Metastable Decay Rates, Asymptotic Expansions, and Analytic Continuation of Thermodynamic Functions

The grand potential P(z)=kT of the cluster model at fugacity z, neglecting interactions between clusters, is deened by a power series P n Q n z n , where Q n , which depends on the temperature T, is thèpartition function' of a cluster of size n. At low temperatures this series has a nite radius of convergence z s. Some theorems are proved showing that if Q n , considered as a function of n, is the Laplace transform of a function with suitable properties then P(z) can be analytically continued into the complex z-plane cut along the real axis from z s to +1 and that (a) the imaginary part of P(z) on the cut is (apart from a relatively unimportant prefactor) equal to the rate of nucleation of the correponding metastable state, as given by Becker-DD oring theory, and 1 (b) the real part of P(z) on the cut is approximately equal to the metastable grand potential as calculated by truncating the divergent power series at its smallest term.