Nonergodicity and central-limit behavior for long-range Hamiltonians

¹  Dipartimento di Fisica e Astronomia, Università di Catania, and INFN sezione di Catania - Via S. Sofia 64, I-95123 Catania, Italy
²  Centro Brasileiro de Pesquisas Fisicas - Rua Xavier Sigaud 150, 22290-180 Rio de Janeiro-RJ, Brazil
³  Santa Fe Institute - 1399 Hyde Park Road, Santa Fe, NM 87501, USA

Received: 27 June 2007
Accepted: 1 September 2007

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.