  We derive a new analytical model for the evolution of a collisional cascade
in a thin annulus around a single central star. In this model, $r_{max}~$ the
size of the largest object declines with time (t); $r_{max} \propto
t^{-\gamma}$, with $\gamma$ = 0.1-0.2. Compared to standard models where
$r_{max}~$ is constant in time, this evolution results in a more rapid decline
of $M_d$, the total mass of solids in the annulus and $L_d$, the luminosity of
small particles in the annulus: $M_d \propto t^{-(\gamma + 1)}~$ and $L_d
\propto t^{-(\gamma/2 + 1)}~~$. We demonstrate that the analytical model
provides an excellent match to a comprehensive suite of numerical coagulation
simulations for annuli at 1 AU and at 25 AU. If the evolution of real debris
disks follows the predictions of the analytical or numerical models, the
observed luminosities for evolved stars require up to a factor of two more mass
than predicted by previous analytical models.
