Journal of Statistical Physics

, Volume 167, Issue 3–4, pp 703–712 | Cite as

Path Selection in a Poisson field

  • Yossi Cohen
  • Daniel H. Rothman


A criterion for path selection for channels growing in a Poisson field is presented. We invoke a generalization of the principle of local symmetry. We then use this criterion to grow channels in a confined geometry. The channel trajectories reveal a self-similar shape as they reach steady state. Analyzing their paths, we identify a cause for branching that may result in a ramified structure in which the golden ratio appears.


Laplacian paths Poisson paths Local symmetry Ramified networks Golden ratio 



We thank M. Z. Bazant, O. Devauchelle, M. Mineev-Weinstein and P. Szymczak for interesting discussions and helpful interactions. This work was supported by U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, Chemical Sciences, Geosciences, and Biosciences Division under Award Number FG02-99ER15004.


  1. 1.
    Saffman PG, Taylor G. The penetration of a fluid into a porous medium or hele-shaw cell containing a more viscous liquid. In: Proceedings of the Royal Society of London A: mathematical, Physical and Engineering Sciences, vol. 245, pp 312–329. The Royal Society, 1958Google Scholar
  2. 2.
    Bensimon, David, Leo, P., Kadanoff, L.P., Liang, S., Shraiman, B.I., Tang, C.: Viscous flows in two dimensions. Rev. Mod. Phys. 58(4), 977 (1986)ADSCrossRefzbMATHGoogle Scholar
  3. 3.
    Barenblatt, G.I., Cherepanov, G.P.: On brittle cracks under longitudinal shear. J. Appl. Math. Mech. 25(6), 1654–1666 (1961)CrossRefzbMATHGoogle Scholar
  4. 4.
    Kessler, D.A., Koplik, J., Levine, H.: Pattern selection in fingered growth phenomena. Adv. Phys. 37(3), 255–339 (1988)ADSCrossRefGoogle Scholar
  5. 5.
    Sander, M.: Diffusion-limited aggregation, a kinetic critical phenomenon. Phys. Rev. Lett. 47(19), 1400 (1981)ADSCrossRefGoogle Scholar
  6. 6.
    Halsey, T.C.: Diffusion-limited aggregation: a model for pattern formation. Phys. Today 53(11), 36–41 (2000)CrossRefGoogle Scholar
  7. 7.
    Niemeyer, L., Pietronero, L., Wiesmann, H.J.: Fractal dimension of dielectric breakdown. Phys. Rev. Lett. 52(12), 1033 (1984)ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    Shelley, M.J., Tian, F.R., Wlodarski, K.: Hele-shaw flow and pattern formation in a time-dependent gap. Nonlinearity 10(6), 1471 (1997)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    McDonald, R., Mineev-Weinstein, M.: Poisson growth. Anal. Math. Phys. 5(2), 193–205 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bazant, M.Z.: Exact solutions and physical analogies for unidirectional flows. Phys. Rev. Fluids 1(2), 024001 (2016)ADSCrossRefGoogle Scholar
  11. 11.
    Cohen, Y., Procaccia, I.: Dynamics of cracks in torn thin sheets. Phys. Rev. E 81(6), 7 (2010)CrossRefGoogle Scholar
  12. 12.
    Cohen, Y., Procaccia, I.: Stress intensity factor of mode-III cracks in thin sheets. Phys. Rev. E 83(2), 026106 (2011)ADSCrossRefGoogle Scholar
  13. 13.
    Petroff, A.P., Devauchelle, O., Kudrolli, A., Rothman, D.H.: Four remarks on the growth of channel networks. C. R. Geosci. 344(1), 33–40 (2012)CrossRefGoogle Scholar
  14. 14.
    Cohen, Y., Devauchelle, O., Seybold, H.F., Robert, S.Y., Szymczak, P., Rothman, D.H.: Path selection in the growth of rivers. Proc. Natl. Acad. Sci. 112(46), 14132–14137 (2015)ADSCrossRefGoogle Scholar
  15. 15.
    Griffith, A.A.: The phenomena of rupture and flow in solids. Philos. Trans. R. Soc. Lond. Ser. A 221, 163–198 (1921)ADSCrossRefGoogle Scholar
  16. 16.
    Irwin, G.R.: Analysis of stresses and strains near the end of a crack traversing a plate. J. Appl. Mech. 137, 16 (1957)Google Scholar
  17. 17.
    Dunne, T.: Formation and controls of channel networks. Prog. Phys. Geogr. 4(2), 211–239 (1980)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Dupuit. J.: Études théoriques et pratiques sur le mouvement des eaux dans les canaux découverts et à travers les terrains perméables. Dunod, 1863Google Scholar
  19. 19.
    Bear, J.: Dynamics of Fluids in Porous Media. Dover, New York (1972)zbMATHGoogle Scholar
  20. 20.
    Polubarinova-Kochina, P.I.A.: Theory of Ground Water Movement. Princeton University Press, Princeton, NJ (1962)zbMATHGoogle Scholar
  21. 21.
    Kadanoff, L.P.: Simulating hydrodynamics: a pedestrian model. J. Stat. Phys. 39(3–4), 267–283 (1985)ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    Kadanoff, L.P.: On complexity. Phys. Today 40(3), 7–9 (1987)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Devauchelle, O., Petroff, A.P., Seybold, H.F., Rothman, D.H.: Ramification of stream networks. Proc. Natl. Acad. Sc. 109(51), 20832–20836 (2012)ADSCrossRefGoogle Scholar
  24. 24.
    Petroff, A.P., Devauchelle, O., Seybold, H., Rothman, D.H.: Bifurcation dynamics of natural drainage networks. Philos. Trans. R. Soc. A 371(2004), 20120365 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Livio, M.: The Golden Ratio: the Story of Phi, the World’s Most Astonishing Number. Random House LLC, New York (2008)zbMATHGoogle Scholar
  26. 26.
    Arneodo, A., Argoul, F., Muzy, J.F., Tabard, M.: Structural five-fold symmetry in the fractal morphology of diffusion-limited aggregates. Physica A 188(1–3), 217–242 (1992)ADSCrossRefGoogle Scholar
  27. 27.
    Broberg, K.B.: Cracks and Fracture. Academic Press, San Diego (1999)Google Scholar
  28. 28.
    Goldstein, R.V., Salganik, R.L.: Brittle-fracture of solids with arbitrary cracks. Int. J. Fract. 10(4), 507–523 (1974)CrossRefGoogle Scholar
  29. 29.
    Hakim, V., Karma, A.: Laws of crack motion and phase-field models of fracture. J. Mech. Phys. Solids 57(2), 342–368 (2009)ADSCrossRefzbMATHGoogle Scholar
  30. 30.
    Devauchelle, O., Szymczak, P., Pecelerowicz, M., Cohen, Y., Seybold, H., Rothman, D.H.: Laplacian networks: growth, local symmetry and shape optimization (in preparation, 2016)Google Scholar
  31. 31.
    Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables, vol. 55. Courier Corporation, New York (1964)zbMATHGoogle Scholar
  32. 32.
    Muskhelishvili, N.I.: Some Basic Problems of the Mathematical Theory of Elasticity. Noordhoff, Groningen (1953)zbMATHGoogle Scholar
  33. 33.
    Katzav, E., Adda-Bedia, M., Arias, R.: Theory of dynamic crack branching in brittle materials. Int. J. Fract. 143(3), 245–271 (2007)CrossRefzbMATHGoogle Scholar
  34. 34.
    Lobkovsky, E.: Unsteady crack motion and branching in a phase-field model of brittle fracture. Phys. Rev. Lett 92(24), 245510 (2004)ADSCrossRefGoogle Scholar
  35. 35.
    Hastings, M.B.: Growth exponents with 3.99 walkers. Phys. Rev. E 64, 046104 (2001)ADSCrossRefGoogle Scholar
  36. 36.
    Carleson, L., Makarov, N.: Laplacian path models. J. d’Anal. Math. 87(1), 103–150 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Gubiec, T., Szymczak, P.: Fingered growth in channel geometry: a Loewner-equation approach. Phys. Rev. E 77(4), 041602 (2008)ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Lorenz Center, Department of Earth Atmospheric, and Planetary SciencesMassachusetts Institute of TechnologyCambridgeUSA

Personalised recommendations