Algebraic Correlation Function and Anomalous Diffusion in the HMF model

TL;DR

This paper investigates anomalous transport in the Hamiltonian Mean-Field model, revealing long-range correlations and a transition between different diffusion regimes, supported by numerical results aligning with kinetic theory predictions.

Contribution

It provides a detailed numerical analysis of correlation functions and diffusion in the HMF model, demonstrating long-range temporal correlations and confirming theoretical anomalous transport exponents.

Findings

Correlation functions decay inversely with time with logarithmic correction.

Transition observed between weak and strong anomalous diffusion.

Long-range correlations persist even at statistical equilibrium.

Abstract

In the quasi-stationary states of the Hamiltonian Mean-Field model, we numerically compute correlation functions of momenta and diffusion of angles with homogeneous initial conditions. This is an example, in a N-body Hamiltonian system, of anomalous transport properties characterized by non exponential relaxations and long-range temporal correlations. Kinetic theory predicts a striking transition between weak anomalous diffusion and strong anomalous diffusion. The numerical results are in excellent agreement with the quantitative predictions of the anomalous transport exponents. Noteworthy, also at statistical equilibrium, the system exhibits long-range temporal correlations: the correlation function is inversely proportional to time with a logarithmic correction instead of the usually expected exponential decay, leading to weak anomalous transport properties.