Adiabatic-invariant change due to separatrix crossing

A slowly varying Hamiltonian with one degree of freedom and nearly closed orbits has an adiabatic invariant. This adiabatic invariant is conserved to all orders in \ensuremath{\epsilon}, the slowness parameter, except for orbits that cross a separatrix. The present work discusses the change of the adiabatic invariant during this crossing process through order \ensuremath{\epsilon}. First, a calculation of the change of the adiabatic invariant is presented. This calculation is general and, hence, encompasses previous results for specific cases. The change in the adiabatic invariant is shown to depend on a maximum of five parameters, which are functions of the Hamiltonian of interest. Second, the statistics of this process are derived. Finally, these results are applied to the motion of a particle in a wave with changing amplitude and phase velocity.