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The potential distribution around growing fractal clusters

Abstract

THE process of diffusion-limited aggregation (DLA) is a common means by which clusters grow from their constituent particles, as exemplified by the formation of soot and the aggregation of colloids in solution. DLA growth is a probabilistic process which results in the formation of fractal (self-similar) clusters. It is controlled by the harmonic measure (the gradient of the electrostatic potential) around the cluster's boundary. Here we show that interactive computer graphics can provide new insight into this potential distribution. We find that points of highest and lowest growth probability can lie unexpectedly close together, and that the lowest growth probabilities may lie very far from the initial seed. Our illustrations also reveal the prevalence of 'fjords' in which the pattern of equipotential lines involves a 'mainstream' with almost parallel walls. We suggest that an understanding of the low values of the harmonic measure will provide new understanding of the growth mechanism itself.

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References

  1. Witten, T. A. & Sander, L. M. Phys. Rev. Lett. 47, 1400–1403 (1981).

    Article  ADS  CAS  Google Scholar 

  2. Pietronero, L. & Tosatti, E. (eds) Fractals in Physics (North-Holland, Amsterdam, 1986).

  3. Feder, J. Fractals (Plenum, New York, 1988).

    Book  Google Scholar 

  4. Meakin, P. in Phase Transitions and Critical Phenomena (eds Domb, C. & Lebowitz, J.) Vol. 12, 355–489 (Academic, New York, 1988).

    Google Scholar 

  5. Vicsek, T. Fractal Growth Phenomena (World Scientific, Singapore, 1989).

    Book  Google Scholar 

  6. Aharony, A. & Feder, J. eds Fractals in Physics (North-Holland, Amsterdam, 1990).

  7. Aharony, A. & Feder, J. Physica D38, 1–398 (1989).

    MathSciNet  Google Scholar 

  8. Mandelbrot, B. B. Fractal Geometry of Nature (Freeman, New York, 1982).

    MATH  Google Scholar 

  9. Kakutani, S. Proc. Imper. Acad. Sci. (Tokyo) 20, 706–714 (1944).

    Article  Google Scholar 

  10. Niemeyer, L., Pietronero, L. & Wiesmann, H. J. Phys. Rev. Lett. 52, 1033–1036 (1984).

    Article  ADS  MathSciNet  Google Scholar 

  11. Pietronero, L., Erzan, A. & Evertsz, C. J. G. Phys. Rev. Lett. 61, 861–864 (1988).

    Article  ADS  CAS  Google Scholar 

  12. Pietronero, L., Erzan, A. & Evertsz, C. S. G. Physica A151, 207–245 (1988).

    Article  MathSciNet  Google Scholar 

  13. Tsallis, C. (ed.) Statistical Physics (North-Holland, Amsterdam, 1990).

  14. Tsallis, C. Physica A163, 1–428 (1990).

    MathSciNet  Google Scholar 

  15. Peitgen, H. O. & Richter, P. H. The Beauty of Fractals (Springer-Verlag, New York, 1986).

    Book  Google Scholar 

  16. Blumenfeld, R. & Aharony, A. Phys. Rev. Lett. 62, 2977–2980 (1989).

    Article  ADS  CAS  Google Scholar 

  17. Carleson, L. & Jones, P. W. Duke math. J. (in the press).

  18. Mandelbrot, B. B. J. Fluid Mech. 62, 331–358 (1974).

    Article  ADS  Google Scholar 

  19. Mandelbrot, B. B. Physica A168, 95–111 (1990).

    Article  MathSciNet  Google Scholar 

  20. Lee, J. & Stanley, H. E. Phys. Rev. Lett. 61, 2945–2948 (1988).

    Article  ADS  CAS  Google Scholar 

  21. Mandelbrot, B. B., Evertsz, C. J. G. & Hayakawa, Y. Phys. Rev. A42, 4528–4536 (1990).

    Article  ADS  CAS  Google Scholar 

  22. Pietronero, L., Evertsz, C. J. G. & Siebesma, A. P. in Stochastic Processes in Physics and Engineering (ed. Albeverio, S. et al.) (Reidel, Dordrecht, 1988).

    MATH  Google Scholar 

  23. Sander, L. M. Sci. Am. 256(1), 82–88 (1987).

    Article  Google Scholar 

  24. Family, F. & Vicsek, T. Computers in Physics, 4, 44–49 (1990).

    Article  ADS  Google Scholar 

  25. Evertsz, C. J. G., Siebesma, A. P. & Oeinck, F. Laplacian Fractal Growth, video (Solid State Physics Laboratory, University of Groningen, 1988).

    Google Scholar 

Download references

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Mandelbrot, B., Evertsz, C. The potential distribution around growing fractal clusters. Nature 348, 143–145 (1990). https://doi.org/10.1038/348143a0

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