  We investigate the form of the momentum distribution function for protons and
electrons in an advection-dominated accretion flow (ADAF). We show that for all
accretion rates, Coulomb collisions are too inefficient to thermalize the
protons. The proton distribution function is therefore determined by the
viscous heating mechanism, which is unknown. The electrons, however, can
exchange energy quite efficiently through Coulomb collisions and the emission
and absorption of synchrotron photons. We find that for accretion rates greater
than \sim 10^{-3} of the Eddington accretion rate, the electrons have a thermal
distribution throughout the accretion flow. For lower accretion rates, the
electron distribution function is determined by the electron's source of
heating, which is primarily adiabatic compression. Using the principle of
adiabatic invariance, we show that an adiabatically compressed collisionless
gas maintains a thermal distribution until the particle energies become
relativistic. We derive a new, non-thermal, distribution function which arises
for relativistic energies and provide analytic formulae for the synchrotron
radiation from this distribution. Finally, we discuss its implications for the
emission spectra from ADAFs.

Astrophysics