Abstract
The formation of branched, ramified, fractal structures in pattern formation limited by diffusion was first pointed out by Witten and Sander in 1981.1 In the intervening years, the study of such patterns has blossomed into a major area of non-linear physics. We have gradually learned, and are still learning, the correct concepts in which to couch quantitative discussion of the problem. One of the key such concepts is the multifractal structure of the growth probability distribution of a diffusion-limited pattern.2,3 In this contribution, I shall argue that this structure, as encoded in the f(α) function of the distribution, provides not merely a rich phenomenological description of the patterns, as has been widely realized. It also gives the foundation on which a more physical scaling picture of diffusion-limited growth may be built.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
T.A. Witten, Jr. and L.M. Sander, Phys. Rev. Lett. 47, 1400 (1981).
T.C. Halsey, P. Meakin, and I. Procaccia, Phys. Rev. Lett. 56, 854 (1986).
T.C. Halsey, M.H. Jensen, L.P. Kadanoff, I. Procaccia, and B. Shraiman, Phys. Rev. A 33, 1141 (1986).
P. Meakin, Phys. Rev. A 27, 1495 (1983).
L. Pietronero and H.J. Wiesmann, J. Stat. Phys. 36, 909 (1984).
L. Niemeyer, L. Pietronero, and H.J. Wiesmann, Phys. Rev. Lett. 52. 1033 (1984).
M. Eden, in Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, Vol. 4, J. Neyman, ed. (University of California Press, Berkeley, 1961) p. 223.
S.R. Forrest and T.A. Witten, Jr., J. Phys. A 12, L109 (1979).
D. Bensimon, L.P. Kadanoff, S. Liang, B. I. Shraiman, and C. Tang, Rev. Mod. Phys. 58, 977 (1986), and references therein.
R. Brady and R.C. Ball, Nature (London) 309, 225 (1984)
M. Matsushita, M. Sano, Y. Hayakawa, H. Honjo, and Y. Sawada, Phys. Rev. Lett. 53, 286 (1984).
P. Meakin, H.E. Stanley, A. Coniglio, and T.A. Witten, Jr., Phys. Rev. A 32, 2364 (1985).
U. Frisch and G. Parisi, in Turbulence and Predictability in Geophysical Fluid Dynamics and Climate Dynamics, Proc. of Int. School of Physics “Enrico Fermi” LXXXVIII, M. Ghil, R. Benzi, and G. Parisi, eds. (North-Holland, Amsterdam, 1985) p. 84.
C. Amitrano, A. Coniglio, and F. di Liberto, Phys. Rev. Lett. 57, 1016 (1986)
Y. Hayakawa, S. Sato, and M. Matsushita, Phys. Rev. A 36, 1963 (1987).
J. Lee and H.E. Stanley, Phys. Rev. Lett. 61, 2945 (1988).
L. Turkevich and H. Scher, Phys. Rev. Lett. 55, 1026 (1985); Phys. Rev. A 33, 786 (1986).
T.C. Halsey, Phys. Rev. A 38, 4749 (1988). Note that the generalization of the Turkevich-Scher law to arbitrary η, Equation (IV.8), is different from that originally proposed in Reference 15.
T.C. Halsey, Phys. Rev. Lett. 59, 2067 (1987).
C. Amitrano, unpublished.
See, e.g., M.H. Jensen, L.P. Kadanoff, and I. Procaccia, Phys. Rev. A 36, 1409 (1987).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1989 Springer Science+Business Media New York
About this chapter
Cite this chapter
Halsey, T.C. (1989). Multifractality, Scaling, and Diffusive Growth. In: Pietronero, L. (eds) Fractals’ Physical Origin and Properties. Ettore Majorana International Science Series. Springer, Boston, MA. https://doi.org/10.1007/978-1-4899-3499-4_9
Download citation
DOI: https://doi.org/10.1007/978-1-4899-3499-4_9
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4899-3501-4
Online ISBN: 978-1-4899-3499-4
eBook Packages: Springer Book Archive