Phase correlations in chaotic dynamics A Shannon entropy measure

TL;DR

This paper introduces Shannon entropy as an effective tool to measure phase correlations in chaotic systems, revealing strong correlations in Hamiltonian phase dynamics and questioning common diffusion approximations.

Contribution

It demonstrates the effectiveness of Shannon entropy in detecting phase correlations in chaotic systems and applies it to Hamiltonian dynamics, providing insights into diffusion processes.

Findings

Entropy accurately detects phase correlations even when weak.

Strong correlations in Hamiltonian phases lead to anomalous diffusion.

Results challenge the validity of certain diffusion approximations.

Abstract

In the present work we investigate phase correlations by recourse to the Shannon entropy. Using theoretical arguments we show that the entropy provides an accurate measure of phase correlations in any dynamical system, in particular when dealing with a chaotic diffusion process. We apply this approach to different low dimensional maps in order to show that indeed the entropy is very sensitive to the presence of correlations among the successive values of angular variables, even when it is weak. Later on, we apply this approach to unveil strong correlations in the time evolution of the phases involved in the Arnold's Hamiltonian that lead to anomalous diffusion, particularly when the perturbation parameters are comparatively large. The obtained results allow us to discuss the validity of several approximations and assumptions usually introduced to derive a local diffusion coefficient in multidimensional near--integrable Hamiltonian systems, in particular the so-called reduced stochasticity approximation.