SELF-CONSISTENT CHECK OF THE VALIDITY OF GIBBS CALCULUS USING DYNAMICAL VARIABLES

The high- and low-energy limits of a chain of coupled rotators are int egrable and correspond respectively to a set of free rotators and to a chain of harmonic oscillators. For intermediate values of the energy, numerical calculations show the agreement of finite time averages of physical observables with their Gibbsian estimate. The boundaries betw een the two integrable limits and the statistical domain are analytica lly computed using the Gibbsian estimates of dynamical observables. Fo r large energies the geometry of nonlinear resonances enables the defi nition of relevant 1.5-degree-of-freedom approximations of the dynamic s. They provide resonance overlap parameters whose Gibbsian probabilit y distribution may be computed. Requiring the support of this distribu tion to be right above the large-scale stochasticity threshold of the 1.5-degree-of-freedom dynamics yields the boundary at the large-energy limit. At the low-energy limit, the boundary is shown to correspond t o the energy where the specific heat departs from that of the correspo nding harmonic chain.