The uphill turtle race: on short time nucleation probabilities

TL;DR

This paper analyzes the early-time behavior of nucleation probabilities by modeling it as diffusion over a barrier, providing explicit formulas and numerical validation, revealing scaling laws for multiple nuclei.

Contribution

It introduces explicit expressions for short-time nucleation probabilities and explores their independence from potential shape at very short times, with new scaling insights for large N.

Findings

Short-time nucleation probability becomes shape-independent.

Average first nucleation time scales as 1/ log N for large N.

Explicit solutions for linear potential wells at all times.

Abstract

The short time behavior of nucleation probabilities is studied by representing nucleation as diffusion in a potential well with escape over a barrier. If initially all growing nuclei start at the bottom of the well, the first nucleation time on average is larger than the inverse nucleation frequency. Explicit expressions are obtained for the short time probability of first nucleation. For very short times these become independent of the shape of the potential well. They agree well with numerical results from an exact enumeration scheme. For a large number N of growing nuclei the average first nucleation time scales as 1/\log N in contrast to the long-time nucleation frequency, which scales as 1/N. For linear potential wells closed form expressions are obtained for all times.