  A spatially extended classical system with metastable states subject to weak
spatiotemporal noise can exhibit a transition in its activation behavior when
one or more external parameters are varied. Depending on the potential, the
transition can be first or second-order, but there exists no systematic theory
of the relation between the order of the transition and the shape of the
potential barrier. In this paper, we address that question in detail for a
general class of systems whose order parameter is describable by a classical
field that can vary both in space and time, and whose zero-noise dynamics are
governed by a smooth polynomial potential. We show that a quartic potential
barrier can only have second-order transitions, confirming an earlier
conjecture [1]. We then derive, through a combination of analytical and
numerical arguments, both necessary conditions and sufficient conditions to
have a first-order vs. a second-order transition in noise-induced activation
behavior, for a large class of systems with smooth polynomial potentials of
arbitrary order. We find in particular that the order of the transition is
especially sensitive to the potential behavior near the top of the barrier.
