  It has been shown recently that the motion of solitons at couplings around a
critical coupling can be reduced to the dynamics of particles (the zeros of the
Higgs field) on a curved manifold with potential. The curvature gives a
velocity dependent force, and the magnitude of the potential is proportional to
the distance from a critical coupling. In this paper we apply this
approximation to determining the equation of state of a gas of vortices in the
Abelian Higgs model. We derive a virial expansion using certain known integrals
of the metric, and the second virial coefficient is calculated, determining the
behaviour of the gas at low densities. A formula for determining higher order
coefficients is given. At low densities and temperatures $T \gg \l$ the
equation of state is of the Van der Waals form $(P+b\frac{N^{2}}{A^{2}})(A-aN)
= NT$ with $a=4\pi$ and $b=-4.89\pi\l$ where $\l$ is a measure of the distance
from critical coupling. It is found that there is no phase transition in a low
density type-II gas, but there is a transition in the type-I case between a
condensed and gaseous state. We conclude with a discussion of the relation of
our results to vortex behaviour in superconductors.
