A subcritical droplet must make a rare thermal fluctuation from the metastable basin over the free-energy barrier $F^*$. Its probability carries the <Boltzmann factor> $e^{-F^*/(k_BT)}$, so the nucleation rate has the <Arrhenius nucleation time>[Arrhenius form]
$$
\Gamma\sim\Gamma_0e^{-F^*/(k_BT)}.
$$
For a volume containing order one candidate subcritical droplet, the waiting time for a supercritical droplet and subsequent macroscopic growth is therefore
$$
\boxed{\tau\sim\tau_0e^{F^*/(k_BT)},}
$$
up to an algebraic kinetic prefactor. This is the <Arrhenius nucleation time>.

Solution

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

Thermal crossing of a nucleation barrier $F^*$ has rate proportional to $e^{-F^*/(k_BT)}$ and mean waiting time proportional to $e^{F^*/(k_BT)}$, up to a kinetic prefactor.


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