Advertisement

Braid theory and Zipf-Mandelbrot relation used in microparticle dynamics

  • K. de Lange KristiansenEmail author
  • G. Helgesen
  • A. T. Skjeltorp
Statistical and Nonlinear Physics

Abstract.

A study is presented of the dynamics of a few body system of microparticles by using rank-ordering statistics in order to gain insight in the magneto-rheological properties of ferrofluids. This dynamical system is made up of micrometer sized plastic spheres dispersed in a ferrofluid driven by external magnetic fields. The world lines of the microspheres are captured and the dynamical modes are described by mathematical braid theory. Rank-ordering statistics on these modes shows a wide power law region consistent with the Zipf-Mandelbrot relation. We have also performed numerical simulations of the experimental system which show results in agreement with the observations.

PACS.

05.45.-a Nonlinear dynamics and chaos 83.10.Pp Particle dynamics 02.10.Kn Knot theory 75.75.+a Magnetic properties of nanostructures 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. R.E. Rosensweig, Ferrohydrodynamics (Cambridge University Press, New York, 1985) Google Scholar
  2. Magnetic fluids and application handbook, edited by B. Berkovski, V. Bashtovoy (Begel House Inc., New York, 1996) Google Scholar
  3. K. Wollny, J. Läuger, S. Huck, Appl. Rheol. 12, 25 (2002) Google Scholar
  4. J.P. McTague, J. Chem. Phys. 51, 133 (1969) CrossRefGoogle Scholar
  5. J. Embs, H.W. Müller, C. Wagner, K. Knorr, M. Lücke, Phys. Rev. E 61, R2196 (2000) Google Scholar
  6. M.I. Shliomis, Soviet Phys. JETP 34, 1291 (1972). Google Scholar
  7. S. Odenbach, H. Störk, J. Magn. Magn. Mater. 183, 188 (1998) CrossRefADSGoogle Scholar
  8. F. Gazeau, C. Baravian, J.-C. Bacri, R. Perzynski, M.I. Shliomis, Phys. Rev. E 56 614 (1997) Google Scholar
  9. A. Zeuner, R. Richter, I. Rehberg, Phys. Rev. E 58 6287 (1998) Google Scholar
  10. A.T. Skjeltorp, Phys. Rev. Lett. 51, 2306 (1983) CrossRefADSGoogle Scholar
  11. P. Pieranski, S. Clausen, G. Helgesen, A.T. Skjeltorp, Phys. Rev. Lett. 77, 1620 (1996) CrossRefADSGoogle Scholar
  12. C. Moore, Phys. Rev. Lett. 70, 3675 (1993) zbMATHMathSciNetCrossRefADSGoogle Scholar
  13. M.A. Berger, Phys. Rev. Lett. 70, 705 (1993) zbMATHMathSciNetCrossRefADSGoogle Scholar
  14. P. L. Boyland, H. Aref, M.A. Stremler, J. Fluid Mech. 403, 277 (2000) zbMATHMathSciNetCrossRefADSGoogle Scholar
  15. An introduction to the Geometry and Topology of Fluid Flows, edited by R.L. Ricca (NATO ASI Series II, Kluwer, 2001), Vol. 47 Google Scholar
  16. E.J. Gumbel, Statistics of Extremes (Columbia University Press, New York, 1958) Google Scholar
  17. G.K. Zipf, Human Behavior and The Principle of Least Effort (Addison-Wesley Press, Massachusetts, 1949) Google Scholar
  18. D. Sornette, L. Knopoff, Y.Y. Kagan, C. Vanneste, J. Geophys. Res. 101, 13883 (1996) CrossRefADSGoogle Scholar
  19. R.N. Mantegna, S.V. Buldyrev, A.L. Goldberger, S. Havlin, C.-K. Peng, M. Simons, H.E. Stanley, Phys. Rev. Lett. 73, 3169 (1994) MathSciNetCrossRefADSGoogle Scholar
  20. B. Mandelbrot, Word 10, 1 (1954) Google Scholar
  21. B.B. Mandelbrot, The Fractal Geometry of Nature (W.H. Freeman, New York, 1982) Google Scholar
  22. C.E. Shannon, Bell Syst. Tech. 27, 379 (1948) MathSciNetGoogle Scholar
  23. J. Ugelstad, P.C. Mørk, K. Herder Kaggerud, T. Ellingsen, A. Berge, Adv. Colloid Int. Sci. 13, 101 (1980) CrossRefGoogle Scholar
  24. Type EMG 909 from Ferrofluidics GmbH, Hohes Gestade 14, 72622 Nürtingen, Germany, with susceptibility χ= 0.8, saturation magnetization Ms = 20 mT, viscosity η= 6 ×10-3 N s m-2, and density ρ= 1020 kg m-3. Google Scholar
  25. S. Clausen, G. Helgesen, A.T. Skjeltorp, Int. J. Bifurcation and Chaos 8, 1383 (1998) zbMATHCrossRefGoogle Scholar
  26. G. Helgesen, P. Pieranski, A.T. Skjeltorp, Phys. Rev. A 42, 7271 (1990) CrossRefADSGoogle Scholar
  27. C.C. Adams, The Knot Book (W.H. Freeman and Company, 1994) Google Scholar
  28. F.A. Garside, Quart. J. Math. 20, 235 (1967) MathSciNetGoogle Scholar
  29. E.A. Elrifai, H.R. Morton, Quart. J. Math. 45, 479 (1994) zbMATHMathSciNetGoogle Scholar
  30. B.B. Mandelbrot, in Proceedings of symposia in applied mathematics vol. XII, edited by R. Jakobson (New York: American Mathematical Society, 1961) Google Scholar
  31. K.D.L. Kristiansen, Ph.D. thesis, University of Oslo, 2005 Google Scholar
  32. K.D.L. Kristiansen, G. Helgesen, A.T. Skjeltorp, Physica A 335, 413 (2004) MathSciNetCrossRefADSGoogle Scholar
  33. E.R. Dufresne, T.M. Squires, M.P. Brenner, D.G. Grier, Phys. Rev. Lett. 85, 3317 (2000) CrossRefADSGoogle Scholar
  34. I. Kanter, D.A. Kessler, Phys. Rev. Lett. 74, 4559 (1995) CrossRefADSGoogle Scholar
  35. C.E. Shannon, Bell Syst. Tech. 30, 50 (1950) MathSciNetGoogle Scholar

Copyright information

© EDP Sciences/Società Italiana di Fisica/Springer-Verlag 2006

Authors and Affiliations

  • K. de Lange Kristiansen
    • 1
    • 2
    Email author
  • G. Helgesen
    • 2
  • A. T. Skjeltorp
    • 1
    • 2
  1. 1.Department of PhysicsUniversity of OsloOsloNorway
  2. 2.Physics DepartmentInstitute for Energy TechnologyKjellerNorway

Personalised recommendations