Resonance overlap, secular effects, and nonintegrability: an approach from ensemble theory

The time evolution of a classical multiresonance nonintegrable Hamiltonian system with few degrees of freedom is analyzed on the ensemble level. Time-dependent perturbation analysis is applied to the Liouville equation to determine the most secular series for the time evolution of the expectation value of some physical observables. In contrast to the so-called lambda(2) t expansion for thermodynamic systems, which is well known in nonequilibrium statistical physics, we find a square root of lambda t expansion in small nonintegrable systems with few degrees of freedom. This asymptotic expansion exists only on the level of ensemble but not on the level of trajectories. Moreover, the time symmetry of this expansion is broken as in nonequilibrium statistical mechanics. The relation of the Chirikov overlapping criterion to our approach is discussed.