Scaling Theories for Anomalous Dynamics on Fractals: Fractons
Many systems in nature exhibit different geometrical structures at different length scales. Although they are homogeneous (on the average) for large length scales, they exhibit self-similarity on short length scales. In the latter case such systems may be modeled by fractal structures.1 On a fractal structure, all the physical properties behave as powers of the relevant length scale. This behavior crosses over to a homogeneous one, similar to that of usual bulk matter, at a crossover length, called ξ. Assuming that ξ is the only relevant length scale in the problem, the dependence on any other length L should arise via the ratio L/ξ. A typical intrinsic quantity χ will have the “fractal” behavior χ~Lx for L«ξ, and will crossover to a size independent behavior, χ~ξx, for L>>ξ.
KeywordsFractal Structure Fractal Dimensionality Crossover Frequency Excitational Mode Large Length Scale
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