[Paper Review] Phase correlations in chaotic dynamics A Shannon entropy measure

P. M. Cincotta, C. M. Giordano|arXiv (Cornell University)|Jul 20, 2018

Statistical Mechanics and Entropy4 citations

TL;DR

This paper introduces a Shannon entropy-based method to detect phase correlations in chaotic dynamical systems, demonstrating that entropy decay rates reveal the presence of weak or strong correlations in angular variables. The approach effectively identifies deviations from normal diffusion—particularly in Arnold’s Hamiltonian—showing that standard approximations like reduced stochasticity fail when correlations are significant.

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.

Motivation & Objective

• To develop a robust, computationally efficient method for detecting phase correlations in chaotic systems, especially where traditional correlation functions are impractical.
• To challenge the validity of the reduced stochasticity approximation in Chirikov's formulation of Arnold diffusion, which assumes uncorrelated phase evolution.
• To provide an alternative measure of chaotic diffusion rate independent of the variance's power-law scaling, using Shannon entropy instead of variance-based methods.
• To investigate how phase correlations affect the accuracy of local diffusion coefficient estimates in near-integrable Hamiltonian systems.
• To demonstrate the method's sensitivity in detecting weak correlations that lead to anomalous diffusion, particularly in low-dimensional maps and Arnold's Hamiltonian.

Proposed method

• Uses Shannon entropy $\mathcal{I}(N) = -\sum_{k=1}^{q} \mu(a_k) \ln \mu(a_k)$ to quantify the information content of phase space partitions, where $\mu(a_k) = n_k/N$ is the occupation probability of cell $a_k$.
• Applies the entropy to time series of angular variables in area-preserving maps and Hamiltonian systems, tracking its decay over time $N$ to infer correlation strength.
• Compares entropy decay to theoretical expectations: inverse time ($\sim t^{-1}$) for ergodic behavior, inverse square ($\sim t^{-2}$) for uniform distribution, and deviations indicating correlations.
• Employs numerical simulations on well-known maps (e.g., standard map, skew-gradient map) to validate the entropy method against known chaotic dynamics.
• Extends the method to Arnold’s Hamiltonian system with parameters $\varepsilon = 0.25$, $\mu = 0.1\varepsilon$, analyzing $I_2$ diffusion and phase evolution.
• Uses the entropy decay rate to assess the validity of the assumption $\langle \cos^2 \theta_2 \rangle \approx Rt/2$ in Chirikov’s model, showing it fails under strong correlations.

Experimental results

Research questions

• RQ1 How can phase correlations in chaotic systems be reliably measured when traditional time-correlation functions are computationally expensive?
• RQ2 To what extent do phase correlations invalidate the reduced stochasticity approximation in Chirikov’s model of Arnold diffusion?
• RQ3 Can Shannon entropy serve as a more robust and independent measure of chaotic diffusion rate than variance-based scaling?
• RQ4 What is the role of weak phase correlations in inducing anomalous diffusion, particularly in systems with slow or sub-diffusive variance growth?
• RQ5 In which parameter regimes of Arnold’s Hamiltonian does the entropy-based method reveal significant deviations from ergodic or random phase behavior?

Key findings

• The Shannon entropy decay rate provides a sensitive and computationally efficient measure of phase correlations, even when they are weak.
• For $\kappa = 9.00$, the entropy decay follows $\sim t^{-1}$, indicating ergodic-like behavior, while for $\kappa = 1.00$, it decays as $\sim t^{-2}$, suggesting uniform phase distribution.
• In Arnold’s Hamiltonian with $\varepsilon = 0.25$, $\mu = 0.1\varepsilon$, and $|\omega_2^0| < 2$, the entropy analysis confirms sub-diffusive behavior, consistent with prior results in CGMB17 and GC18.
• The assumption $\langle \cos^2 \theta_2 \rangle \approx Rt/2$ in Chirikov’s model is invalid when strong phase correlations are present, particularly for $\omega_2 \lesssim 1$.
• The map (25) is a reliable model for diffusion along the chaotic layer of resonance $\omega_1 = 0$ only when $\mu\varepsilon \ll \varepsilon \ll 1$ and $\omega_2$ is far from 0 and 1.
• For $\omega_2 \gg 1$, the reduced stochasticity approximation becomes plausible, indicating that the model’s validity is parameter-dependent and correlation-sensitive.