Abstract:
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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. |