Random Matrices and Statistical Mechanics of Charged Particle Systems

我们研究带电粒子系统统计力学中的两个悬而未决的问题。第一个等价于随机矩阵的局部特征值统计量的普适性猜想，这是一个均衡问题。第二个问题是从一个微观模型--一个非平衡问题--构造相对论弗拉索夫动力学理论。对于第一个问题，我们使用一种新的、完全解析的策略，将酉阵的已知普适性推广到其他类型的随机矩阵：实对称矩阵、复法阵和四元数自对偶矩阵。这一策略适用于光谱的整体和边缘。我们还将我们的方法应用于研究具有相关结构的超导的Laughlin波函数。对于第二个问题，我们利用最近奠定的相对论多粒子理论的微观动力学基础，以弱大数定律的形式建立了相对论Vlasov动力学理论的第一个导数。我们还以中心极限定理的形式研究了它在极限附近的涨落。普适性猜想是概率论和数学物理中的一个子领域，目前是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，到数学方面的深层数字理论含义--这反过来又在密码学和相关领域中有应用。到目前为止，这个猜想对于最简单的矩阵类型已经得到了证明，但对于更一般的矩阵，到目前为止还难以得到证明。带电粒子系统的相对论Vlasov理论构成了大部分高温等离子体物理的动力学基础，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的研究。它的微观原子基础到目前为止一直没有得到研究人员的支持，这将使人们第一次有可能系统地研究弗拉索夫理论的准确性，特别是计算其领先的修正。

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.