We present analytical derivations of the diffusion rates, ratchet currents, and time scales of a new ratchet in a fully chaotic Hamiltonian system, introduced in Phys. Rev. Lett. 89, 194102 (2002), with a proposed implementation using atoms in pulsed standing waves of light. The origin of this type of ratchet current is in asymmetric momentum diffusion rates which result when a double-well lattice is pulsed with unequal kick periods. The form of the new short-time correlations which modify the diffusion rates are derived. The resulting formulas for the classical energy diffusion rates are shown to give good agreement with numerical simulations. A closed analytical formula for the ratchet current is also obtained, which predicts correctly the current magnitudes and current reversals. The characteristic ratchet time, a classical time scale associated with the momentum-diffusion ratchet is derived analytically. The competition between the ratchet time tr, and the quantum break time t* is investigated further.