  Black holes that accrete far below the Eddington limit are believed to
accrete through a geometrically thick, optically thin, rotationally supported
plasma that we will refer to as a radiatively inefficient accretion flow
(RIAF). RIAFs are typically collisionless in the sense that the Coulomb mean
free path is large compared to $GM/c^2$, and relativistically hot near the
event horizon. In this paper we develop a phenomenological model for the plasma
in RIAFs, motivated by the application to sources such as Sgr A* and M87. The
model is derived using Israel-Stewart theory, which considers deviations up to
second order from thermal equilibrium, but modified for a magnetized plasma.
This leads to thermal conduction along magnetic field lines and a difference in
pressure, parallel and perpendicular to the field lines (which is equivalent to
anisotrotropic viscosity). In the non-relativistic limit, our model reduces to
the widely used Braginskii theory of magnetized, weakly collisional plasmas. We
compare our model to the existing literature on dissipative relativistic
fluids, describe the linear theory of the plasma, and elucidate the physical
meaning of the free parameters in the model. We also describe limits of the
model when the conduction is saturated and when the viscosity implies a large
pressure anisotropy. In future work, the formalism developed in this paper will
be used in numerical models of RIAFs to assess the importance of non-ideal
processes for the dynamics and radiative properties of slowly accreting black
holes.
