为止，更一般的矩阵的证明还很难。带电粒子系统的相对论弗拉索夫理论构成了高温等离子体物理学的大部分动力学基础，其应用范围从热核聚变研究到空间等离子体研究，例如关于太阳风和磁暴。它的微观原子基础，迄今为止一直困扰着研究人员，这将使人们有可能首次系统地研究弗拉索夫理论的准确性，特别是计算其主要修正。