  We study with lattice Monte Carlo simulations the interactions and
macroscopic behaviour of a large number of vortices in the 3-dimensional U(1)
gauge+Higgs field theory, in an external magnetic field. We determine
non-perturbatively the (attractive or repelling) interaction energy between two
or more vortices, as well as the critical field strength H_c, the
thermodynamical discontinuities, and the surface tension related to the
boundary between the Meissner phase and the Coulomb phase in the type I region.
We also investigate the emergence of vortex lattice and vortex liquid phases in
the type II region. For the type I region the results obtained are in
qualitative agreement with mean field theory, except for small values of H_c,
while in the type II region there are significant discrepancies. These findings
are relevant for superconductors and some models of cosmic strings, as well as
for the electroweak phase transition in a magnetic field.
