  The dynamics of an interface between the normal and superconducting phases
under nonstationary external conditions is studied within the framework of the
time-dependent Ginzburg-Landau equations of superconductivity, modified to
include thermal fluctuations. An equation of motion for the interface is
derived in two steps. First, the method of matched asymptotic expansions is
used to derive a diffusion equation for the magnetic field in the normal phase,
with nonlinear boundary conditions at the interface. These boundary conditions
are a continuity equation which relates the gradient of the field at the
interface to the normal velocity of the interface, and a modified Gibbs-Thomson
boundary condition for the field at the interface. Second, the boundary
integral method is used to integrate out the magnetic field in favor of an
equation of motion for the interface. This equation of motion, which is highly
nonlinear and nonlocal, exhibits a diffusive instability (the Mullins-Sekerka
instability) when the superconducting phase expands into the normal phase
(i.e., when the external field is reduced below the critical field). In the
limit of infinite diffusion constant the equation of motion becomes local in
time, and can be derived variationally from a static energy functional which
includes the bulk free energy difference between the two phases, the
interfacial energy, and a long range self-interaction of the interface of the
Biot-Savart form. In this limit the dynamics is identical
