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Diffusion and relaxation of the fractional order in fractal media in the classical and quantum cases

  • Elementary Particle Physics and Field Theory
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Russian Physics Journal Aims and scope

Two model examples of the application of fractional calculus are considered. The Riemann–Liouville fractional derivative with 0 < α ≤ 1 was used. The solution of a fractional equation, which describes anomalous relaxation and diffusion in an isotropic fractal space, has been obtained in the form of the product of a Fox function by a Mittag-Leffler function. The solution is simpler than that given in Ref. 6 and it generalizes the result reported in Ref. 7. For the quantum case, a solution of the generalized Neumann–Kolmogorov fractional quantum-statistical equation has been obtained for an incomplete statistical operator which describes the random walk of a quantum spin particle, retarded in traps over a fractal space. The solution contains contributions from quantum Mittag-Leffler (nonharmonic) fractional oscillations, anomalous relaxation, noise fractional oscillations, and exponential fractional diffusion oscillation damping.

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Authors and Affiliations

  1. Perm State Technical University, Perm, Russia

    V. S. Kirchanov

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  1. V. S. Kirchanov
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Correspondence to V. S. Kirchanov.

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Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 4, pp. 15–23, April, 2009.

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Kirchanov, V.S. Diffusion and relaxation of the fractional order in fractal media in the classical and quantum cases. Russ Phys J 52, 343–353 (2009). https://doi.org/10.1007/s11182-009-9245-0

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  • DOI: https://doi.org/10.1007/s11182-009-9245-0

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