Nonergodicity and Central Limit Behavior for Long-range Hamiltonians

TL;DR

This study investigates the applicability of the Central Limit Theorem in a long-range interacting Hamiltonian system, revealing Gaussian behavior in chaotic regimes and q-Gaussian attractors in weakly chaotic states, highlighting nonergodicity effects.

Contribution

It provides the first molecular dynamics evidence of the CLT and its generalizations in long-range Hamiltonian systems, linking chaos levels to distribution types.

Findings

Gaussian PDFs in chaotic regimes at equilibrium

q-Gaussian attractors in metastable states

Ergodicity verified only in strongly chaotic conditions

Abstract

We present a molecular dynamics test of the Central Limit Theorem (CLT) in a paradigmatic long-range-interacting many-body classical Hamiltonian system, the HMF model. We calculate sums of velocities at equidistant times along deterministic trajectories for different sizes and energy densities. We show that, when the system is in a chaotic regime (specifically, at thermal equilibrium), ergodicity is essentially verified, and the Pdfs of the sums appear to be Gaussians, consistently with the standard CLT. When the system is, instead, only weakly chaotic (specifically, along longstanding metastable Quasi-Stationary States), nonergodicity (i.e., discrepant ensemble and time averages) is observed, and robust $q$-Gaussian attractors emerge, consistently with recently proved generalizations of the CLT.