  We describe gravitational stirring models of planetary debris disks using a
new multi-annulus planetesimal evolution code. The current code includes
gravitational stirring and dynamical friction; future studies will include
coagulation, fragmentation, Poynting-Robertson drag, and other physical
processes. We use the results of our calculations to investigate the physical
conditions required for small bodies in a planetesimal disk to reach the
shattering velocity and begin a collisional cascade. Our results demonstrate
that disks composed primarily of bodies with a single size will not undergo a
collisional cascade which produces small dust grains at 30-150 AU on timescales
of 1 Gyr or smaller. Disks with a size distribution of bodies reach conditions
necessary for a collisional cascade in 10 Myr to 1 Gyr if the disk is at least
as massive as a minimum mass solar nebula and if the disk contains objects with
radii of 500 km or larger. The estimated 500 Myr survival time for these disks
is close to the median age of roughly 400 Myr derived for nearby stars with
dusty disks.
