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Fractional Anomalous Diffusion

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An Introduction to Anomalous Diffusion and Relaxation

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Abstract

In this chapter, we discuss the general problem of fractional diffusion equation in connection with the anomalous behavior. We consider first the fundamental solution for the space-time fractional diffusion equation involving the Caputo operator in the time derivatives and the Riesz–Feller operator in the space derivative. The solution of the Cauchy problem can be expressed in terms of a Mellin–Barnes representation for the Green’s function. Subsequently, we discuss the more general case of the space-time diffusion equation involving composite fractional time derivative together with the Riesz–Feller space fractional derivative. Finally, we investigate diffusive phenomena governed by fractional diffusion equation in the presence of a spatial dependent diffusion coefficient and external forces. These problems are chosen here to illustrate the broadness of the application of statistical mechanics tools and the fractional formalism discussed in the previous chapters.

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References

  1. F. Mainardi, Y. Luchko, G. Pagnini, The fundamental solution of the space-time fractional diffusion equation. Fract. Calc. Appl. Anal. 4, 153–192 (2001)

    MathSciNet  MATH  Google Scholar 

  2. J.F. Kelly, M.M. Meerschaert, Space-time duality and high-order fractional diffusion Phys. Rev. E 99, 022122 (2019)

    Google Scholar 

  3. W. Wang, E. Barkai, Fractional advection-diffusion-asymmetry equation. Phys. Rev. Lett. 125, 240606 (2020)

    Article  MathSciNet  Google Scholar 

  4. T. Kosztołowicz, A. Dutkiewicz, Subdiffusion equation with Caputo fractional derivative with respect to another function. Phys. Rev. E 104, 014118 (2021)

    Google Scholar 

  5. R.M. de Moraes, L.C.S.M. Ozelim, A.L.B. Cavalcante, Generalized skewed model for spatial-fractional advective-dispersive phenomena. Sustainability 14, 4024 (2022)

    Article  Google Scholar 

  6. R.L. Magin, E.K. Lenzi, Slices of the anomalous phase cube depict regions of sub- and super-diffusion in the fractional diffusion equation. Mathematics 9, 1481 (2021)

    Article  Google Scholar 

  7. F. Mainardi, G. Pagnini, R.K. Saxena, Fox H functions in fractional diffusion. J. Comput. Appl. Math. 178, 321–331 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  8. E. Scalas, R. Gorenflo, F. Mainardi, M. Raberto, Revisiting the derivation of the fractional diffusion equation. Fractals 11, 281–289 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  9. R. Garra, E. Orsingher, F. Polito, Fractional diffusions with time-varying coefficients. J. Math. Phys. 56, 093301 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  10. M. Bologna, A. Svenkeson, B.J. West, P. Grigolini, Diffusion in heterogeneous media: an iterative scheme for finding approximate solutions to fractional differential equations with time-dependent coefficients. J. Comput. Phys. 293, 297–311 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  11. M. Bologna, B.J. West, P. Grigolini, Renewal and memory origin of anomalous diffusion: a discussion of their joint action. Phys. Rev. E 88, 062106 (2013)

    Article  Google Scholar 

  12. K.S. Fa, E.K. Lenzi, Time-fractional diffusion equation with time-dependent diffusion coefficient. Phys. Rev. E 72, 011107 (2005)

    Article  Google Scholar 

  13. A.A. Kilbas, M. Saigo, Fractional integrals and derivatives of Mittag-Leffler type functions. Dokl. Akad. Nauk Belarusi 39, 22–26 (1995)

    MathSciNet  Google Scholar 

  14. A.A. Kilbas, M. Saigo, On solution of integral equation of Abel-Volterra type. Differ. Integr. Equ. 8, 993–1011 (1995)

    MathSciNet  MATH  Google Scholar 

  15. A.A. Kilbas, M. Saigo, Solution of Abel integral equations of the second kind and differential equations of fractional order. Dokl. Akad. Nauk Belarusi 39, 29–34 (1995)

    MathSciNet  Google Scholar 

  16. E. Capelas de Oliveira, F. Mainardi, J. Vaz Jr., Fractional models of anomalous relaxation based on the Kilbas and Saigo function. Meccanica 49, 2049–2060 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  17. F. Mainardi, G. Pagnini, R. Gorenflo, Some aspects of fractional diffusion equations of single and distributed order. Appl. Math. Comput. 187, 295–305 (2007)

    MathSciNet  MATH  Google Scholar 

  18. C. Cattaneo, Sulla conduzione del calore. Atti Semin. Mat. Fis. Univ. Modena 3, 83–101 (1948)

    MathSciNet  MATH  Google Scholar 

  19. A. Sapora, M. Codegone, G. Barbero, L.R. Evangelista, Adsorption-desorption phenomena and diffusion of neutral particles in the hyperbolic regime. J. Phys. A Math. Theor. 47, 015002 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  20. F. Mainardi, G. Pagnini, The role of the Fox-Wright functions in fractional sub-diffusion of distributed order. J. Comput. Appl. Math. 207, 245–257 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  21. Z. Tomovski, T. Sandev, R. Metzler, J. Dubbeldam, Generalized space-time fractional diffusion equation with composite fractional time derivative. Phys. A 391, 2527–2542 (2012)

    Article  MathSciNet  Google Scholar 

  22. A.M. Mathai, R.K. Saxena, H.J. Haubold, The H-Function: Theory and Applications (Springer, Heidelberg, 2009)

    MATH  Google Scholar 

  23. A. Liemert, A. Kienle, Fundamental solution of the tempered fractional diffusion equation. J. Math. Phys. 56, 113504 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  24. F. Sabzikar, M.M. Meerschaert, J. Chen, Tempered fractional calculus. J. Comput. Phys. 293, 14–28 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  25. B. Baeumer, M.M. Meerschaert, Tempered stable Lévy motion and transient superdiffusion. J. Comput. Appl. Math. 233, 2438–2448 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  26. E.K. Lenzi, L.R. da Silva, T. Sandev, R.S. Zola, Solutions for a fractional diffusion equation in heterogeneous media. J. Stat. Mech. Theory Exp. 3, 033205 (2019)

    Google Scholar 

  27. A.A. Tateishi, H.V. Ribeiro, E.K. Lenzi, The role of fractional time-derivative operators on anomalous diffusion. Front. Phys. 5, 52 (2017)

    Article  Google Scholar 

  28. L.R. Evangelista, E.K. Lenzi, Fractional Diffusion Equations and Anomalous Diffusion (Cambridge University Press, Cambridge, 2018)

    Book  MATH  Google Scholar 

  29. R. Metzler, J. Klafter, The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339, 1–77 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  30. J. Klafter, I.M. Sokolov, First Steps in Random Walks: From Tools to Applications (Oxford University Press, Oxford, 2011)

    Book  MATH  Google Scholar 

  31. K.M. Saadab, J.F. Gómez-Aguilarc, Analysis of reaction diffusion system via a new fractional derivative with non-singular kernel. Phys. A 509, 703 (2018)

    Article  MathSciNet  Google Scholar 

  32. F. Gao, Y.X. Jun, S. Mohyud-Din, On linear viscoelasticity within general fractional derivatives without singular kernel. Therm. Sci. 21, 197 (2017)

    Article  Google Scholar 

  33. K. Saad, Comparing the Caputo, Caputo-Fabrizio and Atangana-Baleanu derivative with fractional order: fractional cubic isothermal auto-catalytic chemical system. Eur. Phys. J. Plus 133, 041102 (2018)

    Article  Google Scholar 

  34. J. Hristov, Derivation of the fractional Dodson equation and beyond: transient diffusion with a non-singular memory and exponentially fading-out diffusivity. Progr. Fract. Differ. Appl. 3, 255–270 (2017)

    Article  Google Scholar 

  35. T. Sandev, R. Metzler, A. Chechkin, From continuous time random walks to the generalized diffusion equation. Fract. Calc. Appl. Anal. 21, 10–28 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  36. A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations (Elsevier, Amsterdam, 2006)

    MATH  Google Scholar 

  37. E.K. Lenzi, L.R. Evangelista, M.K. Lenzi, H.V. Ribeiro, E.C. de Oliveira, Solutions for a non-Markovian diffusion equation. Phys. Lett. A 374, 4193–4198 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  38. X. Jiang, M. Xu, The time fractional heat conduction equation in the general orthogonal curvilinear coordinate and the cylindrical coordinate systems. Phys. A 389, 3368–3374 (2010)

    Article  MathSciNet  Google Scholar 

  39. M.J. Saxton, K. Jacobson, Single-particle tracking: applications to membrane dynamics. Annu. Rev. Biophys. Biomol. Struct. 26, 373–399 (1997)

    Article  Google Scholar 

  40. D. Wirtz, Particle-tracking microrheology of living cells: principles and applications. Annu. Rev. Biophys. 38, 301–326 (2009)

    Article  Google Scholar 

  41. N. Gal, D. Lechtman-Goldstein, D. Weihs, Particle tracking in living cells: a review of the mean square displacement method and beyond. Rheol. Acta 52, 425–443 (2013)

    Article  Google Scholar 

  42. N. Hozé, D. Holcman, Statistical methods for large ensembles of super-resolution stochastic single particle trajectories in cell biology. Annu. Rev. Stat. Appl. 4, 189–223 (2017)

    Article  Google Scholar 

  43. F. Höfling, T. Franosch, Anomalous transport in the crowded world of biological cells. Rep. Prog. Phys. 76, 046602 (2013)

    Article  MathSciNet  Google Scholar 

  44. C. Manzo, M.F. Garcia-Parajo, A review of progress in single particle tracking: from methods to biophysical insights. Rep. Prog. Phys. 78, 124601 (2015)

    Article  Google Scholar 

  45. H. Shen, L.J. Tauzin, R. Baiyasi, W. Wang, N. Moringo, B. Shuang, C.F. Landes, Single particle tracking: from theory to biophysical applications. Chem. Rev. 117, 7331–7376 (2017)

    Article  Google Scholar 

  46. E. Zagato, K. Forier, T. Martens, K. Neyts, J. Demeester, S. De Smedt, K. Remaut, K. Braeckmans, Single-particle tracking for studying nanomaterial dynamics: applications and fundamentals in drug delivery. Nanomedicine 9, 913–927 (2014)

    Article  Google Scholar 

  47. J. Kärger, D.M. Ruthven, D.N. Theodorou, Diffusion in Nanoporous Materials (Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim, 2012)

    Book  Google Scholar 

  48. J. Kärger, D. Ruthven, Diffusion in nanoporous materials: fundamental principles, insights and challenges. New J. Chem. 40, 4027–4048 (2016)

    Article  Google Scholar 

  49. M. Caputo, M. Fabrizio, A new definition of fractional derivative without singular kernel. Progr. Fract. Differ. Appl. 1, 73–85 (2015)

    Google Scholar 

  50. J. Hristov, Transient heat diffusion with a non-singular fading memory: from the Cattaneo constitutive equation with Jeffrey’s kernel to the Caputo-Fabrizio time-fractional derivative. Therm. Sci. 20, 757–762 (2016)

    Article  Google Scholar 

  51. J.F. Gómez-Aguilar, A. Atangana, Fractional Hunter-Saxton equation involving partial operators with bi-order in Riemann-Liouville and Liouville-Caputo sense. Eur. Phys. J. Plus 132, 100 (2017)

    Article  Google Scholar 

  52. J.F. Gómez-Aguilar, Space-time fractional diffusion equation using a derivative with nonsingular and regular kernel. Phys. A 465, 562–572 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  53. A. Atangana, D. Baleanu, New fractional derivative with non-local and non-singular kernel. Therm. Sci. 20, 763–769 (2016)

    Article  Google Scholar 

  54. T. Abdeljawada, D. Baleanu, Integration by parts and its applications of a new nonlocal fractional derivative with Mittag-Leffler nonsingular kernel. J. Nonlinear Sci. Appl. 10, 1098–1107 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  55. Y. Liang, S. Wang, W. Chen, Z. Zhou, R.L. Magin, A survey of models of ultraslow diffusion in heterogeneous materials. Appl. Mech. Rev. 71, 040802 (2019)

    Article  Google Scholar 

  56. Y. Liang, W. Chen, A non-local structural derivative model for characterization of ultraslow diffusion in dense colloids. Commun. Nonlinear Sci. 56, 131–137 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  57. I. Podlubny, Fractional Differential Equations (Academic Press, San Diego, 1999)

    MATH  Google Scholar 

  58. J. Hristov, Derivatives with non-singular kernels from the Caputo-Fabrizio definition and beyond: appraising analysis with emphasis on diffusion models, in Frontiers in Fractional Calculus, ed. by S. Bhalekar (Bentham Science Publishers, Sharjah, 2017), pp. 235–295

    Google Scholar 

  59. V.E. Tarasov, No violation of the Leibniz rule. No fractional derivative. Commun. Nonlinear Sci. Numer. Simul. 18, 2945–2948 (2013)

    Google Scholar 

  60. V.E. Tarasov, Non chain rule for fractional derivatives. Commun. Nonlinear Sci. Numer. Simul. 62, 157–163 (2018)

    Article  MathSciNet  Google Scholar 

  61. V.E. Tarasov, Caputo-Fabrizio operator in terms of integer derivatives: memory or distributed lag? Comput. Appl. Math. 38, 113 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  62. A. Fernandez, M.A. Özarslan, D. Baleanu, On fractional calculus with general analytic kernels. Appl. Math. Comput. 354, 248–265 (2019)

    MathSciNet  MATH  Google Scholar 

  63. V. Méndez, D. Campos, Characterization of stationary states in random walks with stochastic resetting. Phys. Rev. E 93, 022106 (2016)

    Article  Google Scholar 

  64. V.P. Shkilev, Continuous-time random walk under time-dependent resetting. Phys. Rev. E 96, 012126 (2017)

    Article  Google Scholar 

  65. J. Mo, A. Simha, M.G. Raizen, Broadband boundary effects on Brownian motion. Phys. Rev. E 92, 062106 (2015)

    Article  Google Scholar 

  66. H. Berry, H. Chaté, Anomalous diffusion due to hindering by mobile obstacles undergoing Brownian motion or Ornstein-Uhlenbeck process. Phys. Rev. E 89, 022708 (2014)

    Article  Google Scholar 

  67. E.F. Koslover, M.D. de la Rosa, A.J. Spakowitz, Crowding and hopping in a protein’s diffusive transport on DNA. J. Phys. A Math. Theor. 50, 074005 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  68. M.R. Evans, S.N. Majumdar, Diffusion with stochastic resetting. Phys. Rev. Lett. 106, 160601 (2011)

    Article  Google Scholar 

  69. G.T. Skalski, J.F. Gilliam, Modeling diffusive spread in a heterogeneous population: a movement study with stream fish. Ecology 81, 1685–1700 (2000)

    Article  Google Scholar 

  70. A. Upadhyaya, J.-P. Rieu, J.A. Glazier, Y. Sawada, Anomalous diffusion and non-Gaussian velocity distribution of Hydra cells in cellular aggregates. Phys. A 293, 549–558 (2001)

    Article  Google Scholar 

  71. S. Mieruszynski, M.A. Digman, E. Gratton, M.R. Jones, Characterization of exogenous DNA mobility in live cells through fluctuation correlation spectroscopy. Sci. Rep. 5, 13848 (2015)

    Article  Google Scholar 

  72. H.V. Ribeiro, A.A. Tateishi, L.G.A. Alves, R.S. Zola, E.K. Lenzi, Investigating the interplay between mechanisms of anomalous diffusion via fractional Brownian walks on a comb-like structure. New J. Phys. 16, 093050 (2014)

    Article  Google Scholar 

  73. L.G. Alves, D.B. Scariot, R.R. Guimaraes, C.V. Nakamura, R.S. Mendes, H.V. Ribeiro, Transient superdiffusion and long-range correlations in the motility patterns of trypanosomatid flagellate protozoa. PLoS One 11, e0152092 (2016)

    Article  Google Scholar 

  74. A.A. Tateishi, E.K. Lenzi, L.R. da Silva, H.V. Ribeiro, S. Picoli Jr., R.S. Mendes, Different diffusive regimes, generalized Langevin and diffusion equations. Phys. Rev. E 85, 011147 (2012)

    Google Scholar 

  75. E. Barkai, R.J. Silbey, Fractional Kramers equation. J. Phys. Chem. B 104, 3866–3874 (2000)

    Article  Google Scholar 

  76. A.V. Chechkin, R. Gorenflo, I.M. Sokolov, Retarding subdiffusion and accelerating superdiffusion governed by distributed-order fractional diffusion equations. Phys. Rev. E 66, 046129 (2002)

    Article  Google Scholar 

  77. E.K. Lenzi, R.S. Mendes, C. Tsallis, Crossover in diffusion equation: anomalous and normal behaviors. Phys. Rev. E 67, 031104 (2003)

    Article  Google Scholar 

  78. X. Bian, C. Kimb, G.E. Karniadakis, 111 years of Brownian motion. Soft Matter 12, 6331 (2016)

    Article  Google Scholar 

  79. R. Huang, I. Chavez, K.M. Taute, B. Lukić, S. Jeney, M.G. Raizen, E.-L. Florin, Direct observation of the full transition from ballistic to diffusive Brownian motion in a liquid. Nat. Phys. 7, 576–580 (2011)

    Article  Google Scholar 

  80. M. Caputo, Mean fractional-order-derivatives differential equations and filters. Ann. Univ. Ferrara Sez. 41, 73–84 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  81. M. Caputo, Distributed order differential equations modelling dielectric induction and diffusion. Fract. Calc. Appl. Anal. 4, 421–442 (2001)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Departamento de Física, Universidade Estadual de Maringá, Maringá, Paraná, Brazil

    Luiz Roberto Evangelista

  2. Departamento de Física, Universidade Estadual de Ponta Grossa, Ponta Grossa, Paraná, Brazil

    Ervin Kaminski Lenzi

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  1. Luiz Roberto Evangelista
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  2. Ervin Kaminski Lenzi
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Correspondence to Luiz Roberto Evangelista .

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Evangelista, L.R., Lenzi, E.K. (2023). Fractional Anomalous Diffusion. In: An Introduction to Anomalous Diffusion and Relaxation. PoliTO Springer Series. Springer, Cham. https://doi.org/10.1007/978-3-031-18150-4_5

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