On the diffusive anomalies in a long-range Hamiltonian system

TL;DR

This paper investigates anomalous super-diffusive behavior in a long-range Hamiltonian system of classical rotors, revealing q-Gaussian distributions and size-dependent relaxation in quasi-stationary states.

Contribution

It demonstrates the emergence of q-Gaussian phase distributions and analyzes the size-dependent relaxation dynamics in long-range rotor systems.

Findings

Phases exhibit super-diffusion in quasi-stationary states.

Phase distributions follow q-Gaussian form with q increasing over time.

Relaxation to equilibrium depends strongly on system size.

Abstract

We scrutinize the anomalies in diffusion observed in an extended long-range system of classical rotors, the HMF model. Under suitable preparation, the system falls into long-lived quasi-stationary states presenting super-diffusion of rotor phases. We investigate the diffusive motion of phases by monitoring the evolution of their probability density function for large system sizes. These densities are shown to be of the $q$-Gaussian form, $P(x)\propto (1+(q-1)[x/\beta]^2)^{1/(1-q)}$, with parameter $q$ increasing with time before reaching a steady value $q\simeq 3/2$. From this perspective, we also discuss the relaxation to equilibrium and show that diffusive motion in quasi-stationary trajectories strongly depends on system size.