Random subdivisions of space originated in a nucleation and growth process are commonly observed in many scientific fields, such us metallurgy, geology, biology and ecology. One of the simplest of these processes is the one generated by an initial random distribution of seeds or nuclei, all growing at the same rate and fixed in space without pushing apart as they grow into contact. The final stage of this process is the well-known Poisson-Voronoi cellular structure or tessellation. Here, we present an analytical exact result for the evolution of the domain size distribution along the transformation process. The calculations are based on a differentiation of the domains by their different number of collisions with surrounding seeds. The method can be easily extended to the calculation of the probability distribution of any other geometrical characteristic, such as the free boundary fraction of the domains. As far as we know, it is the first time that an exact result is given for this classical system.

Peer Reviewed