Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 344 / 3 / f / Solution

A subcritical droplet must make a rare thermal fluctuation from the metastable basin over the free-energy barrier $F^{\ast }$. Its probability carries the Boltzmann factor $e^{-F^{\ast }/(k_{B}T)}$, so the nucleation rate has the Arrhenius formFor a volume containing order one candidate subcritical droplet, the waiting time for a supercritical droplet and subsequent macroscopic growth is thereforeup to an algebraic kinetic prefactor. This is the Arrhenius nucleation time.