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Representation of random walk in fractal space-time

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  • Kanno, Ryutaro

Abstract

To analyze the anomalous diffusion on a fractal structure with fractal in the time axis, we propose a statistical representation given by a path integral method in arbitrary fractal space-time. Using the method, we can understand easily several properties of the non-Gaussian-type behavior, and a differential equation for the path integral is derived. Finally, to check the validity of this theory, analytical results in this paper are applied to the random walk on the two-dimensional Sierpinski carpet, which agree precisely with numerical results by Monte Carlo simulations in the paper of Fujiwara and Yonezawa [Phys. Rev. E 51 (1995) 2277].

Suggested Citation

  • Kanno, Ryutaro, 1998. "Representation of random walk in fractal space-time," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 248(1), pages 165-175.
    • Handle: RePEc:eee:phsmap:v:248:y:1998:i:1:p:165-175
    DOI: 10.1016/S0378-4371(97)00422-6
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    References listed on IDEAS

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    1. West, Bruce J. & Shlesinger, Michael, 1984. "Random walk model of impact phenomena," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 127(3), pages 490-508.
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    Cited by:

    1. Kritika, & Agarwal, Ritu & Purohit, Sunil Dutt, 2020. "Mathematical model for anomalous subdiffusion using comformable operator," Chaos, Solitons & Fractals, Elsevier, vol. 140(C).
    2. Saifullah, Sayed & Ali, Amir & Franc Doungmo Goufo, Emile, 2021. "Investigation of complex behaviour of fractal fractional chaotic attractor with mittag-leffler Kernel," Chaos, Solitons & Fractals, Elsevier, vol. 152(C).
    3. Atangana, Abdon & Khan, Muhammad Altaf, 2019. "Validity of fractal derivative to capturing chaotic attractors," Chaos, Solitons & Fractals, Elsevier, vol. 126(C), pages 50-59.
    4. Chen, Wen & Hei, Xindong & Sun, Hongguang & Hu, Dongliang, 2018. "Stretched exponential stability of nonlinear Hausdorff dynamical systems," Chaos, Solitons & Fractals, Elsevier, vol. 109(C), pages 259-264.
    5. Atangana, Abdon, 2017. "Fractal-fractional differentiation and integration: Connecting fractal calculus and fractional calculus to predict complex system," Chaos, Solitons & Fractals, Elsevier, vol. 102(C), pages 396-406.
    6. 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.
    7. Rubayyi T. Alqahtani & Shabir Ahmad & Ali Akgül, 2022. "On Numerical Analysis of Bio-Ethanol Production Model with the Effect of Recycling and Death Rates under Fractal Fractional Operators with Three Different Kernels," Mathematics, MDPI, vol. 10(7), pages 1-23, March.
    8. Gómez-Aguilar, J.F., 2020. "Chaos and multiple attractors in a fractal–fractional Shinriki’s oscillator model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 539(C).
    9. 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).

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