Skip to main content
SpringerLink
Log in

potential theory and analytic properties of self-similar fractal and multifractal distributions

  • Articles
  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

By the use of recursion relations and analytic techniques we deduce general analytic results pertaining to the electrostatic potential, moments, and Fourier transform of exactly self-similar fractal and multifractal charge distributions. Three specific examples are given: the binomial distribution on the middle-third Cantor set, which is a multifractal distribution, the uniform distribution on the Menger sponge, which illustrates the added complication of higher dimensionality, and the uniform distribution on the von Koch snowflake, which illustrates the effect of rotations in the defining transformations.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. P. Carruthers,Astrophys. J. 380:24 (1991).

    Google Scholar 

  2. J. E. Avron and B. Simon,Phys. Rev. Lett. 46:1166 (1981).

    Google Scholar 

  3. C. J. G. Evertz and B. B. Mandelbrot,J. Phys. A 25:1781 (1992).

    Google Scholar 

  4. B. Sapoval, T. Gobron, and A. Margolina,Phys. Rev. Lett. 67:2974 (1991).

    Google Scholar 

  5. D. Berger, S. Chamaly, M. Perreau, D. Mercier, P. Monceau, and J.-C. S. Levy,J. Phys. I 1:1433 (1991).

    Google Scholar 

  6. M. V. Berry,J. Phys. A 12:781 (1979).

    Google Scholar 

  7. C. Allain and M. Cloitre,Phys. Rev. A 36:5751 (1987).

    Google Scholar 

  8. C. Allain and M. Cloitre,Physica A 157:352 (1989).

    Google Scholar 

  9. P. D'Antonio and J. Konnert,J. Audio Eng. Soc. 40:117 (1992).

    Google Scholar 

  10. K. Falconer,Fractal Geometry and Its Applications (Wiley, Chichester, 1990).

    Google Scholar 

  11. N. S. Landkof,Foundations of Modern Potential Theory (Springer-Verlag, Berlin, 1972).

    Google Scholar 

  12. D. Bessis, J. S. Geronimo, and P. Moussa,J. Stat. Phys. 34:75 (1984).

    Google Scholar 

  13. C. P. Dettmann and N. E. Frankel,J. Phys. A 26:1009 (1993).

    Google Scholar 

  14. F. Hille and J. D. Tamarkin,Am. Math. Monthly 36:255 (1929).

    Google Scholar 

  15. F. Oberhettinger,Tables of Mellin Transforms (Springer-Verlag, Berlin, 1974).

    Google Scholar 

  16. B. B. Mandelbrot,The Fractal Geometry of Nature (Freeman, New York, 1977).

    Google Scholar 

  17. C. P. Dettmann and N. E. Frankel,Fractals Vol. 1, No. 2, (1993), in press.

Download references

Author information

Authors and Affiliations

  1. School of Physics, University of Melbourne, 3052, Parkville, Victoria, Australia

    C. P. Dettmann & N. E. Frankel

Authors
  1. C. P. Dettmann
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. N. E. Frankel
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Dettmann, C.P., Frankel, N.E. potential theory and analytic properties of self-similar fractal and multifractal distributions. J Stat Phys 72, 241–275 (1993). https://doi.org/10.1007/BF01048049

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01048049

Key words

Search

Navigation