Skip to main content
SpringerLink
Log in
Book cover

Chaotic Flows pp 129–144Cite as

Fractal Objects and Scaling

  • Chapter
  • First Online:

  • 1043 Accesses

Part of the book series: Springer Series in Synergetics ((SSSYN,volume 10))

Abstract

Many physical systems do exhibit self-similarity, although, of course, within some finite range of scales. The list includes Brownian motion, turbulent flows, porous media, polymers, clusters, etc. The geometry of these systems, often based on random processes, is complicated. The concept of fractal dimension helps to express, model, and comprehend both the geometrical complexity and its physical consequence [157–164]. Furthermore, fractal concepts and scaling laws establish similarities between correlation effects and growth phenomena in a variety of equilibrium (such as percolation) and far-from-equilibrium (such as diffusion-limited aggregation and viscous fingering) processes. This connection is of heuristic significance, because presently there is no first-principle theory to describe, for example, turbulence, turbulent transport, or diffusion-limited aggregation, which are markedly far-from-equilibrium and nonlocal processes.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   54.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. E.W. Montroll and M. F. Shlesinger On the Wonderful World of Random Walks, in Studies in Statistical Mechanics, vol 11, (Elsevier Science Publishers, Amsterdam 1984), p. 1

    Google Scholar 

  2. M.F. Shiesinger, G.M. Zaslavsky, Levy Flights and Related Topics in Physics (Springer, Berlin, 1995)

    Book  Google Scholar 

  3. D. Sornette, Critical Phenomena in Natural Sciences (Springer, Berlin, 2006)

    MATH  Google Scholar 

  4. B.B. Mandelbrot, The Fractal Geometry of Nature (Freemen, San Francisco, 1982)

    MATH  Google Scholar 

  5. A. Bunde, S. Havlin (eds.), Fractals and Disordered Systems (Springer, Berlin, 1995)

    Google Scholar 

  6. A. Bunde, S. Havlin (eds.), Fractals in Science (Springer, Berlin, 1996)

    Google Scholar 

  7. J. Feder, Fractals, Department of Physics University of Oslo, Norway (Plenum, New York, 1988)

    Google Scholar 

  8. J.-F. Gouyet, Physics and Fractal Structure (Springer, Berlin, 1996)

    Google Scholar 

  9. H.M. Hastings, G. Sugihara, Fractals (Oxford University Press, Oxford, 1993)

    Google Scholar 

  10. L.S. Liebovitch, Fractals and Chaos Simplified for the Life Sciences (Oxford University Press, Oxford, 1998)

    MATH  Google Scholar 

  11. M. Schroeder, Fractals, Chaos, Power Laws. Minutes from an Infinite Paradise (W.H. Freeman and Company, New York, 2001)

    MATH  Google Scholar 

  12. P.G. De Gennes, Introduction to Polymer Dynamics (Cambridge, 1990)

    Google Scholar 

  13. P.G. De Gennes, Scaling Concepts in Polymer Physics, (Cornell University Press, 1979)

    Google Scholar 

  14. M. Kleman, O.D. Lavrentovich, Soft Matter Physics (Springer, Berlin, 2003)

    Google Scholar 

  15. V.E. Kravtsov, I.V. Lerner, V.I. Udson, J. Exp. Theor. Phys. 91(2(8)), 569 (1986)

    Google Scholar 

  16. P.B. Rhines, J. Fluid Mech. 69(Part 3), 417–443 (1975)

    Article  ADS  MATH  Google Scholar 

  17. M.B. Isichenko, J. Kalda, J. Nonlinear Sci. 1, 255 (1991)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  18. M.B. Isichenko, J. Kalda, J. Nonlinear Sci. 1, 375 (1991)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  19. M. Mitsugu, S. Ouchi, K. Honda, J. Phys. Sos. Japan 60, 2109 (1991)

    Article  ADS  Google Scholar 

  20. S. Isogami, M. Matsushita, J. Phys. Sos. Japan 61, 1445 (1992)

    Article  ADS  Google Scholar 

  21. A. Bunde, J.F. Gouet, J. Phys. A: Math. Gen. 18, L285 (1985)

    Article  ADS  Google Scholar 

  22. H. Saleur, B. Duplantier, Phys. Rev. Lett. 58, 2325 (1987)

    Article  MathSciNet  ADS  Google Scholar 

  23. B. Sapoval, B. Rosso, J. Gouyet, J. Phys. Lett. 46, 149 (1985)

    Article  Google Scholar 

  24. J. Kondev, C.L. Henley, Phys. Rev. Lett. 74, 4580 (1995)

    Article  ADS  Google Scholar 

  25. J. Kondev, C.L. Henley, D.G. Salinas, Phys. Rev. E 61, 104 (2000)

    Article  ADS  Google Scholar 

  26. J. Kondev, Phys. Rev. Lett. 86, 5890 (2001)

    Article  ADS  Google Scholar 

  27. J. Kalda, Phys. Rev. E 64, 020101(R) (2001)

    Article  ADS  Google Scholar 

  28. J. Kalda, Phys. Rev. Lett. 90, 118501–1 (2003)

    Article  ADS  Google Scholar 

  29. O.G. Bakunin, Chaos Solitons & Fractals 23, 1703 (2005)

    ADS  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Plasma Physics Department, “Kurchatov Institute”, Kurchatova Square 1, 123182, Moscow, Russia

    Oleg G. Bakunin

Authors
  1. Oleg G. Bakunin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Oleg G. Bakunin .

Further Reading

Further Reading

1.1 Chaos and Fractals

  • J.B. Bassingthwaighte, L.S. Liebovitch, B.J. West, Fractal Physiology (Oxford University Press, Oxford, 1994)

  • H.V. Beijeren (ed.), Fundamental Problems in Statistical Mechanics (North Holland, Amsterdam, 1990)

  • A. Bunde, S. Havlin (eds.), Fractals and Disordered Systems (Springer, Berlin, 1995)

  • A. Bunde, S. Havlin (eds.), Fractals in Science (Springer, Berlin, 1996)

  • J. Feder, Fractals (Plenum, New York, 1988). Department of Physics University of Oslo, Norway

  • J.-F. Gouyet, Physics and Fractal Structure (Springer, Berlin, 1996)

  • H.M. Hastings, G. Sugihara, Fractals (Oxford University Press, Oxford, 1993)

  • L.S. Liebovitch, Fractals and Chaos Simplified for the Life Sciences (Oxford University Press, Oxford, 1998)

  • B.B. Mandelbrot, The Fractal Geometry of Nature (Freemen, San Francisco, CA, 1982)

  • R.A. Meyers, Encyclopedia of Complexity and Systems Science (Springer, Berlin, 2009)

  • G. Nicolis, Foundations of Complex Systems (World Scientific, Singapore, 2007)

  • M. Schroeder, Fractals, Chaos, Power Laws. Minutes from an Infinite Paradise (W.H Freeman, New York, 2001)

  • S.H. Strogatz, Nonlinear Dynamics and Chaos (Perseus, Cambridge, 2000)

1.2 Diffusion and Fractals

  • R. Badii et al. (eds.), Complexity Hierarchical Structures and Scaling in Physics (Cambridge University Press, Cambridge, 1997)

  • D. Ben-Avraham, S. Havlin, Diffusion and Reactions in Fractals and Disordered Systems (Cambridge University Press, Cambridge, 1996)

  • G.P. Bouchaud, A. Georges, Phys. Rep. 195, 132–292 (1990)

  • A. Bovier, Statistical Mechanics of Disordered Systems. A Mathematical Perspective (Cambridge University Press, Cambridge, 2006)

  • O. Coussy, Mechanics and Physics of Porous Solids (Wiley, New York, 2010)

  • P.G. De Gennes, Introduction to Polymer Dynamics (Cambridge University Press, Cambridge, 1990)

  • P.G. De Gennes, Scaling Concepts in Polymer Physics (Cornell University Press, Ithaca, NY, 1979)

  • J.W. Haus, K.W. Kehr, Phys. Rep. 150, 263 (1987)

  • M. Kleman, O.D. Lavrentovich, Soft Matter Physics (Springer, Berlin, 2003)

  • R. Metzler, J. Klafter, Phys. Rep. 339, 1 (2000)

  • M.E. Raikh, I.M. Ruzin, in Mesoscopic Phenomena in Solids, ed. by B.L. Altshuler, P.A. Lee, R.A. Webb (North-Holland, Amsterdam, 1991)

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Bakunin, O.G. (2011). Fractal Objects and Scaling. In: Chaotic Flows. Springer Series in Synergetics, vol 10. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-20350-3_8

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-20350-3_8

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-20349-7

  • Online ISBN: 978-3-642-20350-3

  • eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)

Publish with us

Policies and ethics

Search

Navigation