AbstractWe 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 , systems of solids with initial eccentricitygenerate a collisional cascade where objects with radiiare ground to dust. This process converts 1–100 km asteroids into 1 μm particles in 102−106yr. Throughout this evolution, the swarm maintains an initially large vertical scale heightH. Adding solids at a rateenables the system to find an equilibrium where the mass in solids is roughly constant. This equilibrium depends onand, the radius of the largest solid added to the swarm. When ≲ 10 km, this equilibrium is stable. For larger, the mass oscillates between high and low states; the fraction of time spent in high states ranges from 100% for largeto much less than 1% for small. During high states, the stellar luminosity reprocessed by the solids is comparable to the excess infrared emission observed in many metallic line white dwarfs.