About the time evolving Voronoi tessellation (Q1181894)

Voronoi diagrams are relevant in different areas of physics such as condensed non-crystalline systems, including liquids, metallic glasses, and oxide glasses. Their construction depends strongly on the algorithm one uses, which can range from classical geometrical methods to construct the vertices of the polyhedra to more complicated procedures that make use of various lemmas and theorems to extract the number of faces. All these algorithms are not very easy to handle: we therefore decided to use more simple arguments on a finite lattice. These have already been introduced by \textit{T. Kiang} [Z. Astrophys. 64, 433 (1966)]. Due to the great improvement in the computing facilities we can handle a \(400\times400\times400\) cubic lattice. In \textit{Kiang} [loc. cit.] the size distribution of random Voronoi segments has also been analyzed and the following approximate distribution function derived \[ p(x)={6^ 6\over 120}\left({x \over \langle x\rangle}\right)^ 5 e^{-6x/\langle x\rangle}, \] where \(p(x)\) is the probability of having a value \(x\). This result is similar to the one derived from \textit{N. F. Mott} [Proc. R. Soc. A 189, 300 (1947)] for the mass fragmentation \[ p(m)={{e^{-(2m/\langle m\rangle)^{0.5}}} \over {\int_ 0^{+\infty} e^{-(2m/\langle m\rangle)^{0.5}}dm}} \] and rederived from \textit{W. Brostow} and \textit{H. C. Rogers} [Mater. Chem. Phys. 12, 499 (1985)] by using the theory for information. By using approximate methods we can easily derive the Voronoid diagrams and the related parameters and we can also explore intermediate situations in which the diagrams are not yet completely developed. This could be the case of foams in which the signals start from the various nuclei at shifted times and therefore generate a dynamical rather than a static situation.