We determine the non-equilibrium grain size distribution during the crystallization of a solid in $d$ dimensions at fixed thermodynamic conditions, for the random nucleation and growth model, and in absence of grain coalescence. Two distinct generalizations of the theory established earlier are considered. A closed analytic expression of the grain size distribution useful for experimental studies is derived for anisotropic growth rates. The main difference from the isotropic growth case is the appearance of a constant prefactor in the distribution. The second generalization considers a Gaussian source term: nuclei are stable when their volume is within a finite range determined by the thermodynamics of the crystallization process. The numerical results show that this generalization does not change the qualitative picture of our previous study. The generalization only affects quantitatively the early stage of crystallization, when nucleation is dominant. The remarkable result of these major generalizations is that the non-equilibrium grain size distribution is robust against anisotropic growth of grains and fluctuations of nuclei sizes.
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