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Time–space fabric underlying anomalous diffusion

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  • Chen, W.

Abstract

This study unveils the time–space transforms underlying anomalous diffusion process. Based on this finding, we present the two hypotheses concerning the effect of fractal time–space fabric on physical behaviors and accordingly derive fractional quantum relationships between energy and frequency, momentum and wavenumber which further give rise to fractional Schrödinger equation. As an alternative modeling approach to the standard fractional derivatives, we introduce the concept of the Hausdorff derivative underlying the Hausdorff dimensions of metric spacetime. And in terms of the proposed hypotheses, the Hausdorff derivative is used to derive a linear anomalous transport–diffusion equation underlying anomalous diffusion process. Its Green’s function solution turn out to be a stretched Gaussian distribution and is compared with that from the Richardson’s turbulence diffusion equation.

Suggested Citation

  • Chen, W., 2006. "Time–space fabric underlying anomalous diffusion," Chaos, Solitons & Fractals, Elsevier, vol. 28(4), pages 923-929.
    • Handle: RePEc:eee:chsofr:v:28:y:2006:i:4:p:923-929
    DOI: 10.1016/j.chaos.2005.08.199
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    References listed on IDEAS

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    1. El Naschie, M.S., 2005. "A guide to the mathematics of E-infinity Cantorian spacetime theory," Chaos, Solitons & Fractals, Elsevier, vol. 25(5), pages 955-964.
    2. A. La Porta & Greg A. Voth & Alice M. Crawford & Jim Alexander & Eberhard Bodenschatz, 2001. "Fluid particle accelerations in fully developed turbulence," Nature, Nature, vol. 409(6823), pages 1017-1019, February.
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    Cited by:

    1. Balankin, Alexander S., 2020. "Fractional space approach to studies of physical phenomena on fractals and in confined low-dimensional systems," Chaos, Solitons & Fractals, Elsevier, vol. 132(C).
    2. Duarte Valério & Manuel D. Ortigueira & António M. Lopes, 2022. "How Many Fractional Derivatives Are There?," Mathematics, MDPI, vol. 10(5), pages 1-18, February.
    3. Atangana, Abdon & Qureshi, Sania, 2019. "Modeling attractors of chaotic dynamical systems with fractal–fractional operators," Chaos, Solitons & Fractals, Elsevier, vol. 123(C), pages 320-337.
    4. Chen, Wen & Liang, Yingjie, 2017. "New methodologies in fractional and fractal derivatives modeling," Chaos, Solitons & Fractals, Elsevier, vol. 102(C), pages 72-77.
    5. Sun, HongGuang & Li, Zhipeng & Zhang, Yong & Chen, Wen, 2017. "Fractional and fractal derivative models for transient anomalous diffusion: Model comparison," Chaos, Solitons & Fractals, Elsevier, vol. 102(C), pages 346-353.
    6. ARAZ, Seda İĞRET, 2020. "Numerical analysis of a new volterra integro-differential equation involving fractal-fractional operators," Chaos, Solitons & Fractals, Elsevier, vol. 130(C).
    7. Shojaeizadeh, T. & Mahmoudi, M. & Darehmiraki, M., 2021. "Optimal control problem of advection-diffusion-reaction equation of kind fractal-fractional applying shifted Jacobi polynomials," Chaos, Solitons & Fractals, Elsevier, vol. 143(C).
    8. Imran, M.A., 2020. "Application of fractal fractional derivative of power law kernel (FFP0Dxα,β) to MHD viscous fluid flow between two plates," Chaos, Solitons & Fractals, Elsevier, vol. 134(C).
    9. Tian, Peibo & Liang, Yingjie, 2022. "Material coordinate driven variable-order fractal derivative model of water anomalous adsorption in swelling soil," Chaos, Solitons & Fractals, Elsevier, vol. 164(C).

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