  We show that, in strongly chaotic dynamical systems, the average particle
velocity can be calculated analytically by consideration of Brownian dynamics
in phase space, the method of images and use of the classical diffusion
equation. The method is demonstrated on the simplified Fermi-Ulam accelerator
model, which has a mixed phase space with chaotic seas, invariant tori and
Kolmogorov-Arnold-Moser (KAM) islands. The calculated average velocities agree
well with numerical simulations and with an earlier empirical theory. The
procedure can readily be extended to other systems including time-dependent
billiards.
