Two-phase model for Black Hole feeding and feedback
  We study effects of AGN feedback outflows on multi-phase inter stellar medium
(ISM) of the host galaxy. We argue that SMBH growth is dominated by accretion
of dense cold clumps and filaments. AGN feedback outflows overtake the cold
medium, compress it, and trigger a powerful starburst -- a positive AGN
feedback. This predicts a statistical correlation between AGN luminosity and
star formation rate at high luminosities. Most of the outflow's kinetic energy
escapes from the bulge via low density voids. The cold phase is pushed outward
only by the ram pressure (momentum) of the outflow. The combination of the
negative and positive forms of AGN feedback leads to an $M-\sigma$ relation
similar to the result of King (2003). Due to porosity of cold ISM in the bulge,
SMBH influence on the low density medium of the host galaxy is significant even
for SMBH well below the $M-\sigma$ mass. The role of SMBH feedback in our model
evolves in space and time with the ISM structure. In the early gas rich phase,
SMBH accelerates star formation in the bulge. During later gas poor
(red-and-dead) phases, SMBH feedback is mostly negative everywhere due to
scarcity of the cold ISM.
