Chaos in the Hamiltonian mean field model

TL;DR

This paper investigates chaos in the Hamiltonian mean-field model, revealing how the largest Lyapunov exponent behaves with system size and energy, and exploring the effects of adding degrees of freedom on chaos.

Contribution

It provides a detailed analysis of Lyapunov exponents in the HMF model, including scaling laws near criticality and the impact of extended degrees of freedom on chaos.

Findings

Largest Lyapunov exponent remains positive as system size grows

Spectrum of Lyapunov exponents includes a bulk and subextensive bands

Chaos persists at finite sizes but vanishes in the Vlasov limit

Abstract

We study the dynamical properties of the canonical ordered phase of the Hamiltonian mean-field (HMF) model, in which $N$ particles, globally-coupled via pairwise attractive interactions, form a rotating cluster. Using a combination of numerical and analytical arguments, we first show that the largest Lyapunov exponent remains strictly positive in the infinite-size limit, converging to its asymptotic value with $1/\ln N$ corrections. We then elucidate the scaling laws ruling the behavior of this asymptotic value in the critical region separating the ordered, clustered phase and the disordered phase present at high energy densities. We also show that the full spectrum of Lyapunov exponents consists of a bulk component converging to the (zero) value taken by a test oscillator forced by the mean field, plus subextensive bands of ${\cal O}(\ln N)$ exponents taking finite values. We finally investigate the robustness of these results by studying a "2D" extension of the HMF model where each particle is endowed with 4 degrees of freedom, thus allowing the emergence of chaos at the level of single particle. Altogether, these results illustrate the subtle effects of global (or long-range) coupling, and the importance of the order in which the infinite-time and infinite-size limits are taken: for an infinite-size HMF system represented by the Vlasov equation, no chaos is present, while chaos exists and subsists for any finite system size.