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Fractal Structures in Condensed Matter Physics

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Encyclopedia of Complexity and Systems Science

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Definition of the Subject

The idea of fractals is based on self‐similarity, which is a symmetry property of a systemcharacterized by invariance under an isotropic scale‐transformation on certain length scales. The term scale‐invariance has the implication that objects look the same on different scales of observations. While the underlyingconcept of fractals is quite simple, the concept is used for an extremely broad range of topics, providing a simple description of highly complexstructures found in nature. The term fractal was first introduced byBenoit B. Mandelbrot in 1975, who gave a definition on fractals in a simple manner “A fractal isa shape made of parts similar to the whole in some way”. Thus far, the concept of fractals has been extensively used to understandthe behaviors of many complex systems or has been applied from physics, chemistry, and biology for applied sciences and technological purposes. Examplesof fractal structures in condensed matter physics are numerous such...

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Abbreviations

Anomalous diffusion:

It is well known that the mean‐square displacement \( { \langle r^{2}(t) \rangle } \) of a diffusing particle on a uniform system is proportional to the time t such as \( { \langle r^{2}(t) \rangle \sim t } \). This is called normal diffusion. Particles on fractal networks diffuse more slowly compared with the case of normal diffusion. This slow diffusion called anomalous diffusion follows the relation given by \( { \langle r^{2}(t) \rangle \sim t^{a} } \), where the condition \( { 0<a<1 } \) always holds.

Brownian motion:

Einstein published the important paper in 1905 opening the way to investigate the movement of small particles suspended in a stationary liquid, the so‐called Brownian motion, which stimulated J. Perrin in 1909 to pursue his experimental work confirming the atomic nature of matter. The trail of a random walker provides an instructive example for understanding the meaning of random fractal structures.

Fractons:

Fractons, excitations on fractal elastic‐networks, were named by S. Alexander and R. Orbach in 1982. Fractons manifest not only static properties of fractal structures but also their dynamic properties. These modes show unique characteristics such as strongly localized nature with the localization length of the order of wavelength.

Spectral density of states:

The spectral density of states of ordinary elastic networks are expressed by the Debye spectral density of states given by \( { D(\omega)\sim\omega^{d-1} } \), where d is the Euclidean dimensionality. The spectral density of states of fractal networks is given by \( { D(\omega)\sim\omega^{d_\mathrm{s}-1} } \), where \( { d_\mathrm{s} } \) is called the spectral or fracton dimension of the system.

Spectral dimension:

This exponent characterizes the spectral density of states for vibrational modes excited on fractal networks. The spectral dimension constitutes the dynamic exponent of fractal networks together with the conductivity exponent and the exponent of anomalous diffusion.

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Authors and Affiliations

  1. Toyota Physical and Chemical Research Institute, Nagakute, Japan

    Tsuneyoshi Nakayama

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Editors and Affiliations

  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Nakayama, T. (2009). Fractal Structures in Condensed Matter Physics. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_229

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