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 defined by a power series Sigma(n) Q(n)z(n), where Q(n), which depends on the temperature T, is the ''partition function'' of a cluster of size n. At low temperatures this series has a finite 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 +infinity 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 corresponding metastable state, as given by Becker-Doring theory, and (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.