  Direct numerical integrations of the two-dimensional Fokker-Planck equation
are carried out for compact objects orbiting a supermassive black hole (SBH) at
the center of a galaxy. As in Papers I-III, the diffusion coefficients
incorporate the effects of the lowest-order post-Newtonian corrections to the
equations of motion. In addition, terms describing the loss of orbital energy
and angular momentum due to the 5/2-order post-Newtonian terms are included. In
the steady state, captures are found to occur in two regimes that are clearly
differentiated in terms of energy, or semimajor axis; these two regimes are
naturally characterized as "plunges" (low binding energy) and "EMRIs," or
extreme-mass-ratio inspirals (high binding energy). The capture rate, and the
distribution of orbital elements of the captured objects, are presented for two
steady-state models based on the Milky Way: one with a relatively high density
of remnants and one with a lower density. In both models, but particularly in
the second, the steady-state energy distribution and the distribution of
orbital elements of the captured objects are substantially different than if
the Bahcall-Wolf energy distribution were assumed. The ability of classical
relaxation to soften the blocking effects of the Schwarzschild barrier is
quantified.These results, together with those of Papers I-III, suggest that a
Fokker-Planck description can adequately represent the dynamics of collisional
loss cones in the relativistic regime.
