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
Witten, T. A. & Sander, L. M. Phys. Rev. Lett. 47, 1400–1403 (1981).
Pietronero, L. & Tosatti, E. (eds) Fractals in Physics (North-Holland, Amsterdam, 1986).
Feder, J. Fractals (Plenum, New York, 1988).
Meakin, P. in Phase Transitions and Critical Phenomena (eds Domb, C. & Lebowitz, J.) Vol. 12, 355–489 (Academic, New York, 1988).
Vicsek, T. Fractal Growth Phenomena (World Scientific, Singapore, 1989).
Aharony, A. & Feder, J. eds Fractals in Physics (North-Holland, Amsterdam, 1990).
Aharony, A. & Feder, J. Physica D38, 1–398 (1989).
Mandelbrot, B. B. Fractal Geometry of Nature (Freeman, New York, 1982).
Kakutani, S. Proc. Imper. Acad. Sci. (Tokyo) 20, 706–714 (1944).
Niemeyer, L., Pietronero, L. & Wiesmann, H. J. Phys. Rev. Lett. 52, 1033–1036 (1984).
Pietronero, L., Erzan, A. & Evertsz, C. J. G. Phys. Rev. Lett. 61, 861–864 (1988).
Pietronero, L., Erzan, A. & Evertsz, C. S. G. Physica A151, 207–245 (1988).
Tsallis, C. (ed.) Statistical Physics (North-Holland, Amsterdam, 1990).
Tsallis, C. Physica A163, 1–428 (1990).
Peitgen, H. O. & Richter, P. H. The Beauty of Fractals (Springer-Verlag, New York, 1986).
Blumenfeld, R. & Aharony, A. Phys. Rev. Lett. 62, 2977–2980 (1989).
Carleson, L. & Jones, P. W. Duke math. J. (in the press).
Mandelbrot, B. B. J. Fluid Mech. 62, 331–358 (1974).
Mandelbrot, B. B. Physica A168, 95–111 (1990).
Lee, J. & Stanley, H. E. Phys. Rev. Lett. 61, 2945–2948 (1988).
Mandelbrot, B. B., Evertsz, C. J. G. & Hayakawa, Y. Phys. Rev. A42, 4528–4536 (1990).
Pietronero, L., Evertsz, C. J. G. & Siebesma, A. P. in Stochastic Processes in Physics and Engineering (ed. Albeverio, S. et al.) (Reidel, Dordrecht, 1988).
Sander, L. M. Sci. Am. 256(1), 82–88 (1987).
Family, F. & Vicsek, T. Computers in Physics, 4, 44–49 (1990).
Evertsz, C. J. G., Siebesma, A. P. & Oeinck, F. Laplacian Fractal Growth, video (Solid State Physics Laboratory, University of Groningen, 1988).
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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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DOI: https://doi.org/10.1038/348143a0
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