Nonergodicity and central limit behavior for systems with long-range interactions

TL;DR

This paper investigates nonergodic behavior in long-range interacting systems, showing that velocity distributions tend to q-Gaussians instead of Gaussians, challenging traditional assumptions of the Central Limit Theorem.

Contribution

It demonstrates the nonergodic nature of quasi-stationary states in the HMF model and reveals the emergence of q-Gaussian attractors in velocity distributions.

Findings

Ensemble and time averages of velocities differ in the HMF model.

Velocity distributions tend to q-Gaussians rather than Gaussians.

Nonergodic behavior affects the applicability of the Central Limit Theorem.

Abstract

In this paper we discuss the nonergodic behavior for a class of long-standing quasi-stationary states in a paradigmatic model of long-range interacting systems, i.e. the HMF model. We show that ensemble averages and time averages for velocities probability density functions (pdfs) do not coincide and in particular the latter exhibit a tendency to converge towards a q-Gaussian attractor instead of the usual Gaussian one predicted by the Central Limit Theorem, when ergodicity applies.