The mean free path approximation and stellar collisions in star
  clusters: Numerical exploration of the analytic rates and the role of
  perturbations on binary star mergers
  In this paper we compute predictions for the number of stellar collisions
derived from analytic models based on the mean free path (MFP) approximation
and compare them to the results of $N$-body simulations. Our goal is to
identify the cluster conditions under which the MFP approximation remains
valid. Adopting a range of particle numbers ($100\leq N\leq5000$) and different
combinations of particle masses and radii, we explore three different channels
leading to stellar collisions, all of which are expected to occur in realistic
stellar environments. At high densities, binaries form from isolated three-body
interactions of single stars. Hence, we consider collisions between single
stars and collisions involving binary stars, after they form in our
simulations. For the latter, we consider two channels for mergers, namely
direct stellar collisions during chaotic single-binary interactions and
perturbation-driven mergers of binaries due to random walks in eccentricity
approaching unity. In the densest systems considered here, a very massive
object is formed at the cluster centre, causing local stellar orbits to become
increasingly Keplerian and the assumptions going into our analytic model to
break down. Before reaching this limit, we obtain excellent agreement between
our theoretical predictions and the simulations: the analytic rates are
typically accurate to within one standard deviation for the entire parameter
space considered here, but the agreement is best for short integration times.
Our results have direct implications for blue straggler formation in dense star
clusters, and stellar mergers in galactic nuclei hosting massive black holes.
