Nucleation and growth in one dimension

We study statistical properties of the Kolmogorov-Avrami-Johnson-Mehl nucleation-and-growth model in one dimension. We obtain exact results for the gap density as well as the island distribution. When all nucleation events occur simultaneously, we show that the island distribution has discontinuous derivatives on the rays x_n(t)=nt, n=1,2,3... . We introduce an accelerated growth mechanism with growth rate increasing linearly with the island size. We solve for the interisland gap density and show that the system reaches complete coverage in a finite time and that the near-critical behavior of the system is robust; i.e., it is insensitive to details such as the nucleation mechanism.