The speed of Arnold diffusion

TL;DR

This paper investigates the rate of Arnold diffusion in nearly integrable Hamiltonian systems, combining numerical experiments with theoretical estimates to understand how the diffusion speed relates to the size of the Hamiltonian's remainder term.

Contribution

It provides a concrete numerical analysis of Arnold diffusion, visualizes the phenomenon in specific variables, and establishes a power-law relationship between diffusion coefficient and the remainder size.

Findings

Diffusion speed D scales as ||R_{opt}||^{2(1+b)} with a small positive b.

Arnold diffusion can be clearly visualized in a specific set of variables.

The study compares numerical results with Nekhoroshev theorem predictions.

Abstract

A detailed numerical study is presented of the slow diffusion (Arnold diffusion) taking place around resonance crossings in nearly integrable Hamiltonian systems of three degrees of freedom in the so-called `Nekhoroshev regime'. The aim is to construct estimates regarding the speed of diffusion based on the numerical values of a truncated form of the so-called remainder of a normalized Hamiltonian function, and to compare them with the outcomes of direct numerical experiments using ensembles of orbits. In this comparison we examine, one by one, the main steps of the so-called analytic and geometric parts of the Nekhoroshev theorem. We are led to two main results: i) We construct in our concrete example a convenient set of variables, proposed first by Benettin and Gallavotti (1986), in which the phenomenon of Arnold diffusion in doubly resonant domains can be clearly visualized. ii) We determine, by numerical fitting of our data the dependence of the local diffusion coefficient "D" on the size "||R_{opt}||" of the optimal remainder function, and we compare this with a heuristic argument based on the assumption of normal diffusion. We find a power law "D\propto ||R_{opt}||^{2(1+b)}", where the constant "b" has a small positive value depending also on the multiplicity of the resonance considered.