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Direct numerical integrations of the two-dimensional Fokker-Planck equationare carried out for compact objects orbiting a supermassive black hole (SBH) atthe center of a galaxy. As in Papers I-III, the diffusion coefficientsincorporate the effects of the lowest-order post-Newtonian corrections to theequations of motion. In addition, terms describing the loss of orbital energyand angular momentum due to the 5/2-order post-Newtonian terms are included. Inthe steady state, captures are found to occur in two regimes that are clearlydifferentiated in terms of energy, or semimajor axis; these two regimes arenaturally characterized as "plunges" (low binding energy) and "EMRIs," orextreme-mass-ratio inspirals (high binding energy). The capture rate, and thedistribution of orbital elements of the captured objects, are presented for twosteady-state models based on the Milky Way: one with a relatively high densityof remnants and one with a lower density. In both models, but particularly inthe second, the steady-state energy distribution and the distribution oforbital elements of the captured objects are substantially different than ifthe Bahcall-Wolf energy distribution were assumed. The ability of classicalrelaxation to soften the blocking effects of the Schwarzschild barrier isquantified.These results, together with those of Papers I-III, suggest that aFokker-Planck description can adequately represent the dynamics of collisionalloss cones in the relativistic regime.

Gravitational Encounters and the Evolution of Galactic Nuclei. IV. Captures Mediated by Gravitational-Wave Energy Loss