Temporal evolution of the domain structure in a Poisson-Voronoi nucleation and growth transformation: results for one and three dimensions.

The distribution of spatial domain structures originated during one- and three-dimensional Poisson-Voronoi transformations are computed analytically extending the recently obtained results for the two-dimensional case. The presented method gives a full description of the developed microstructure and is valid for tessellations of any dimensionality. The temporal and spatial dependences of the domain structure are completely discriminated and separated, showing the existence of geometric configurations independent of time. A single computation of the probability distribution of these geometric configurations allows us to calculate the total free-boundary and size probability distributions at any desired time. The obtained results show full agreement with stochastic simulations and reproduce completely the previously existing partial results. A discussion about the potential applications of the method to the calculation of other geometrical properties and the characteristics of the final static structure leading to a Gamma distribution of sizes is also presented.