Role of an intermediate state in homogeneous nucleation

We explore the role of an intermediate state (phase) in homogeneous nucleation by examining the decay process through a double-humped potential barrier. We analyze the one-dimensional Fokker-Planck (FP) equations with the fourth- and sixth-order Landau potentials. In the low-temperature case, we apply the WKB method to the FP equation and obtain an analytic expression for the decay rate which is accurate for a wide range of depth and curvature of the middle well. In the case of a deep middle well, it reduces to an extended Kramers formula, in which the barrier height in the original formula is replaced by the arithmetic mean height of the higher (outer) and lower (inner) barriers, and the curvature of the initial well in the original one is replaced by the geometric mean curvature of the initial and intermediate wells. In the case of a shallow middle well, the Kramers escape rate is evaluated also within the standard framework of the mean-first-passage-time problem, whose result is consistent with our WKB analysis. Criteria for enhancement of the decay rate are revealed.