Goodness of Ergodic Adiabatic Invariants

For a "slowly" time-dependent Hamiltonian system exhibiting ergodic motion, the volume inside the hypersurface on which the Hamiltonian equals a constant is an adiabatic invariant. It is shown that the error in the constant is diffusive and scales as ${\left(\frac{{\tau }_C}{\tau }\right)}^{\frac{1}{2}}$, where ${\tau }_C$ is a certain correlation time of the ergodic motion, and $\tau$ is the time scale over which the Hamiltonian changes.