  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.
