Role of an intermediate state in homogeneous nucleation

We explore the role of an intermediate state (phase) in homogeneous nucleation phenomenon by examining the decay process through a doubly-humped potential barrier. As a generic model we use the fourth- and sixth-order Landau potentials and analyze the Fokker-Planck equation for the one-dimensional thermal diffusion in the system characterized by a triple-well potential. In the low temperature case we apply the WKB method to the decay process and obtain 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 a doubly-humped-barrier counterpart of the Kramers escape rate: the barrier height and the curvature of an initial well in the Kramers rate are replaced by the arithmetic mean of higher(or outer) and lower(or inner) partial barriers and the geometric mean of curvatures of the initial and intermediate wells, respectively. It seems to be a universal formula. In the case of a shallow-enough middle well, Kramers escape rate is alternatively evaluated within the standard framework of the mean-first-passage time problem, which certainly supports the WKB result. The criteria whether or not the existence of an intermediate state can enhance the decay rate are revealed.