  We consider the long-term collisional and dynamical evolution of solid
material orbiting in a narrow annulus near the Roche limit of a white dwarf.
With orbital velocities of 300 km/sec, systems of solids with initial
eccentricity $e \gtrsim 10^{-3}$ generate a collisional cascade where objects
with radii $r \lesssim$ 100--300 km are ground to dust. This process converts
1-100 km asteroids into 1 $\mu$m particles in $10^2 - 10^6$ yr. Throughout this
evolution, the swarm maintains an initially large vertical scale height $H$.
Adding solids at a rate $\dot{M}$ enables the system to find an equilibrium
where the mass in solids is roughly constant. This equilibrium depends on
$\dot{M}$ and $r_0$, the radius of the largest solid added to the swarm. When
$r_0 \lesssim$ 10 km, this equilibrium is stable. For larger $r_0$, the mass
oscillates between high and low states; the fraction of time spent in high
states ranges from 100% for large $\dot{M}$ to much less than 1% for small
$\dot{M}$. During high states, the stellar luminosity reprocessed by the solids
is comparable to the excess infrared emission observed in many metallic line
white dwarfs.
