  In this paper we investigate the relationship between the mixing times of the
Glauber dynamics of a statistical mechanical system with its thermodynamic
equilibrium structure. For this we consider the mean-field Blume-Capel model,
one of the simplest statistical mechanical models that exhibits the following
intricate phase transition structure: within a two dimensional parameter space
there exists a curve at which the model undergoes a second-order, continuous
phase transition, a curve where the model undergoes a first-order,
discontinuous phase transition, and a tricritical point which separates the two
curves. We determine the interface between the regions of slow and rapid
mixing. In order to completely determine the region of rapid mixing, we employ
a novel extension of the path coupling method, successfully proving rapid
mixing even in the absence of contraction between neighboring states.
