Adiabatic propagation of distributions: Exactly solvable models

We study the analytical structure of corrections to perfect adiabatic evolution associated with an ensemble of classical ergodic Hamiltonians with specific correlation properties, distributed at inital time, e.g., over a single energy shell. In particular, we aim to check the prediction of the multiple-time-scale method concerning the structure of energy moments that measure the extent of violation of an ergodic adiabatic invariant when the slowness parameter is small but finite. Solving exactly for the evolution of the phase space density, we find the explicit form of the energy moments for an infinite one-dimensional system of harmonic oscillators with time-decaying couplings. A comparison with the multiple-time-scale method shows its restricted applicability to a marginal limit of a vanishing slow time scale.