Adiabatic invariants and some statistical properties of the time dependent linear and nonlinear oscillators

Abstract

We consider 1D time dependent Hamilton systems and the time evolution of initial microcanonical distributions. In linear oscillator (LO) the distribution of energy is always arcsine distribution, and the adiabatic invariant at the average energy (AIAE)(and thus the entropy) always increases. In nonlinear (quartic) oscillator there are regimes of slow driving where the AIAE can decrease, but increases for faster driving. Near the adiabatic regime the distribution is similar to arcsine distribution; in general it depends on the dynamical details. We also consider parametrically kicked systems. We prove for all homogeneous power-law potentials that in a single parametric kick the AIAE always increases. The approximation of one kick is good for times up to one oscillation period. In LO only, due to isochronicity, an initial kick disperses the microcanonical distribution, but an antikick at the right phase can restore it. The periodic parametric kicking is also studied. (It is my great pleasure to dedicate this work to Professor Hermann Haken on occastion of his 85th birthday.).