On "Ergodicity and Central Limit Theorem in Systems with Long-Range Interactions" by Figueiredo et al

TL;DR

This paper defends the applicability of the $q$-generalized Central Limit Theorem to the Hamiltonian Mean Field model, showing that $q$-Gaussian attractors can arise in finite systems with specific initial conditions before reaching equilibrium.

Contribution

It refutes recent criticisms and demonstrates that $q$-Gaussian attractors are valid for finite-size long-range interacting systems under certain initial conditions.

Findings

$q$-Gaussian-like curves are possible as attractors.

Such attractors occur for specific initial conditions.

They appear during the quasi-stationary states before equilibrium.

Abstract

In the present paper we refute the criticism advanced in a recent preprint by Figueiredo et al [1] about the possible application of the $q$-generalized Central Limit Theorem (CLT) to a paradigmatic long-range-interacting many-body classical Hamiltonian system, the so-called Hamiltonian Mean Field (HMF) model. We exhibit that, contrary to what is claimed by these authors and in accordance with our previous results, $q$-Gaussian-like curves are possible and real attractors for a certain class of initial conditions, namely the one which produces nontrivial longstanding quasi-stationary states before the arrival, only for finite size, to the thermal equilibrium.