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

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  • 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
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    References listed on IDEAS

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    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. 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.
    3. 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.
    4. Zanette, Damián H., 1998. "Macroscopic current in fractional anomalous diffusion," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 252(1), pages 159-164.
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    Cited by:

    1. 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.
    2. 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.
    3. 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.
    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.

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