Conference paper
Part of the NATO Science Series II book series (NAII, volume 232)


We investigate systems of nature driven by combinations of diffusive growth, size fragmentation and fragment coagulation. In particular we derive and solve analytically rate equations for the size distribution of fragments and demonstrate the applicability of our models in very different systems of nature, ranging from the distribution of ice crystal sizes from the Greenland ice sheet to the length distribution of α-helices in proteins. Initially, we consider processes where coagulation is absent. In this case the diffusion-fragmentation equation can be solved exactly in terms of Bessel functions. Introducing the coagulation term, the full non-linear model can be mapped exactly onto a Riccati equation that has various asymptotic solutions for the distribution function. In particular, we find a standard exponential decay, exp(–x), for large x, and observe a crossover from the Bessel function for intermediate values of x.


Merging Process Crystal Size Distribution Vertical Size Fragmentation Rate Helix Length 
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. 1.
    D. Dahl-Jensen, N. Gundestrup, H. Miller, O.Watanabe, S.J. Johnsen, J.P Steffensen, H.B. Clausen, A. Svensson, and L.B. Larsen, Annals of Glaciology 35, 1–5, 2002.CrossRefADSGoogle Scholar
  2. 2.
    A. Svensson, K.G. Schmidt, D. Dahl-Jensen, S.J. Johnsen, Y. Wang, J. Kipfstuhl, and T. Thorsteinsson, Annals of Glaciology 37,113–118 (2003).CrossRefADSGoogle Scholar
  3. 3.
    N. P. Louat, Acta Metallurgica 22, 721 (1974).CrossRefGoogle Scholar
  4. 4.
    R. B. Alley, J. H. Perepezko and C. R. Bentley, J. Glaciol. 32(112), 425–433 (1986).ADSGoogle Scholar
  5. 5.
    Paterson, W.S.B. The Physics of Glaciers, 480 pp., Pergamon, New York (1994).Google Scholar
  6. 6.
    J. Mathiesen, J. Ferkingho.-Borg, M.H. Jensen, M. Levinsen, P. Olesen, D. Dahl-Jensen and A. Svensson, J. Glaciol. 50, 325 (2004).CrossRefADSGoogle Scholar
  7. 7.
    J. Ferkinghoff-Borg, M. H. Jensen, J. Mathiesen, P. Olesen and K. Sneppen, Phys. Rev. Lett. 91, 266103 (2003).CrossRefADSGoogle Scholar
  8. 8.
    P. Olesen, J. Ferkinghoff-Borg, M. H. Jensen and J. Mathiesen. Phys. Rev. E in print, cond-mat/0411514.Google Scholar
  9. 9.
    Laurençot P., Steady states for a fragmentation equation with size diffusion, to appear in a volume of Banach Center Publications.Google Scholar
  10. 10.
    A. Fersht, Structure and Mechanism in Protein Science, W.H Freeman and Company (1999).Google Scholar
  11. 11.
    Penel, S., Morisson, R. G., Mortishire-Smith, R. J. and Doig, A. J. J. Mol. Biol., 293, 1211–1219 (1999).CrossRefGoogle Scholar
  12. 12.
    S. Miyazawa and R.L. Jernigen, Macromolecules 18, 534 (1985); S. Miyazawa and R.L. Jernigen, J. Mol. Biol. 256, 623 (1996)CrossRefADSGoogle Scholar
  13. 13.
    A. J. Doig, A. Chakrabartty, T. M. Klingler and R. Baldwin, Biochemistry 33, 3369 (1994).CrossRefGoogle Scholar
  14. 14.
    M. von Smoluchowski, Phys. Z. 17, 557, 585 (1916).ADSGoogle Scholar
  15. 15.
    M. von Smoluchowski, Z. Physik. Chem. 92, 129 (1917).Google Scholar
  16. 16.
    F. Leyvraz, Phys. Rep. 383, 95 (2003).CrossRefADSGoogle Scholar
  17. 17.
    Z. A. Melzak, Trans. Amer. Math. Soc 85 547 (1957).CrossRefMathSciNetzbMATHGoogle Scholar
  18. 18.
    R. M. Zi, J. Stat. Phys. 23, 241 (1980).CrossRefADSGoogle Scholar
  19. 19.
    S. K. Frielander. Smoke, dust and haze: fundamentals of aerosol dynamics, Wiley, New York (1977).Google Scholar
  20. 20.
    J. Silk and S. D. White, Astrophysical J. 223 L59 (1978).CrossRefADSGoogle Scholar
  21. 21.
    A. Okubo, Adv. Biophys. 22, 1 (1986).CrossRefGoogle Scholar
  22. 22.
    S. Gueron and S. A. Levin, Math Biosci. 128, 243 (1995).CrossRefzbMATHGoogle Scholar
  23. 23.
    R. L. Drake, A general mathematical survey of the coagulation equation. In G.M. Hidy and J.R. Brock, editors, Topics in Current Aerosol Research (Part 2), pages 201–376. Pergamon Press, Oxford, 1972.Google Scholar
  24. 24.
    P. Laurençot and S. Mischler, On coalescence equations and related models in “Modeling and computational methods for kinetic equations”, Editors Degond, Pierre; Pareschi, Lorenzo and Russo, Giovanni, in the Series Modeling and Simulation in Science, Engineering and Technology (MSSET), Birkhauser (2004)Google Scholar
  25. 25.
    H. Amann, H., Arch. Rat. Mech. Anal. 151, 339 (2003); H. Amann. and Weber, Adv. Math. Sci. Appl. 11, 227–263 (2001); P. Laurençot and S. Mischler, Arch. Rat. Mech. Anal. 162(1), 45 (2002).CrossRefGoogle Scholar

Copyright information

© Springer 2006

Authors and Affiliations

    • 1
    • 1
    • 1
    • 2
  1. 1.Niels Bohr InstituteCopenhagenDenmark
  2. 2.NTNU, Institutt for fysikkTrondheim

Personalised recommendations