Apparent Fractal Dimensions in the HMF Model

Abstract

We show that recent observations of fractal dimensions in the μ‐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.