Kolmogorov stochasticity parameter as a measure of quantum chaos

TL;DR

This paper introduces the Kolmogorov stochasticity parameter as a novel measure to distinguish between integrable and chaotic quantum systems, revealing distinct probability distributions and scaling behaviors.

Contribution

It proposes using the Kolmogorov stochasticity parameter to classify quantum systems by their classical dynamics and analyzes its probability distribution and scaling properties.

Findings

PDF of integrable systems is identical and differs from chaotic systems

The stochasticity parameter scales as n^(-α) with energy level index n

Stochastic probability jumps significantly when systems become chaotic

Abstract

We propose the Kolmogorov stochasticity parameter, $\lambda$ for energy level spectra to classify quantum systems with corresponding classical dynamics ranging from integrable to chaotic. We also study the probability distribution function (PDF) of $\lambda$. Remarkably, the PDF of all the integrable systems studied here is the same and is found to be completely different from the PDF of chaotic systems. We also note that $\lambda_n$ for $n$ energy levels scales as $\lambda_n \sim n^{-\alpha}$. Furthermore, with $\alpha$, the stochastic probability (calculated from PDF) is seen to jump by about an order of magnitude as the systems turn chaotic.