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Fractional model equation for anomalous diffusion


  • Metzler, Ralf
  • Glöckle, Walter G.
  • Nonnenmacher, Theo F.


In recent years the phenomenon of anomalous diffusion has attracted more and more attention. One of the main impulses was initiated by de Gennes' idea of the “ant in the labyrinth”. Several authors presented asymptotic probability density functions for the location of a random walker on a fractal object. As this density function and the time dependence of its second moment are now well established, a modified diffusion equation providing the correct result is formulated. The parameters of this fractional partial differential equation are uniquely determined by the fractal Hausdorff dimension of the underlying object and the anomalous diffusion exponent. The presented equation reduces exactly to the ordinary isotropic diffusion equation by appropriate choice of the parameters. A closed form solution is given in terms of Fox's H-function. In the asymptotic case a “halved” diffusion equation can be established. Furthermore, the differences to equations considered previously are discussed.

Suggested Citation

  • Metzler, Ralf & Glöckle, Walter G. & Nonnenmacher, Theo F., 1994. "Fractional model equation for anomalous diffusion," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 211(1), pages 13-24.
  • Handle: RePEc:eee:phsmap:v:211:y:1994:i:1:p:13-24
    DOI: 10.1016/0378-4371(94)90064-7

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    Cited by:

    1. Gorenflo, Rudolf & Mainardi, Francesco & Moretti, Daniele & Pagnini, Gianni & Paradisi, Paolo, 2002. "Fractional diffusion: probability distributions and random walk models," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 305(1), pages 106-112.
    2. Saenko, Viacheslav V., 2016. "The influence of the finite velocity on spatial distribution of particles in the frame of Levy walk model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 444(C), pages 765-782.
    3. repec:eee:apmaco:v:257:y:2015:i:c:p:374-380 is not listed on IDEAS
    4. Paradisi, Paolo & Cesari, Rita & Mainardi, Francesco & Tampieri, Francesco, 2001. "The fractional Fick's law for non-local transport processes," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 293(1), pages 130-142.
    5. Scalas, Enrico, 2006. "The application of continuous-time random walks in finance and economics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 362(2), pages 225-239.
    6. Krasowska, Monika & Strzelewicz, Anna & Rybak, Aleksandra & Dudek, Gabriela & Cieśla, Michał, 2016. "Structure and transport properties of ethylcellulose membranes with different types and granulation of magnetic powder," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 452(C), pages 241-250.
    7. Essex, Christopher & Schulzky, Christian & Franz, Astrid & Hoffmann, Karl Heinz, 2000. "Tsallis and Rényi entropies in fractional diffusion and entropy production," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 284(1), pages 299-308.
    8. Pavlos, G.P. & Karakatsanis, L.P. & Iliopoulos, A.C. & Pavlos, E.G. & Xenakis, M.N. & Clark, Peter & Duke, Jamie & Monos, D.S., 2015. "Measuring complexity, nonextensivity and chaos in the DNA sequence of the Major Histocompatibility Complex," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 438(C), pages 188-209.
    9. repec:eee:apmaco:v:257:y:2015:i:c:p:467-486 is not listed on IDEAS

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