  The accretion rates needed to fuel the central black hole in a galaxy can be
achieved via viscous torques in thick disks and rings, which can be resolved by
millimetre interferometry within the inner ~20pc of the active galaxy NGC1068
at comparable scales and sensitivity to single dish observations of the
Circumnuclear Disk (CND) in the Galactic Center. To interpret observations of
these regions and determine the physical properties of their gas distribution,
we present a modelling effort that includes (i) a simple dynamical simulations
involving partially inelastic collisions between disk gas clouds, (ii) an
analytical model of a turbulent clumpy gas disk calibrated by the dynamical
model and observations, (iii) local turbulent and cosmic ray gas heating and
cooling via H2O, H2, and CO emission, and (iv) determination of the molecular
abundances. We also consider photodissociation regions (PDR) where gas is
directly illuminated by the central engine. We compare the resulting model
datacubes of the CO, HCN, HCO+, and CS brightness temperatures to available
observations. In both cases the kinematics can be explained by one or two
clouds colliding with a pre-existing ring, in a prograde sense for the CND and
retrograde for NGC1068. And, with only dense disk clouds, the line fluxes can
be reproduced to within a factor of about two. To avoid self-absorption of the
intercloud medium, turbulent heating at the largest scales, comparable to the
disk height, has to be decreased by a factor of 50-200. Our models indicate
that turbulent mechanical energy input is the dominant gas heating mechanism
within the thick gas disks. In N1068, while the bulk of the AGN X-ray radiation
is absorbed in a layer of Compton-thick gas inside the dust sublimation radius,
the optical/UV radiation may enhance the molecular line emission from
photodissociation regions by ~50% at the inner edge of the gas ring.
