Resonance overlap, secular effects and non-integrability: An approach from ensemble theory

The time evolution of classical multi-resonance non-integrable Hamiltonian systems with few degrees of freedom (the small Poincaré system) is analyzed on the ensemble level. In such systems, one encounters the small denominator problem in the traditional approach of trajectory dynamics. By applying the time-dependent perturbation analysis to the Liouville equation we can determine the most secular effects for the time evolution of the expectation value of some physical observables.; For the case of the large Poincaré system studied in non-equilibrium statistical mechanics with infinite degrees of freedom it is well known that the spectrum of the Liouville operator is continuous so that under the integration over the wave vector the small denominator can be treated as a distribution. On the other hand, the spectrum of the Liouville operator is discrete in the small Poincaré system. Therefore, it is necessary in this case to perform an ensemble average over the continuous action variables for the small denominator to be treated as a distribution. In contrast to the so called λ 2t-limit (the Van Hove limit) in non-equilibrium statistical mechanics for the large Poincaré system, we find $l$t-limit in the small Poincaré system. This shows that the resonance effect in the small Poincaré system is much stronger than in the large Poincaré system. In this limit, the time symmetry is broken as in non-equilibrium statistical mechanics. These secular effects exist only on the level of ensemble but not on the level of trajectory.; We are able to distinguish contributions from individual resonances and from the interference between the resonances. Since the interference is responsible for the non-integrability, one can now treat quantitatively the non-integrable effects on the level of ensemble. Our treatment of the interference effect naturally leads to the Chirikov overlapping criterion for the onset of global chaos. Comparison of our theoretical prediction to numerical simulation is excellent in the asymptotic time scale t ∼ 1/$l$.