Temporal evolution of the domain structure in a Poisson-Voronoi transformation

Abstract

The temporal evolution of the domain size and free boundary distributions is calculated for a Poisson–Voronoi transformation. In this kind of transformation a set of randomly distributed domain seeds start growing simultaneously, all with equal isotropic growth rate, occupying the original untransformed space. At the end of the transformation, all the space is occupied and the final configuration is the well-known Poisson–Voronoi tessellation. In this work, the temporal evolution of the domain structure in a twodimensional transformation is obtained by means of a calculation method presented recently (Pineda et al 2007 Phys. Rev. E 75 040107(R)). The method is based on the differentiation of the domains through their number of extended collisions. It is found that the probability distribution of geometrical configurations for domains with a certain number of extended collisions is time invariant throughout the transformation. The calculation of these time-invariant probability distributions allows us to obtain the probability density function of any geometric characteristic of the domains at any finite time during the transformation. In this work this is applied to obtain the size and the free boundary fraction distributions. As far as we are aware, this is the first time that an analytical solution has been obtained for this system.