  We treat analytically a model that captures several features of the
phenomenon of spatially inhomogeneous reversal of an order parameter. The model
is a classical Ginzburg-Landau field theory restricted to a bounded
one-dimensional spatial domain, perturbed by weak spatiotemporal noise having a
flat power spectrum in time and space. Our analysis extends the Kramers theory
of noise-induced transitions to the case when the system acted on by the noise
has nonzero spatial extent, and the noise itself is spatially dependent. By
extending the Langer-Coleman theory of the noise-induced decay of a metastable
state, we determine the dependence of the activation barrier and the Kramers
reversal rate prefactor on the size of the spatial domain. As this is increased
from zero and passes through a certain critical value, a transition between
activation regimes occurs, at which the rate prefactor diverges. Beyond the
transition, reversal preferentially takes place in a spatially inhomogeneous
rather than in a homogeneous way. Transitions of this sort were not discovered
by Langer or Coleman, since they treated only the infinite-volume limit. Our
analysis uses higher transcendental functions to handle the case of finite
volume. Similar transitions between activation regimes should occur in other
models of metastable systems with nonzero spatial extent, perturbed by weak
noise, as the size of the spatial domain is varied.
