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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. 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.
    3. 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.
    4. 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.
    5. 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).
    6. 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).
    7. Atangana, Abdon & Khan, Muhammad Altaf, 2019. "Validity of fractal derivative to capturing chaotic attractors," Chaos, Solitons & Fractals, Elsevier, vol. 126(C), pages 50-59.
    8. 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.
    9. 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).

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