  We study the classical two-dimensional Coulomb gas model for thermal vortex
fluctuations in thin superconducting/superfluid films by Monte Carlo simulation
of a grand canonical vortex ensemble defined on a continuum. The
Kosterlitz-Thouless transition is well understood at low vortex density, but at
high vortex density the nature of the phase diagram and of the vortex phase
transition is less clear. From our Monte Carlo data we construct phase diagrams
for the 2D Coulomb gas without any restrictions on the vortex density. For
negative vortex chemical potential (positive vortex core energy) we always find
a Kosterlitz-Thouless transition. Only if the Coulomb interaction is
supplemented with a short-distance repulsion, a first order transition line is
found, above some positive value of the vortex chemical potential.
