PHASE-TRANSITIONS IN A VORTEX GAS

It has been shown recently that the motion of solitons at couplings ar ound 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 vi rial expansion using certain known integrals of the metric, and the se cond virial coefficient is calculated, determining the behaviour of th e gas at low densities. A formula for determining higher-order coeffic ients is given, At low densities and temperatures T much greater than lambda the equation of state is of the Van der Waals form (P + bN(2)/A (2)) (A - aN) = NT with a = 4 pi and b = -4.89 pi lambda where lambda 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 i s a transition in the type-I case between a condensed and gaseous stat e. We conclude with a discussion of the relation of our results to vor tex behaviour in superconductors.