Nonergodicity and Central Limit Behavior for Long-range Hamiltonians

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.

💡 Research Summary

This paper investigates the interplay between ergodicity, chaos, and the Central Limit Theorem (CLT) in the Hamiltonian Mean Field (HMF) model, a prototypical long‑range interacting many‑body system. By performing extensive molecular‑dynamics simulations for various system sizes (N = 10^3–10^5) and energy densities, the authors compute time series of particle velocities at equally spaced intervals and form partial sums S_M = Σ_{k=1}^{M} v_i(t_k). After appropriate rescaling, the probability density functions (PDFs) of these sums are compared with the Gaussian distribution predicted by the standard CLT and with the q‑Gaussian distribution that emerges from the generalized CLT associated with non‑extensive statistical mechanics.

The study distinguishes two dynamical regimes. In the chaotic, thermal‑equilibrium regime (energy density above the critical value), the system exhibits strong mixing, and time averages coincide with ensemble averages, confirming ergodicity. The PDFs of S_M converge rapidly to a Gaussian shape, in full agreement with the classical CLT. Conversely, when the system is prepared at energy densities below the critical point, it becomes trapped in long‑lived quasi‑stationary states (QSS). In this weakly chaotic regime, the dynamics are only partially mixing; trajectories remain correlated over very long times, leading to a clear breakdown of ergodicity. The PDFs of the velocity sums no longer follow a Gaussian law but are accurately described by q‑Gaussians with q ≈ 1.3–1.5. The q‑parameter approaches unity as the system eventually relaxes to equilibrium, consistent with the theoretical expectation that the generalized CLT reduces to the ordinary CLT in the limit of vanishing correlations.

A systematic finite‑size analysis shows that the lifetime of the QSS grows proportionally to N, and the deviation from Gaussianity becomes more pronounced for larger systems. This scaling behavior underscores the intrinsic role of long‑range interactions in sustaining non‑ergodic, non‑Gaussian statistics. The numerical findings are in line with recent rigorous proofs of the generalized CLT, which allow for correlated variables and predict q‑Gaussian attractors under suitable conditions.

The authors conclude that the HMF model provides a clean laboratory for observing the transition from standard to generalized CLT behavior as the degree of chaos and ergodicity varies. Their results have broader implications for a wide class of physical systems with long‑range forces—such as self‑gravitating gases, nonneutral plasmas, and vortex dynamics—where non‑ergodic phases and anomalous statistical distributions are frequently reported. The work suggests that non‑extensive statistical mechanics, embodied in the q‑Gaussian formalism, offers a robust framework for describing the statistical properties of such systems when traditional Boltzmann‑Gibbs theory fails.