Sub-diffusive behavior in the Standard Map

TL;DR

This paper studies sub-diffusive behavior in the Standard Map, linking stickiness phenomena to anomalous diffusion through a CTRW model and numerical approximation of the Perron-Frobenius operator, confirming anomalous transport in Hamiltonian systems.

Contribution

It introduces a CTRW-based characterization of stickiness in the Standard Map and employs a variant Ulam method to numerically compute the anomalous diffusion exponent.

Findings

Stickiness causes sub-diffusive behavior in the Standard Map.

The CTRW model effectively describes anomalous diffusion.

Numerical methods accurately estimate the diffusion exponent.

Abstract

In this work, we investigate the presence of sub-diffusive behavior in the Chirikov-Taylor Standard Map. We show that the stickiness phenomena, present in the mixed phase space of the map setup, can be characterized as a Continuous Time Random Walk model and connected to the theoretical background for anomalous diffusion. Additionally, we choose a variant of the Ulam method to numerically approximate the Perron-Frobenius operator for the map, allowing us to calculate the anomalous diffusion exponent via an eigenvalue problem, compared to the solution of the Fractional Diffusion Equation. The results here corroborate other findings in the literature of anomalous transport in Hamiltonian maps and can be suitable to describe transport properties of other dynamical systems.