A new theory for anomalous diffusion with a bimodal flux distribution

  • L. BevilacquaEmail author
  • A. C. N. R. Galeão
  • J. G. Simas
  • Ana Paula Rio Doce
Technical Paper


This paper deals with a new governing equation for anomalous diffusion encompassing a large spectrum of phenomena with particular attention on delaying processes. The analysis starts with a discrete approach and a law of evolution introducing a partial retention of the diffusing particles at each time step. The resulting differential equation assuming that the concentration function belongs to class C3 is a fourth-order differential equation. To fit this result into the framework of a new theory, a bi-modal flux distribution for the diffusion process associated with two energy states is proposed. The first energy state is related to the set of particles flowing according to the Fick’s law and the complementary set follows a new law. Two key parameters are introduced, namely, a parameter β indicating the fraction of the particles in the principal energy state and a parameter R controlling the effect of the secondary flux. Some examples are presented characterizing different types of phenomena as function of the relative values of β and R. The necessary conditions for the retention behavior are discussed for some particular cases.


Anomalous diffusion Multiflux Retention Fourth order PDE 



The results presented in this paper could not be achieved without the support of the National Research Council (CNPq) through the Research Fellowship Program, and the Research Project : 480865/2009-4. We are also indebted with the State of Rio de Janeiro Foundation, Research project: E-26/101.728/2010, for the scholarship granted to one of the authors of this paper.


  1. 1.
    Islam MA (2004) Einstein-Smulochowski diffusion equation: a discussion. Phys Scripta 70((2–3)):120–125CrossRefzbMATHGoogle Scholar
  2. 2.
    Philibert J (2006) One and half century of diffusion: Fick, Einstein, before and beyond, 4 6.1–6.19Google Scholar
  3. 3.
    Brandani S, Jama M, Ruthven D (2000) Diffusion, self-diffusion and counter-diffusion of benzene and p-xylene in silicalite. Microporous Mesopor Mat 35–36:283–300CrossRefGoogle Scholar
  4. 4.
    Crank J, Henry ME (1949) Diffusion in media with variable properties. Part I. The effect of a variable diffusion coefficient on the rates of absorption and desorption. Trans Faraday Soc 45:636–650CrossRefGoogle Scholar
  5. 5.
    Crank J, Henry ME (1949) Diffusion in media with variable properties. Part II. The effect of a variable diffusion coefficient on the concentration-distance relationship in the non-steady state. Trans Faraday Soc 45:1119–1130CrossRefGoogle Scholar
  6. 6.
    Hall LD (1953) An analytical method of calculating diffusion coefficients. J Chem Phys 21(1):87–89CrossRefGoogle Scholar
  7. 7.
    Küntz M, Lavallée P (2004) Anomalous diffusion is the rule in concentration-dependent diffusion processes. J Phys D Appl Phys 37(1):L5–L8CrossRefGoogle Scholar
  8. 8.
    Vanaparthy SH, Barthe C, Meiburg E (2006) Density-driven instabilities in capillary tubes: influence of a variable diffusion coefficient, Phys. Fluids 18 (n.4): 048101-1Google Scholar
  9. 9.
    Atsumi H (2002) Hydrogen bulk retention in graphite and kinetics of diffusion. J Nucl Mater 307–311:1466–1470CrossRefGoogle Scholar
  10. 10.
    Atsumi H, Tanabe T, Shikama T (2011) Hydrogen behavior in carbon and graphite before and after neutron irradiation—trapping, diffusion and the simulation of bulk retention. J Nucl Mater 417:633–636CrossRefGoogle Scholar
  11. 11.
    D`Angelo MV, Fontana E, Chertcoff R, Rosen M (2003) Retention phenomena in non-Newtonian fluids flow. Physics A 327:44–48CrossRefGoogle Scholar
  12. 12.
    Deleersnijder E, Beckers J-M, Delhez EJM (2006) The residence time of setting in the surface mixed layer. Environ Fluid Mech 6:25–42CrossRefGoogle Scholar
  13. 13.
    Huang J-C, Madey R (1982) Effect of liquid-phase diffusion resistance on retention time in gas-liquid chromatography. Anal Chem 54(2):326–328CrossRefGoogle Scholar
  14. 14.
    Kindler K, Khalili A, Stocker R (2010) Diffusion-limited retention of porous particles at density interfaces. Proc Natl Acad Sci USA 107:22163–22168CrossRefGoogle Scholar
  15. 15.
    Ferreira JA, Branco JR, Silva P (2010) Non-Fickian delay reaction–diffusion equations: theoretical and numerical study. Appl Numer Math 60(5):531–549MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Fort J, Méndez V (2002) Wavefronts in time-delayed reaction-diffusion systems. Theory and comparison to experiment. Rep Prog Phys 65:895–954CrossRefzbMATHGoogle Scholar
  17. 17.
    Jianhong W, Xingfu Z (2001) Traveling waves fronts of reaction-diffusion systems with delay. J Dyn Differ Equ 13(3):651–686CrossRefzbMATHGoogle Scholar
  18. 18.
    Sen S, Ghosh P, Riaz SS, Ray DS (2009) Time-delay-induced instabilities in reaction-diffusion system. Phys Rev E 80:046212CrossRefGoogle Scholar
  19. 19.
    Zanette TH (1999) Statistical-thermodynamical foundations of anomalous diffusion. Braz J Phys 29(1):108–124CrossRefGoogle Scholar
  20. 20.
    Cohen DS, Murray JM (1981) A generalized model for growth and dispersal in a population. J Math Biol 12:237–249MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Kurzynski M, Palacz K, Chełminiak P (1998) Time course of reactions controlled and gated by intramolecular dynamics of proteins: Predictions of the model of random walk on fractal lattices. Proc Natl Acad Sci USA 95((n. 20)):11685–11690 BiophysicsCrossRefGoogle Scholar
  22. 22.
    Mainardi F (1996) The fundamental solutions for fractional diffusive-wave equation. Appl Math Lett 9((n. 6)):23–28MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Scherer R, Kalla SL, Boyadjiev L, Al-Saqabi B (2008) Numerical treatment of fractional heat equations. Appl Numer Math 58(8):1212–1223MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Bevilacqua L, Galeão ACNR, Costa FP (2011) On the significance of higher order terms in diffusion processes. J Braz Soc Mech Sci 34(2):166–175CrossRefGoogle Scholar
  25. 25.
    Bevilacqua L, Galeão ACNR, Costa FP (2011) A new analytical formulation of retention effects on particle diffusion processes. An Acad Bras Cienc 83(4):1443–1464CrossRefGoogle Scholar
  26. 26.
    Osborne WA, Jackson LC (1914) Counter diffusion in aqueous solution. Biochem J 8(3):246–249Google Scholar
  27. 27.
    Schaefer KE (1974) Present status of underwater medicine. Review of some challenging problems. Cell Mol Life Sci 30(3):217–221MathSciNetCrossRefGoogle Scholar
  28. 28.
    Broadbridge P (2008) Entropy diagnostics for fourth order partial differential equations in conservation form. Entropy 10:365–379MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Simas (2012) Resolução Numérica para o Problema de Difusão com Retenção. MSc Thesis, LNCC/MCTGoogle Scholar

Copyright information

© The Brazilian Society of Mechanical Sciences and Engineering 2013

Authors and Affiliations

  • L. Bevilacqua
    • 1
    Email author
  • A. C. N. R. Galeão
    • 2
  • J. G. Simas
    • 2
  • Ana Paula Rio Doce
    • 1
  1. 1.Instituto Alberto Coimbra, COPPE, Ilha do FundãoUniversidade Federal do Rio de JaneiroRio de JaneiroBrazil
  2. 2.Laboratório Nacional de Computação CientíficaPetrópolisBrazil

Personalised recommendations