The goodness of ergodic adiabatic invariants

Abstract

For a “slowly” time-dependent Hamiltonian system exhibiting chaotic motion that ergodically covers the energy surface, the phase space volume enclosed inside this surface is an adiabatic invariant. In this paper we examine, both numerically and theoretically, how the error in this “ergodic adiabatic invariant” scales with the slowness of the time variation of the Hamiltonian. It is found that under certain circumstances, the error is diffusive and scales likeT ^−1/2, whereT is the characteristic time over which the Hamiltonian changes. On the other hand, for other cases (where motion in the Hamiltonian has a long-time 1/t tail in a certain correlation function), the error scales like [T ⁻¹ ln(T)]^1/2. Both of these scalings are verified by numerical experiments. In the situation where invariant tori exist amid chaos, the motion may not be fully ergodic on the entire energy surface. The ergodic adiabatic invariant may still be useful in this case and the circumstances under which this is so are investigated numerically (in particular, the islands have to be small enough).