标题： 多维混沌 I：经典力学和量子力学 显示英文标题

标题： Multi-dimensional chaos I: Classical and quantum mechanics

摘要： 我们引入了多维混沌的概念，该概念适用于由多个动力学变量的不规则函数描述的过程。 我们将这一概念用于对弹球系统上经典和量子散射的解释。 在前一种情况下，通过散射角和碰撞次数的二维图示进行说明。 我们为不同圆盘几何结构的量子微分截面绘制了类似的图案。 我们发现，S矩阵的本征值根据随机矩阵理论（RMT）中的圆正交系综（COE）分布，前提是设置是不对称的且波数足够大。 然后，我们考虑电荷随机分布在平面上的电势作为一个简化模型，该模型推广了从漏磁环面的散射。 我们提出了几种分析该函数极值之间间隔的方法。 我们表明，即使电荷的位置服从泊松分布，这些间隔也遵循排斥性的高斯β系综分布。 引入并确定了谱形因子的一个推广。 我们将这些方法应用于混沌S矩阵和量子弹球散射的情况。 最近邻极值点之间的间隔以及相邻间隔之间的比率分别符合逻辑斯蒂分布和Beta分布。 我们推测其与随机张量理论之间可能存在某种关系。 显示英文摘要

摘要： We introduce the notion of multi-dimensional chaos that applies to processes described by erratic functions of several dynamical variables. We employ this concept in the interpretation of classical and quantum scattering off a pinball system. In the former case it is illustrated by means of two-dimensional plots of the scattering angle and of the number of bounces. We draw similar patterns for the quantum differential cross section for various geometries of the disks. We find that the eigenvalues of the S-matrix are distributed according to the Circular Orthogonal Ensemble (COE) in random matrix theory (RMT), provided the setup be asymmetric and the wave-number be large enough. We then consider the electric potential associated with charges randomly located on a plane as a toy model that generalizes the scattering from a leaky torus. We propose several methods to analyze the spacings between the extrema of this function. We show that these follow a repulsive Gaussian beta-ensemble distribution even for Poisson-distributed positions of the charges. A generalization of the spectral form factor is introduced and determined. We apply these methods to the case of a chaotic S-matrix and of the quantum pinball scattering. The spacings between nearest neighbor extrema points and ratios between adjacent spacings follow a logistic and Beta distributions correspondingly. We conjecture about a potential relation with random tensor theory.