Multi-dimensional chaos I: Classical and quantum mechanics

TL;DR

This work introduces multi-dimensional chaos by studying erratic, two-variable observables in classical and quantum scattering, using the three-disk pinball and a leaky-torus-inspired toy model. It develops and applies several 2D spacing measures (all spacings, nearest-neighbor, consecutive-path spacings, and axis-projected spacings) and a 2D spectral form factor to quantify chaotic structure, linking classical fractal patterns and quantum S-matrix statistics to random matrix theory. A key finding is that quantum S-matrix eigenvalues follow COE in asymmetric, high-k regimes, while symmetric configurations yield Poisson-like statistics; peak spacings in the quantum pinball differential cross section show logistic and Beta-type distributions for NN and path spacings, respectively. The paper also introduces a two-dimensional random-matrix toy model and its analytic spacings, connects these results to a two-matrix random-model framework, and speculates on a map to random tensor theory, suggesting a broader framework for multi-variable chaotic phenomena with potential implications for high-dimensional quantum systems and string-theoretic scattering.

Abstract

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.

Figures (25)

• Figure 1: Symmetric setup for the three-disk pinball system using two angles to parameterize the initial condition. We plot two trajectories at fixed $\theta$, differing by $\delta\phi < 10^{-7}$.
• Figure 2: The scattering angle (top) is an erratic function of both angles, $\theta$ (left), and $\phi$ (right), when the other is kept fixed. The regions where the function is erratic correspond to regions where the number of collisions (bottom) is large.
• Figure 3: Successively zooming in on erratic regions in the plot reveals a self-similar structure.
• Figure 4: Sine of scattering angle as a function of $\theta$ and $\phi$.
• Figure 5: Number of collisions as a function of both angles. The regions where the number of collisions is large, and there is erratic behavior follows line patterns.