Article Outline
Glossary
Definition of the Subject
Introduction
Random Walks and Normal Diffusion
Anomalous Diffusion
Anomalous Diffusion on Fractal Structures
Percolation Clusters
Scaling of PDF and Diffusion Equationsl on Fractal Lattices
Further Directions
Bibliography
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Abbreviations
- Anomalous diffusion :
-
An essentially diffusive process in which the mean squared displacement grows, however, not as \({\langle \mathbf{R}^2 \rangle \propto t}\) like in normal diffusion, but as \({\langle \mathbf{R}^2 \rangle \propto t^\alpha}\) with \({\alpha \neq 0}\), either asymptotically faster than in normal diffusion (\({\alpha > 1}\), superdiffusion) or asymptotically slower (\({\alpha < 1}\), subdiffusion).
- Comb model :
-
A planar network consisting of a backbone (spine) and teeth (dangling ends). A popular simple model showing anomalous diffusion.
- Fractional diffusion equation:
-
A diffusion equation for a non-Markovian diffusion process, typically with a memory kernel corresponding to a fractional derivative. A mathematical instrument which adequately describes many processes of anomalous diffusion, for example the continuous‐time random walks (CTRW).
- Walk dimension:
-
The fractal dimension of the trajectory of a random walker on a network. The walk dimension is defined through the mean time T or the mean number of steps n which a walker needs to leave for the first time the ball of radius R around the starting point of the walk: \({T \propto R^{d_{\text{w}}}}\).
- Spectral dimension:
-
A property of a fractal structure which substitutes the Euclidean dimension in the expression for the probability to be at the origin at time t, \({P(\boldsymbol{0},t) \propto t^{-d_{\text{s}}/2}}\). Defines the behavior of the network Laplace operator.
- Alexander–Orbach conjecture:
-
A conjecture that the spectral dimension of the incipient infinite cluster in percolation is equal to 4/3 independently on the Euclidean dimension. The invariance of spectral dimension is only approximate and holds within \({2{\%}}\) accuracy even in \({d=2}\). The value of 3/4 is achieved in \({d > 6}\) and on trees.
- Compact visitation:
-
A property of a random walk to visit practically all sites within the domain of the size of the mean squared displacement. The visitation is compact if the walk dimension \({d_{\text{w}}}\) exceeds the fractal dimension of the substrate \({d_{\text{f}}}\).
Bibliography
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Additional Reading
The present article gave a brief overview of what is known about the diffusion on fractal networks, however this overview is far from covering all the facets of the problem. Thus, we only discussed unbiased diffusion (the effects of bias may be drastic due to e. g. stronger trapping in the dangling ends), and considered only the situations in which the waiting time at all nodes was the same (we did not discuss e. g. the continuous‐time random walks on fractal networks), as well as left out of attention many particular systems and applications. Several review articles can be recommended as a further reading, some of them already mentioned in the text. One of the best-known sources is [10] being a reprint of the text from the “sturm und drang” period of investigation of fractal geometries. A lot of useful information on random walks models in general an on walks on fractals is contained in the review by Haus and Kehr from approximately the same time. General discussion of the anomalous diffusion is contained in the work by Bouchaud and Georges. The classical review of the percolation theory is given in the book of Stauffer and Aharony. Some additional information on anomalous diffusion in percolation systems can be found in the review by Isichenko. A classical source on random walks in disordered systems is the book by Hughes.
Haus JW, Kehr K (1987) Diffusion in regular and disordered lattices. Phys Rep 150:263–406
Bouchaud JP, Georges A (1990) Anomalous diffusion in disordered media – statistical mechanisms, models and physical applications. Phys Rep 195:127–293
Stauffer D, Aharony A (2003) Introduction to Percolation Theory. Taylor & Fransis, London
Isichenko MB (1992) Percolation, statistical topography, and transport in random‐media. Rev Mod Phys 64:961–1043
Hughes BD (1995) Random Walks and random Environments. Oxford University Press, New York
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Sokolov, I.M. (2012). Anomalous Diffusion on Fractal Networks. In: Meyers, R. (eds) Mathematics of Complexity and Dynamical Systems. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1806-1_2
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