  There is currently debate over whether the dust content of planetary systems
is stochastically regenerated or originates in planetesimal belts evolving in
steady state. In this paper a simple model for the steady state evolution of
debris disks due to collisions is developed and confronted with the properties
of the emerging population of 7 sun-like stars that have hot dust <10AU. The
model shows there is a maximum possible disk mass at a given age, since more
massive primordial disks process their mass faster. The corresponding maximum
dust luminosity is f_max=0.00016r^(7/3)/t_age. The majority (4/7) of the hot
disks exceed this limit by >1000 and so cannot be the products of massive
asteroid belts, rather the following systems must be undergoing transient
events characterized by an unusually high dust content near the star: eta
Corvi, HD69830, HD72905 and BD+20307. It is also shown that the hot dust cannot
originate in a recent collision in an asteroid belt, since there is also a
maximum rate at which collisions of sufficient magnitude to reproduce a given
dust luminosity can occur. Further it is shown that the planetesimal belt
feeding the dust in these systems must be located further from the star than
the dust, typically at >2AU. Other notable properties of the 4 hot dust systems
are: two also have a planetesimal belt at >10AU (eta Corvi and HD72905); one
has 3 Neptune mass planets at <1AU (HD69830); all exhibit strong silicate
features in the mid-IR. We consider the most likely origin for the dust in
these systems to be a dynamical instability which scattered planetesimals
inwards from a more distant planetesimal belt in an event akin to the Late
Heavy Bombardment in our own system, the dust being released from such
planetesimals in collisions and possibly also sublimation.
