IDEAS home Printed from https://ideas.repec.org/a/eee/phsmap/v293y2001i1p130-142.html
   My bibliography  Save this article

The fractional Fick's law for non-local transport processes

Author

Listed:
  • Paradisi, Paolo
  • Cesari, Rita
  • Mainardi, Francesco
  • Tampieri, Francesco

Abstract

Fick's law is extensively adopted as a model for standard diffusion processes. However, requiring separation of scales, it is not suitable for describing non-local transport processes. We discuss a generalized non-local Fick's law derived from the space-fractional diffusion equation generating the Lévy–Feller statistics. This means that the fundamental solutions can be interpreted as Lévy stable probability densities (in the Feller parameterization) with index α (1<α⩽2) and skewness θ (|θ|⩽2−α). We explore the possibility of defining an equivalent local diffusivity by displaying a few numerical case studies concerning the relevant quantities (flux and gradient). It turns out that the presence of asymmetry (θ≠0) plays a fundamental role: it produces shift of the maximum location of the probability density function and gives raise to phenomena of counter-gradient transport.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:phsmap:v:293:y:2001:i:1:p:130-142
    DOI: 10.1016/S0378-4371(00)00491-X
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S037843710000491X
    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/S0378-4371(00)00491-X?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. Gorenflo, Rudolf & Fabritiis, Gianni De & Mainardi, Francesco, 1999. "Discrete random walk models for symmetric Lévy–Feller diffusion processes," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 269(1), pages 79-89.
    3. Zanette, Damián H., 1998. "Macroscopic current in fractional anomalous diffusion," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 252(1), pages 159-164.
    4. West, Bruce J. & Seshadri, V., 1982. "Linear systems with Lévy fluctuations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 113(1), pages 203-216.
    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. Hernandez-Martinez, Eliseo & Valdés-Parada, Francisco & Alvarez-Ramirez, Jose & Puebla, Hector & Morales-Zarate, Epifanio, 2016. "A Green’s function approach for the numerical solution of a class of fractional reaction–diffusion equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 121(C), pages 133-145.
    2. Qi, Haitao & Jiang, Xiaoyun, 2011. "Solutions of the space-time fractional Cattaneo diffusion equation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 390(11), pages 1876-1883.
    3. 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.
    4. Néel, Marie-Christine & Abdennadher, Ali & Solofoniaina, Joelson, 2008. "A continuous variant for Grünwald–Letnikov fractional derivatives," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(12), pages 2750-2760.

    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. 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.
    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. 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.
    4. West, Bruce J., 1996. "Lévy statistics of water wave forces," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 230(3), pages 359-363.
    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. 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.
    7. 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.
    8. 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.
    9. 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.
    10. 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).
    11. 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.
    12. 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.
    13. Roscani, Sabrina D. & Bollati, Julieta & Tarzia, Domingo A., 2018. "A new mathematical formulation for a phase change problem with a memory flux," Chaos, Solitons & Fractals, Elsevier, vol. 116(C), pages 340-347.
    14. Silvia Vitali & Iva Budimir & Claudio Runfola & Gastone Castellani, 2019. "The Role of the Central Limit Theorem in the Heterogeneous Ensemble of Brownian Particles Approach," Mathematics, MDPI, vol. 7(12), pages 1-9, November.
    15. Hongler, Max-Olivier & Filliger, Roger & Blanchard, Philippe, 2006. "Soluble models for dynamics driven by a super-diffusive noise," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 370(2), pages 301-315.
    16. 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.
    17. Vyacheslav Svetukhin, 2021. "Nucleation Controlled by Non-Fickian Fractional Diffusion," Mathematics, MDPI, vol. 9(7), pages 1-11, March.
    18. 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.
    19. 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.
    20. Janicki, Aleksander & Weron, Aleksander, 1995. "Computer simulation of attractors in stochastic models with α-stable noise," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 39(1), pages 9-19.

    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:293:y:2001:i:1:p:130-142. 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.