Metastability and weak mixing in classical long-range many-rotator systems

We perform a molecular dynamical study of the isolated $d=1$ classical Hamiltonian $H=1/2{{∑}}ᵢ₌₁N{L}ᵢ²+{{∑}}_{i{≠}j}[1{-}cos({{θ}}ᵢ{-}{{θ}}ⱼ)]{/r}ᵢⱼ^{{α}};({α}>~0),$ known to exhibit a second order phase transition, being disordered for $u{≡}U/N\~N>~{u}c({α},d)$ and ordered otherwise $[U{≡}$ total energy and $\~N{≡}{(N}^{1{-}{α}/d}{-}{α}/d)/(1{-}{α}/d)].$ We focus on the nonextensive case ${α}/d<~1$ and observe that, for $u<{u}c,$ a basin of attraction exists for the initial conditions for which the system quickly relaxes onto a long standing metastable state (whose duration presumably diverges with N-like $√\~N)$ which eventually crosses over to the microcanonical Boltzmann-Gibbs stable state. It is exhibited that the appropriately scaled maximal Lyapunov exponent ${{λ}}_{u<{u}c}ᵐᵃˣ(metastable){∝}{N}^{{-}{{κ}}metastable};({{→}}{N}{∞}),$ where, for all values of ${α}/d,$ ${{κ}}metastable$ numerically coincides with one third of its value for $u>{u}c,$ hence decreases from 1/9 to zero when ${α}/d$ increases from zero to unity, remaining zero thereafter. This simple connection between anomalies above and below the critical point reinforces the nonextensive universality scenario.