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Effect of empty configurations in the fixed scale transformation theory of fractal growth

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  • Vespignani, A.
  • Pietronero, L.

Abstract

The fixed scale transformation theory of fractal growth is based on two essential points. The first is the identification of the basic configurations that appear in the process of fine (or coarse) graining. The second is the calculation of the probability distribution for these configurations obtained by the fixed point of a transformation based on the dynamical evolution. Configurations that are completely empty cannot appear in the fine graining but they can be generated by the dynamical evolution. In the original approach these effects where neglected under the assumptions that they would give rise only to higher-order corrections. Here we show how to generalize the FST method to include empty configurations and how to derive from this the fractal dimension. The result is that the fractal dimension is lowered by about one per cent from the third order value obtained previously. This shows that the effect of the environment can be properly described in terms of boundary condition effects. In addition it shows the flexibility of the FST method that can be easily extended to study new effects in a systematic way.

Suggested Citation

  • Vespignani, A. & Pietronero, L., 1990. "Effect of empty configurations in the fixed scale transformation theory of fractal growth," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 168(2), pages 723-735.
    • Handle: RePEc:eee:phsmap:v:168:y:1990:i:2:p:723-735
    DOI: 10.1016/0378-4371(90)90027-P
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