DCS Seminar #5 Time-dependent Perturbation Analysis of Small Nonintegrable Hamiltonian System

2010-6-9 16:30 - 2010-6-9 17:30
Faculty of Science Building #5 Room 302
Chun Biu LI, Associate Professor, RIES, Hokkaido University
It is well known in classical perturbation theory of Hamiltonian system that resonances give rise to the small denominator problem which spoils the convergence of the perturbation series on the trajectory level. In order to resolve this difficulty, we analyze the dynamics of the classical multi-resonance nonintegrable Hamiltonian systems with few degrees of freedom on the ensemble level.

By employing the time-dependent perturbation analysis to the Liouville equation and treating appropriately the small denominators as poles in a Cauchy integral, we are able to extract from the resonance singularities the analytic 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 (lambda: perturbation expansion parameter) for thermodynamic systems, which is well known in nonequilibrium statistical physics, we find a (lambda^0.5)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 will be discussed.