Voronoi tessellations: The classical theory of moments for structure description of granular materials

Authors: O.I. Gerasymov, N.N. Khudyntsev

Year: 2015

Issue: 19

Pages: 170-175


The theoretical description of the local structure of granular materials has been performed by means of Voronoi method. The detailed investigation of structure transformations has been carried on with help of Voronoi tessellation supplemented by direct modeling of the relevant distribution function in terms of classical moments theory. Analytical expression for distribution function of Voronoi figures has been constructed with the help of Nevanlinna’s formula from theory of orthogonal polynomials .Proposed approach permit to avoid the problem of week argumentation of applicability the statistical mechanics methods for description of the structure and physical properties of granular materials. We show that generated ordering in local structure are escorted by appearing of particular symmetries in Voronoi diagrams. We perform a numerical simulations of structural configurations in 2D system of hard discs. Proposed algorithm allow us to prove theoretical predictions about existence of correlations between configurational ordering and symmetry breaking in Voronoy tessellations. We study these effects in the vicinity of jammed states. Obtained results shows that criticality in structurisation (formation of jammed states)connected with particular behavior of the first two moments of Voronoi figures distribution function. We show nonhomogeneous character of jammed states in which kinematic freedom degrees become frozen. Namely, coexisting ordered domains which has a different symmetries in grain configurations are observed. Therefore given analysis fulfill the basis of research in the area of granular physics which are mostly based on the concepts of probabilistic stereology and do not use methods from statistical mechanics which in the case of granular materials are not enough argumeneted.

Tags: classical theory of moments; granular materials; jammed states; Voronoi figures


  1. Jaeger H.M., Nagel S.R., Behringer R.P. Rev. Mod. Phys., 1996, no. 68, pp. 1259.
  2. Duran J. Sands, Powders and Grains. New York: Springer, 2000.
  3. Kadanoff L. Rev.Mod.Phys., 1999, no. 71, pp. 435.
  4. De Gennes P.G., Rev. Mod. Phys., 1999, no. 71, pp. 374.
  5. Bideau D., Gervois A., Oger L., Troadec J.P. J.Physique, 1986, no. 47, pp. 1697.
  6. Lumay J., Vandewalle N. Phys. Rev. Lett., 2005, no. 95, pp. 028002.
  7. Voronoi G., Reine Angew J. Math., 1908, no. 134, pp. 198.
  8. Quickenden T.I., Tan G.K. Journal of Collloid and Interface Science , 1974, no. 48, pp. 382.
  9. Aste T., Di Matteo T. Phys. Rev., 2008, no. E77, pp. 021309.
  10. Edwards S.F. Physica, 2005, no. A353 , pp. 114.
  11. Achiezer N.I., Classical problem of moments. Мoscow: Nauka, 1961. 600 p.
  12. Gerasymov O.I. Ukr.Journ.Phys., 2010, vol.55, no. 5, pp. 560.
Download full text (PDF)