  We study the time dependence of the grain size distribution N(r,t) during
crystallization of a d-dimensional solid. A partial differential equation
including a source term for nuclei and a growth law for grains is solved
analytically for any dimension d. We discuss solutions obtained for processes
described by the Kolmogorov-Avrami-Mehl-Johnson model for random nucleation and
growth (RNG). Nucleation and growth are set on the same footing, which leads to
a time-dependent decay of both effective rates. We analyze in detail how model
parameters, the dimensionality of the crystallization process, and time
influence the shape of the distribution. The calculations show that the
dynamics of the effective nucleation and effective growth rates play an
essential role in determining the final form of the distribution obtained at
full crystallization. We demonstrate that for one class of nucleation and
growth rates the distribution evolves in time into the logarithmic-normal
(lognormal) form discussed earlier by Bergmann and Bill [J. Cryst. Growth 310,
3135 (2008)]. We also obtain an analytical expression for the finite maximal
grain size at all times. The theory allows for the description of a variety of
RNG crystallization processes in thin films and bulk materials. Expressions
useful for experimental data analysis are presented for the grain size
distribution and the moments in terms of fundamental and measurable parameters
of the model.
