  Models of the zodiacal cloud's thermal emission and sporadic meteoroids
suggest Jupiter-family comets (JFCs) as the dominant source of interplanetary
dust. However, comet sublimation is insufficient to sustain the quantity of
dust presently in the inner solar system, suggesting that spontaneous
disruptions of JFCs may supply the zodiacal cloud. We present a model for the
dust produced in comet fragmentations and its evolution. Using results from
dynamical simulations, the model follows individual comets drawn from a size
distribution as they evolve and undergo recurrent splitting events. The
resulting dust is followed with a kinetic model which accounts for the effects
of collisional evolution, Poynting-Robertson drag, and radiation pressure. This
allows to model the evolution of both the size distribution and radial profile
of dust, and we demonstrate the importance of including collisions (both as a
source and sink of dust) in zodiacal cloud models. With physically-motivated
free parameters this model provides a good fit to zodiacal cloud observables,
supporting comet fragmentation as the plausibly dominant dust source. The model
implies that dust in the present zodiacal cloud likely originated primarily
from disruptions of $\sim$ 50 km comets, since larger comets are ejected before
losing all their mass. Thus much of the dust seen today was likely deposited as
larger grains $\sim$ 0.1 Myr in the past. The model also finds the dust level
to vary stochastically; e.g., every $\sim$ 50 Myr large (>100 km) comets with
long dynamical lifetimes inside Jupiter cause dust spikes with order of
magnitude increases in zodiacal light brightness lasting $\sim$ 1 Myr. If
exozodiacal dust is cometary in origin, our model suggests it should be
similarly variable.
