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Anomalous transport equations and their application to fractal walking

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  • Uchaikin, Vladimir V.

Abstract

Generalized transport equations covering both normal transport processes (such as neutron transport) and anomalous ones including Lévy walk and trapping are derived. The behaviours of the spatial distribution and mean-square displacement are investigated for fractal space–time processes. Exact solutions for the spatial distribution have been found in the one-dimensional case and numerical results are obtained for the case when the free paths and waiting times obey the one-sided stable distributions with the same characteristic exponent α=β=1/2. The principal result of the performed analysis is that the finite speed of a moving particle has a direct bearing on the spatial distribution. In the case of anomalous diffusion this effect does not vanish with time and has an influence on the asymptotic behaviour of the distribution form in contrast with normal diffusion when the speed affects only diffusivity but does not affect the form.

Suggested Citation

  • Uchaikin, Vladimir V., 1998. "Anomalous transport equations and their application to fractal walking," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 255(1), pages 65-92.
    • Handle: RePEc:eee:phsmap:v:255:y:1998:i:1:p:65-92
    DOI: 10.1016/S0378-4371(98)00047-8
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    Citations

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    Cited by:

    1. 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.
    2. Rukolaine, Sergey A., 2016. "Generalized linear Boltzmann equation, describing non-classical particle transport, and related asymptotic solutions for small mean free paths," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 450(C), pages 205-216.
    3. Uchaikin, V.V. & Sibatov, R.T., 2017. "Fractional derivatives on cosmic scales," Chaos, Solitons & Fractals, Elsevier, vol. 102(C), pages 197-209.
    4. Viacheslav V. Saenko & Vladislav N. Kovalnogov & Ruslan V. Fedorov & Dmitry A. Generalov & Ekaterina V. Tsvetova, 2022. "Numerical Method for Solving of the Anomalous Diffusion Equation Based on a Local Estimate of the Monte Carlo Method," Mathematics, MDPI, vol. 10(3), pages 1-19, February.
    5. Viacheslav V. Saenko & Vladislav N. Kovalnogov & Ruslan V. Fedorov & Yuri E. Chamchiyan, 2021. "Numerical Solution to Anomalous Diffusion Equations for Levy Walks," Mathematics, MDPI, vol. 9(24), pages 1-17, December.

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