Calculation of Superdiffusion for the Chirikov-Taylor Model

TL;DR

This paper analytically derives the superdiffusion coefficient in the Chirikov-Taylor model, revealing how elliptic islands cause enhanced diffusion, supported by numerical simulations.

Contribution

It introduces a differential form of the Perron-Frobenius operator that separates normal diffusion from superdiffusion, enabling analytical calculation of the superdiffusion coefficient.

Findings

Superdiffusion coefficient derived analytically with a Schloemilch series

Exponent for divergence is β=3/2

Numerical simulations confirm analytical results

Abstract

It is widely known that the paradigmatic Chirikov-Taylor model presents enhanced diffusion for specific intervals of its stochasticity parameter due to islands of stability, which are elliptic orbits surrounding accelerator mode fixed points. In contrast with normal diffusion, its effect has never been analytically calculated. Here, we introduce a differential form for the Perron-Frobenius evolution operator in which normal diffusion and superdiffusion are treated separately through phases formed by angular wave numbers. The superdiffusion coefficient is then calculated analytically resulting in a Schloemilch series with an exponent $\beta=3/2$ for the divergences. Numerical simulations support our results.