  We study a charged scalar field in a bulk 3+1 dimensional anti-deSitter
spacetime with a planar black hole background metric. Through the AdS/CFT
correspondence this is equivalent to a strongly coupled field theory in 2+1
dimensions describing a superconductor. We use the gradient flow method and
solve the flow equations numerically between two fixed points: a vacuum
solution and a hairy black hole solution. We study the corresponding flow on
the boundary between a normal metal phase and a superconducting phase. We show
how the gradient flow moves fields between two fixed points in a way that
minimizes the free energy of the system. At the fixed points of the flow the
AdS/CFT correspondence provides an equivalence between the Euclidean on-shell
action in the bulk and the free energy of the boundary, but it does not tell us
about fields away from equilibrium. However, we can formally link static
off-shell configurations in the bulk and in the boundary at the same point
along the flow. For quasi-static evolution at least, it may be reasonable to
think of this link as an extension of the AdS/CFT correspondance.
