Maximum Entropy and Equations of State for Random Cellular Structures

  • N. Rivier
Part of the Fundamental Theories of Physics book series (FTPH, volume 39)


Random, space-filling cellular structures (biological tissues, metallurgical grain aggregates, foams, etc) are investigated. Maximum entropy inference under a few constraints yields structural equations of state, relating the size of cells to their topological shape. These relations are known empirically as Lewis’s law in Botany, or Desch’s relation in Metallurgy. Here, the functional form of the constraints is not known a priori, and one takes advantage of this arbitrariness to increase the entropy further. The resulting structural equations of state are independent of priors, they are measurable experimentally and constitute therefore a direct test for the applicability of MaxEnt inference (given that the structure is in statistical equilibrium, a fact which can be tested by another simple relation (Aboav’s law)).


Cellular Network Detailed Balance Microscopic Parameter Human Amnion Topological Shape 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Aboav, D.A. (1970) ‘The arrangement of grains in a polycrystal’, Metallogr. 3, 383–390.CrossRefGoogle Scholar
  2. Boots, B.N (1987) ‘Edge length properties of random Voronoi polygons’, Metallogr.20, 231–236.CrossRefGoogle Scholar
  3. Crain, I.K. (1978) ‘The Monte-Carlo generation of random polygons’, Comput. and Geosc. 4, 131–134.CrossRefGoogle Scholar
  4. Desch, C.H. (1919) ‘The solidification of metals from the liquid state’, J. Inst. Metals 22, 241–276.Google Scholar
  5. Fougere, P.F. (1988) ‘Maximum Entropy calculations on a discrete probability space’, in G.J. Erickson and C.R. Smith (eds.), Maximum-Entropy and Bayesian Methods in Science and Engineering (Vol. 1), 205–234.Google Scholar
  6. Glazier, J.A., Gross, S.P., and Stavans, J. (1987) ‘Dynamics of two-dimensional soap froths’, Phys. Rev. A 36, 306–312.CrossRefGoogle Scholar
  7. Jaynes, E.T. (1957) ‘Information theory and statistical mechanics’, reprinted in R.D. Rosenkrantz (ed.), E.T. Jaynes: Papers on Probability Statistics and Statistical Physics, Reidel, Dordrecht (1983), 4–16.Google Scholar
  8. Jaynes, E.T. (1978) ‘Where do we stand on Maximum Entropy?’, ibid., 210–314.Google Scholar
  9. Jaynes, E.T. (1979) ‘Concentration of distributions at entropy maxima’, ibid., 315–336.Google Scholar
  10. Lewis, F.T. (1928) ‘The correlation between cell division and the shapes and sizes of prismatic cells in the epidermis of cucumis’, Anat. Record 38, 341–376.CrossRefGoogle Scholar
  11. Lewis, F.T. (1931) ‘A comparison between the mosaic of polygons in a film of artificial emulsion and the pattern of simple epithelium in surface view (cucumber epidermis and human amnion)’, Anat. Record 50, 235–265.CrossRefGoogle Scholar
  12. Mombach, J.C.M., Vasconcellos, M.A.Z., and de Almeida, R.M.C. (1989) ‘Arrangement of cells in vegetable tissues’, J. Phys. D, to appear.Google Scholar
  13. von Neumann, J. (1952) ‘Discussion — Shape of metal grains’, in Metal Interfaces, Amer. Soc. Metals, Cleveland, 108–110.Google Scholar
  14. Rivier, N. (1983a) ‘On the structure of random tissues or froths, and their evolution’, Phil. Mag. B47, L45–49.Google Scholar
  15. Rivier, N. (1983b) ‘Topological structure of glasses’, in V. Vitek (ed.), Amorphous Materials: Modeling of Structure and Properties, The Metall. Soc. of AIME, Warrendale, 81–97.Google Scholar
  16. Rivier, N. (1985) ‘Statistical crystallography. Structure of random cellular networks’, Phil. Mag. B52, 795–819.Google Scholar
  17. Rivier, N. (1988) ‘Statistical geometry of tissues’, in I. Lamprecht and A.I. Zotin (eds.), Thermodynamics and Pattern Formation in Biology, de Gruyter, Berlin, 415–446.Google Scholar
  18. Rivier, N. and Lissowski, A. (1982) ‘On the correlation between sizes and shapes of cells in epithelial mosaics’, J. Phys. A15, L143–148.MathSciNetGoogle Scholar
  19. Smoljaninov, V.V. (1980) Mathematical Models of Biological Tissues, Nauka, Moscow (in Russian).Google Scholar
  20. Srolovitz, D.J., Anderson, M.P., Sahni, P.S., and Grest, G.S. (1984) ‘Computer simulation of grain growth — II’. Acta Metall. 32, 793–802.CrossRefGoogle Scholar
  21. Telley, H. (1989) Modélisation et Simulation Bidimensionnelle de la Croissance des Polycristaux, PhD Thesis, EPFL, Lausanne.Google Scholar
  22. Truesdell, C. (1980) The Tragicomical History of Thermodynamics 1822–1854, Springer, Berlin.zbMATHGoogle Scholar
  23. Weaire, D and Rivier, N. (1984) ‘Soap, cells and statistics — Random patterns in two dimensions’, Contemp. Physics. 25, 59–99.CrossRefGoogle Scholar

Copyright information

© Kluwer Academic Publishers 1990

Authors and Affiliations

  • N. Rivier
    • 1
  1. 1.Materials Science DivisionArgonne National LaboratoryArgonneUSA

Personalised recommendations