  The size distribution in the cold classical Kuiper belt can be approximated
by two idealized power laws: one with steep slope for radii R>R* and one with
shallow slope for R<R*, where R*~25-50 km. Previous works suggested that the
SFD roll-over at R* can be the result of extensive collisional grinding in the
Kuiper belt that led to the catastrophic disruption of most bodies with R<R*.
Here we use a new code to test the effect of collisions in the Kuiper belt. We
find that the observed roll-over could indeed be explained by collisional
grinding provided that the initial mass in large bodies was much larger than
the one in the present Kuiper belt, and was dynamically depleted. In addition
to the size distribution changes, our code also tracks the effects of
collisions on binary systems. We find that it is generally easier to dissolve
wide binary systems, such as the ones existing in the cold Kuiper belt today,
than to catastrophically disrupt objects with R~R*. Thus, the binary survival
sets important limits on the extent of collisional grinding in the Kuiper belt.
We find that the extensive collisional grinding required to produce the SFD
roll-over at R* would imply a strong gradient of the binary fraction with R and
separation, because it is generally easier to dissolve binaries with small
components and/or those with wide orbits. The expected binary fraction for R<R*
is <0.1. The present observational data do not show such a gradient. Instead,
they suggest a large binary fraction of ~0.4 for R=30-40 km. This may indicate
that the roll-over was not produced by disruptive collisions, but is instead a
fossil remnant of the KBO formation process.
