E. Bennaim et Pl. Krapivsky, NUCLEATION AND GROWTH IN ONE-DIMENSION, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics, 54(4), 1996, pp. 3562-3568
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 nucle
ation events occur simultaneously, we show that the island distributio
n 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 densi
ty 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.