  The distribution function describing the acceleration of relativistic
particles in an advection-dominated accretion disk is analyzed using a
transport formalism that includes first-order Fermi acceleration, advection,
spatial diffusion, and the escape of particles through the upper and lower
surfaces of the disk. When a centrifugally-supported shock is present in the
disk, the concentrated particle acceleration occurring in the vicinity of the
shock channels a significant fraction of the binding energy of the accreting
gas into a population of relativistic particles. These high-energy particles
diffuse vertically through the disk and escape, carrying away both energy and
entropy and allowing the remaining gas to accrete. The dynamical structure of
the disk/shock system is computed self-consistently using a model previously
developed by the authors that successfully accounts for the production of the
observed relativistic outflows (jets) in M87 and \SgrA. This ensures that the
rate at which energy is carried away from the disk by the escaping relativistic
particles is equal to the drop in the radial energy flux at the shock location,
as required for energy conservation. We investigate the influence of advection,
diffusion, and acceleration on the particle distribution by computing the
nonthermal Green's function, which displays a relatively flat power-law tail at
high energies. We also obtain the energy distribution for the particles
escaping from the disk, and we conclude by discussing the spectrum of the
observable secondary radiation produced by the escaping particles.
