Macroscopic behavior of systems with an axial dynamic preferred direction

Regular Article


We present the derivation of the macroscopic equations for systems with an axial dynamic preferred direction. In addition to the usual hydrodynamic variables, we introduce the time derivative of the local preferred direction as a new variable and discuss its macroscopic consequences including new cross-coupling terms. Such an approach is expected to be useful for a number of systems for which orientational degrees of freedom are important including, for example, the formation of dynamic macroscopic patterns shown by certain bacteria such a Proteus mirabilis. We point out similarities in symmetry between the additional macroscopic variable discussed here, and the magnetization density in magnetic systems as well as the so-called \(\hat l\) vector in superfluid 3He-A. Furthermore we investigate the coupling to a gel-like system for which one has the strain tensor and relative rotations between the new variable and the network as additional macroscopic variables.


Prefer Direction Coupling Term Proteus Mirabilis Relative Rotation Momentum Density 
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  1. 1.
    Y. Shimada, A. Nakahara, M. Matsushita, T. Matsuyama, J. Phys. Soc. Jpn. 64, 1896 (1995).ADSCrossRefGoogle Scholar
  2. 2.
    K. Watanabe, J. Wakita, H. Itoh, H. Shimada, S. Kurosu, T. Ikeda, Y. Yamazaki, T. Matsuyama, M. Matsushita, J. Phys. Soc. Jpn. 71, 650 (2002).ADSCrossRefGoogle Scholar
  3. 3.
    H. Honda, H. Nagashima, S. Asakura, J. Mol. Biol. 191, 131 (1986).CrossRefGoogle Scholar
  4. 4.
    D. Humphrey, C. Duggan, D. Saha, D. Smith, J. Kaes, Nature (London) 416, 413 (2002).ADSCrossRefGoogle Scholar
  5. 5.
    D. Bray, Cell Movements (Garland, New York, 1992).Google Scholar
  6. 6.
    O. Rauprich, M. Matsushita, C.J. Weijer, F. Siegert, S.E. Esipov, J.A. Shapiro, J. Bacteriol. 178, 6525 (1996).Google Scholar
  7. 7.
    A. Nakahara, Y. Shimada, J. Wakita, M. Matsushita, T. Matsuyama, J. Phys. Soc. Jpn. 65, 2700 (1996).ADSCrossRefGoogle Scholar
  8. 8.
    X.-L. Wu, A. Libchaber, Phys. Rev. Lett. 84, 3017 (2000).ADSCrossRefGoogle Scholar
  9. 9.
    L. Cisneros, C. Dombrowski, R.E. Goldstein, J.O. Kessler, Phys. Rev. E 73, 030901 (2006).ADSCrossRefGoogle Scholar
  10. 10.
    P.C. Martin, O. Parodi, P.S. Pershan, Phys. Rev. A 6, 2401 (1972).ADSCrossRefGoogle Scholar
  11. 11.
    D. Forster, Hydrodynamic Fluctuations, Broken Symmetry, and Correlation Functions (W.A. Benjamin, Reading, Mass., 1975).Google Scholar
  12. 12.
    P.G. de Gennes, The Physics of Liquid Crystals (Clarendon Press, Oxford, 1975).Google Scholar
  13. 13.
    H. Pleiner, H.R. Brand in Pattern Formation in Liquid Crystals, edited by A. Buka, L. Kramer (Springer, New York, 1996) p. 15.CrossRefGoogle Scholar
  14. 14.
    R. Graham, Phys. Rev. Lett. 33, 1431 (1974).ADSCrossRefGoogle Scholar
  15. 15.
    H.R. Brand, M. Dörfle, R. Graham, Ann. Phys. (New York) 119, 434 (1979).ADSCrossRefGoogle Scholar
  16. 16.
    D. Collin, G.K. Auernhammer, O. Gavat, P. Martinoty, H.R. Brand, Macromol. Rapid Commun. 24, 737 (2003).CrossRefGoogle Scholar
  17. 17.
    Z. Varga, J. Feher, G. Filipcsei, L. Zrinyi, Macromol. Symp. 200, 93 (2003).CrossRefGoogle Scholar
  18. 18.
    S. Bohlius, H.R. Brand, H. Pleiner, Phys. Rev. E 70, 061411 (2004).ADSCrossRefGoogle Scholar
  19. 19.
    K. Kruse, J.F. Joanny, F. Julicher, J. Prost, K. Sekimoto, Eur. Phys. J. E 16, 5 (2005).CrossRefGoogle Scholar
  20. 20.
    S. Muhuri, M. Rao, S. Ramaswamy, Europhys. Lett. 78, 48002 (2007).ADSCrossRefGoogle Scholar
  21. 21.
    D. Svenšek, H. Pleiner, H.R. Brand, in preparation (2011).Google Scholar
  22. 22.
    J.K. Parrish, L. Edelstein-Keshet, Science 284, 99 (1999).ADSCrossRefGoogle Scholar
  23. 23.
    J.H. Tien, S.A. Levin, D. Rubenstein, Evol. Ecol. Res. 6, 555 (2004).Google Scholar
  24. 24.
    M. Ballerini et al., Anim. Behav. 76, 201 (2008).CrossRefGoogle Scholar
  25. 25.
    M. Ballerini et al., Proc. Nat. Acad. Sci. U.S.A. 105, 1232 (2008).ADSCrossRefGoogle Scholar
  26. 26.
    S. Gueron, K. Levit-Gurevich, Biophys. J. 74, 1658 (1998).CrossRefGoogle Scholar
  27. 27.
    C. Brennen, H. Winet, Annu. Rev. Fluid Mech. 9, 339 (1977).ADSCrossRefGoogle Scholar
  28. 28.
    D. Bray, Cell Movements, 2nd edition (Garland, New York, 2001).Google Scholar
  29. 29.
    H. Pleiner, D. Svenšek, H.R. Brand, in preparation (2011).Google Scholar
  30. 30.
    H. Pleiner, H.R. Brand, Europhys. Lett. 9, 243 (1989).ADSCrossRefGoogle Scholar
  31. 31.
    H.R. Brand, H. Pleiner, F. Ziebert, Phys. Rev. E 74, 021713 (2006).ADSCrossRefGoogle Scholar
  32. 32.
    H.R. Brand, P.E. Cladis, H. Pleiner, Phys. Rev. E 79, 032701 (2009).ADSCrossRefGoogle Scholar
  33. 33.
    P.G. de Gennes, in Liquid Crystals of One- and Two-Dimensional Order, edited by W. Helfrich, G. Heppke (Springer, New York, 1980).Google Scholar
  34. 34.
    H.R. Brand, H. Pleiner, Eur. Phys. J. E 31, 37 (2010).CrossRefGoogle Scholar
  35. 35.
    H.R. Brand, H. Pleiner, Physica A 208, 359 (1994).ADSCrossRefGoogle Scholar
  36. 36.
    H. Pleiner, H.R. Brand, Europhys. Lett. 89, 26003 (2010).ADSCrossRefGoogle Scholar
  37. 37.
    H.R. Brand, H. Pleiner, Phys. Rev. A 37, 2736 (1988).ADSCrossRefGoogle Scholar
  38. 38.
    D. Svenšek, H. Pleiner, H.R. Brand, Phys. Rev. Lett. 96, 140601 (2006).ADSCrossRefGoogle Scholar
  39. 39.
    D. Svenšek, H. Pleiner, H.R. Brand, Phys. Rev. E 78, 021703 (2008).ADSCrossRefGoogle Scholar
  40. 40.
    Y. Tabe, H. Yokoyama, Nat. Mater. 2, 806 (2003).ADSCrossRefGoogle Scholar
  41. 41.
    Y. Tabe, invited talk IL8 at the European Conference on Liquid Crystals 2007, ECLC abstract CD-ROM, issued by Universidade de Lisboa, Faculdade de Sciencias et Technologia (2007) p. 58.Google Scholar
  42. 42.
    T.C. Lubensky, Mol. Cryst. Liq. Cryst. 23, 99 (1973).CrossRefGoogle Scholar
  43. 43.
    M. Liu, Phys. Rev. Lett. 43, 1740 (1979).ADSCrossRefGoogle Scholar
  44. 44.
    P.E. Cladis, Y. Couder, H.R. Brand, Phys. Rev. Lett. 55, 2945 (1985).ADSCrossRefGoogle Scholar
  45. 45.
    P.E. Cladis, P.L. Finn, H.R. Brand, Phys. Rev. Lett. 75, 1518 (1995).ADSCrossRefGoogle Scholar
  46. 46.
    O. Lehmann, Ann. Phys. (Leipzig) 2, 649 (1900).ADSGoogle Scholar

Copyright information

© EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  1. 1.Theoretische Physik IIIUniversität BayreuthBayreuthGermany
  2. 2.Max-Planck-Institute for Polymer ResearchMainzGermany
  3. 3.Department of Physics, Faculty of Mathematics and PhysicsUniversity of LjubljanaLjubljanaSlovenia

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