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Theory of Laplacian fractals: Diffusion limited aggregation and dielectric breakdown model

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  • Pietronero, L.
  • Erzan, A.
  • Evertsz, C.

Abstract

We describe a new theoretical approach that clarifies the origin of fractal structures in irreversible growth models based on the Laplace equation and a stochastic field. This new theory provides a systematic method for the calculation of the fractal dimension D and of the multifractal spectrum of the growth probability (ƒ(α)). A detailed application to the dielectric breakdown model and diffusion limited aggregation in two dimensions is presented. Our approach exploits the scale invariance of the Laplace equation that implies that the structure is self-similar both under growth and scale transformation. This allows one to introduce a Fixed Scale Transformation (instead of coarse graining as in the renormalization group theory) that defines a functional equation for the fixed point of the distribution of basic diagrams used in the coarse graining process. For the calculation of the matrix elements of this transformation one has to consider an infinite, but rapidly convergent, number of processes that occurs outside a considered diagram.

Suggested Citation

  • Pietronero, L. & Erzan, A. & Evertsz, C., 1988. "Theory of Laplacian fractals: Diffusion limited aggregation and dielectric breakdown model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 151(2), pages 207-245.
    • Handle: RePEc:eee:phsmap:v:151:y:1988:i:2:p:207-245
    DOI: 10.1016/0378-4371(88)90014-3
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    Cited by:

    1. Meneveau, Charles & Chhabra, Ashvin B., 1990. "Two-point statistics of multifractal measures," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 164(3), pages 564-574.
    2. Marsili, M. & Pietronero, L., 1991. "Fixed scale transformation approach to the multifractcal properties of the growth probabilities in the dielectric breakdown model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 175(1), pages 31-46.
    3. Vanderzande, Carlo, 1992. "Fractal dimensions of Potts clusters," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 185(1), pages 235-239.
    4. Meakin, Paul, 1992. "Simplified diffusion-limited aggregation models," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 187(1), pages 1-17.
    5. Tolman, Susan & Meakin, Paul, 1989. "Two, three and four-dimensional diffusion-limited aggregation models," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 158(3), pages 801-816.
    6. Sidoretti, S. & Vespignani, A., 1992. "Fixed scale transformation applied to cluster-cluster aggregation in two and three dimensions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 185(1), pages 202-210.
    7. Evertsz, Carl J.G. & Mandelbrot, Benoit B., 1992. "Self-similarity of harmonic measure on DLA," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 185(1), pages 77-86.
    8. Lee, Sung Jong & Halsey, Thomas C., 1990. "Some results on multifractal correlations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 164(3), pages 575-592.
    9. Marsili, M. & Pietronero, L., 1991. "Properties of the growth probability for the dielectric breakdown model in cylinder geometry," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 175(1), pages 9-30.
    10. Vergassola, M. & Vespignani, A., 1991. "Non-conservative character of the intersection of self-similar cascades," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 174(2), pages 425-437.
    11. Martinez-Saito, Mario, 2022. "Discrete scaling and criticality in a chain of adaptive excitable integrators," Chaos, Solitons & Fractals, Elsevier, vol. 163(C).

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