Apparent fractal dimensions in the HMF model

TL;DR

This paper demonstrates that observed fractal dimensions in the phase space of long-range interacting Hamiltonian systems are artifacts of finite system size and resolution, and that in the continuum limit, sets retain their initial non-fractal structure.

Contribution

It provides numerical evidence that in the Vlasov limit, sets preserve their initial geometry, challenging previous interpretations of fractal structures in such systems.

Findings

Finite $N$ and resolution effects cause apparent fractal dimensions.

In the continuum limit, sets maintain their initial non-fractal structure.

Analysis applies to the HMF model and is supported by studies of the Chirikov standard map.

Abstract

We show that recent observations of fractal dimensions in the $\mu$-space of $N$-body Hamiltonian systems with long-range interactions are due to finite $N$ and finite resolution effects. We provide strong numerical evidence that, in the continuum (Vlasov) limit, a set which initially is not a fractal (e.g. a line in 2D) remains such for all finite times. We perform this analysis for the Hamiltonian Mean Field (HMF) model, which describes the motion of a system of $N$ fully coupled rotors. The analysis can be indirectly confirmed by studying the evolution of a large set of initial points for the Chirikov standard map.