IDEAS home Printed from https://ideas.repec.org/a/eee/phsmap/v444y2016icp765-782.html
   My bibliography  Save this article

The influence of the finite velocity on spatial distribution of particles in the frame of Levy walk model

Author

Listed:
  • Saenko, Viacheslav V.

Abstract

Levy walk at the finite velocity is considered. To analyze the spatial and temporal characteristics of this process, the method of moments has been used. The asymptotic distributions of the moments (at t→∞) have been obtained for N dimensional case where the free path of particles demonstrates the power-law distribution pξ(x)=αx0αx−α−1, x→∞, 0<α<2. The three regimes of distribution have been distinguished: ballistic, diffusion and asymptotic. Introduction of the finite velocity requires considering of two problems: propagation with distribution at the finite mathematical expectation of the free path (1<α<2) and propagation with distribution at the infinite mathematical expectation of the free path of the particle (0<α<1). In the case 1<α<2, the asymptotic distribution is described by the Levy stable law and the effect of the finite velocity is reduced to a decrease of diffusivity. At 0<α<1, the situation is quite different. Here, the asymptotic distribution exhibits a U-or W-shape and is described as the ballistic regime of distribution. The obtained moments allow to reconstruct the distribution densities of particles in one-dimensional and three-dimensional cases.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:phsmap:v:444:y:2016:i:c:p:765-782
    DOI: 10.1016/j.physa.2015.10.046
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S037843711500905X
    Download Restriction: Full text for ScienceDirect subscribers only. Journal offers the option of making the article available online on Science direct for a fee of $3,000
    File URL: https://libkey.io/10.1016/j.physa.2015.10.046?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Metzler, Ralf & Glöckle, Walter G. & Nonnenmacher, Theo F., 1994. "Fractional model equation for anomalous diffusion," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 211(1), pages 13-24.
    2. 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.
    3. Klafter, J. & Zumofen, G. & Shlesinger, M.F., 1993. "Lévy walks in dynamical systems," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 200(1), pages 222-230.
    4. Liu, Jian & Bao, Jing-Dong, 2013. "Continuous time random walk with jump length correlated with waiting time," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(4), pages 612-617.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Pereira, A.P.P. & Fernandes, J.P. & Atman, A.P.F. & Acebal, J.L., 2018. "Parameter calibration between models and simulations: Connecting linear and non-linear descriptions of anomalous diffusion," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 509(C), pages 369-382.
    2. 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.
    3. 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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. 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.
    2. Fernando Alcántara-López & Carlos Fuentes & Rodolfo G. Camacho-Velázquez & Fernando Brambila-Paz & Carlos Chávez, 2022. "Spatial Fractional Darcy’s Law on the Diffusion Equation with a Fractional Time Derivative in Single-Porosity Naturally Fractured Reservoirs," Energies, MDPI, vol. 15(13), pages 1-11, July.
    3. 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.
    4. Uchaikin, V.V. & Sibatov, R.T., 2017. "Fractional derivatives on cosmic scales," Chaos, Solitons & Fractals, Elsevier, vol. 102(C), pages 197-209.
    5. Razminia, Kambiz & Razminia, Abolhassan & Baleanu, Dumitru, 2019. "Fractal-fractional modelling of partially penetrating wells," Chaos, Solitons & Fractals, Elsevier, vol. 119(C), pages 135-142.
    6. Scalas, Enrico, 2006. "The application of continuous-time random walks in finance and economics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 362(2), pages 225-239.
    7. Pavlos, G.P. & Karakatsanis, L.P. & Iliopoulos, A.C. & Pavlos, E.G. & Xenakis, M.N. & Clark, Peter & Duke, Jamie & Monos, D.S., 2015. "Measuring complexity, nonextensivity and chaos in the DNA sequence of the Major Histocompatibility Complex," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 438(C), pages 188-209.
    8. Claudia A. Pérez-Pinacho & Cristina Verde, 2022. "A Note on an Integral Transformation for the Equivalence between a Fractional and Integer Order Diffusion Model," Mathematics, MDPI, vol. 10(5), pages 1-13, February.
    9. Vyacheslav Svetukhin, 2021. "Nucleation Controlled by Non-Fickian Fractional Diffusion," Mathematics, MDPI, vol. 9(7), pages 1-11, March.
    10. Dmitry Zhukov & Konstantin Otradnov & Vladimir Kalinin, 2024. "Fractional-Differential Models of the Time Series Evolution of Socio-Dynamic Processes with Possible Self-Organization and Memory," Mathematics, MDPI, vol. 12(3), pages 1-19, February.
    11. Essex, Christopher & Schulzky, Christian & Franz, Astrid & Hoffmann, Karl Heinz, 2000. "Tsallis and Rényi entropies in fractional diffusion and entropy production," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 284(1), pages 299-308.
    12. Afzaal Mubashir Hayat & Muhammad Bilal Riaz & Muhammad Abbas & Moataz Alosaimi & Adil Jhangeer & Tahir Nazir, 2024. "Numerical Solution to the Time-Fractional Burgers–Huxley Equation Involving the Mittag-Leffler Function," Mathematics, MDPI, vol. 12(13), pages 1-22, July.
    13. Duan, Jun-Sheng & Wang, Zhong & Liu, Yu-Lu & Qiu, Xiang, 2013. "Eigenvalue problems for fractional ordinary differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 46(C), pages 46-53.
    14. Lenzi, M.K. & Lenzi, E.K. & Guilherme, L.M.S. & Evangelista, L.R. & Ribeiro, H.V., 2022. "Transient anomalous diffusion in heterogeneous media with stochastic resetting," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 588(C).
    15. Satin, Seema E. & Parvate, Abhay & Gangal, A.D., 2013. "Fokker–Planck equation on fractal curves," Chaos, Solitons & Fractals, Elsevier, vol. 52(C), pages 30-35.
    16. Qureshi, Sania & Bonyah, Ebenezer & Shaikh, Asif Ali, 2019. "Classical and contemporary fractional operators for modeling diarrhea transmission dynamics under real statistical data," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 535(C).
    17. Paradisi, Paolo & Cesari, Rita & Mainardi, Francesco & Tampieri, Francesco, 2001. "The fractional Fick's law for non-local transport processes," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 293(1), pages 130-142.
    18. Lenzi, E.K. & Mendes, R.S. & Gonçalves, G. & Lenzi, M.K. & da Silva, L.R., 2006. "Fractional diffusion equation and Green function approach: Exact solutions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 360(2), pages 215-226.
    19. Gorenflo, Rudolf & Mainardi, Francesco & Moretti, Daniele & Pagnini, Gianni & Paradisi, Paolo, 2002. "Fractional diffusion: probability distributions and random walk models," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 305(1), pages 106-112.
    20. Razminia, Kambiz & Razminia, Abolhassan & Torres, Delfim F.M., 2015. "Pressure responses of a vertically hydraulic fractured well in a reservoir with fractal structure," Applied Mathematics and Computation, Elsevier, vol. 257(C), pages 374-380.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:phsmap:v:444:y:2016:i:c:p:765-782. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.journals.elsevier.com/physica-a-statistical-mechpplications/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.