Nucleation and growth in one dimension, part I: The generalized Kolmogorov-Johnson-Mehl-Avrami model

TL;DR

This paper extends the one-dimensional KJMA model to include arbitrary nucleation rates, providing analytical solutions and improved simulations, with applications to DNA replication.

Contribution

It introduces a generalized 1D KJMA model with arbitrary nucleation rates and offers analytical solutions alongside enhanced simulation algorithms.

Findings

Analytical expressions for time-dependent distributions derived.

Simulation results match analytical predictions.

Model applicable to DNA replication studies.

Abstract

Motivated by a recent application of the Kolmogorov-Johnson-Mehl-Avrami (KJMA) model to the study of DNA replication, we consider the one-dimensional version of this model. We generalize previous work to the case where the nucleation rate is an arbitrary function $I(t)$ and obtain analytical results for the time-dependent distributions of various quantities (such as the island distribution). We also present improved computer simulation algorithms to study the 1D KJMA model. The analytical results and simulations are in excellent agreement.