Generalized Cattaneo–Maxwell diffusion equation with fractional derivatives. Dispersion relations

The new non-Markovian diffusion equations of ions in spatially heterogeneous environment with fractal structure and generalized Cattaneo--Maxwell diffusion equation with taking into account the space-time nonlocality are obtained.  Dispersion relations for the Cattaneo--Maxwell-type diffusion equation with taking into account the space-time nonlocality in fractional derivatives are found.  The frequency spectrum, phase and group velocities are calculated.  It is shown that it has a wave behavior with discontinuities, which are also manifested in behavior of the phase velocity.

generalized diffusion equation
nonequilibrium statistical operator
Renyi statistics
multifractal time
spatial fractality
nonlocality of space-time
  1. Oldham K. B., Spanier J.  The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order.  Dover Books on Mathematics, Dover Publications (2006).
  2. Samko S. G., Kilbas A. A., Marichev O. I.  Fractional Integrals and Derivatives: Theory and Applications.  Gordon and Breach Science Publishers (1993).
  3. Podlubny I., Kenneth V. T. E.  Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications.  Mathematics in Science and Engineering 198, Academic Press (1998).
  4. Mandelbrot B. B.  The fractal geometry of nature.  W. H. Freeman and Company (1982).
  5. Uchaikin V. V.  Fractional Derivatives Method.  Artishock-Press, Uljanovsk (2008), (in Russian).
  6. Sahimi M.  Non-linear and non-local transport processes in heterogeneous media: from long-range correlated percolation to fracture and materials breakdown.  Physics Reports. 306 (4--6), 213--395 (1998).
  7. Korošak D., Cvikl B., Kramer J., Jecl R., Prapotnik A.  Fractional calculus applied to the analysis of spectral electrical conductivity of clay–water system.  Journal of Contaminant Hydrology. 92 (1–2), 1--9 (2007).
  8. Metzler R., Klafter J.  The random walk's guide to anomalous diffusion: a fractional dynamics approach.  Physics Reports. 339 (1), 1--77 (2000).
  9. Hilfer R.  Fractional Time Evolution, chapter II, pp. 87--130.  World Scientific, Singapore, New Jersey, London, Hong Kong (2000).
  10. Bisquert J., Garcia-Belmonte G., Fabregat-Santiago F., Ferriols N. S., Bogdanoff P., Pereira E. C.  Doubling Exponent Models for the Analysis of Porous Film Electrodes by Impedance.  Relaxation of TiO$_2$ Nanoporous in Aqueous Solution.  The Journal of Physical Chemistry. 104 (10), 2287--2298 (2000).
  11. Bisquert J., Compte A.  Theory of the electrochemical impedance of anomalous diffusion.  Journal of Electroanalytical Chemistry. 499 (1), 112--120 (2001).
  12. Kosztołowicz T., Lewandowska K. D.  Hyperbolic subdiffusive impedance.  Journal of Physics A: Mathematical and Theoretical. 42 (5), 055004 (2009).
  13. Pyanylo Y. D., Prytula M. G., Prytula N. M., Lopuh N. B.  Models of mass transfer in gas transmission systems.  Mathematical Modeling and Computing. 1 (1), 84--96 (2014).
  14. Zhokh A., Trypolskyi A., Strizhak P.  Relationship between the anomalous diffusion and the fractal dimension of the environment.  Chemical Physics. 503, 71--76 (2018).
  15. Zhokh A. A., Strizhak P. E.  Effect of zeolite ZSM-5 content on the methanol transport in the ZSM-5/alumina catalysts for methanol-to-olefin reaction.  Chemical Engineering Research and Design. 127, 35--44 (2017).
  16. Zhokh A., Strizhak P.  Non-Fickian diffusion of methanol in mesoporous media: Geometrical restrictions or adsorption-induced?  The Journal of Chemical Physics. 146 (12), 124704 (2017).
  17. Scher H., Montroll E. W.  Anomalous transit-time dispersion in amorphous solids.  Phys. Rev. B. 12 (6), 2455--2477 (1975).
  18. Berkowitz B., Scher H.  Theory of anomalous chemical transport in random fracture networks.  Phys. Rev. E. 57 (5), 5858--5869 (1998).
  19. Bouchaud J. P., Georges A.  Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications.  Physics Reports. 195 (4), 127--293 (1990).
  20. Nigmatullin R. R.  To the Theoretical Explanation of the "Universal Response''.  Physica Status Solidi (B). 123 (2), 739--745 (1984).
  21. Nigmatullin R. R.  On the Theory of Relaxation for Systems with "Remnant'' Memory.  Physica Status Solidi (B). 124 (1), 389--393 (1984).
  22. Nigmatullin R. R.  The realization of the generalized transfer equation in a medium with fractal geometry.  Physica Status Solidi (B). 133 (1), 425--430 (1986).
  23. Nigmatullin R. R.  Fractional integral and its physical interpretation.  Theoretical and Mathematical Physics. 90 (3), 242--251 (1992).
  24. Nigmatullin R. R., Ryabov Y. E.  Cole--Davidson dielectric relaxation as a self-similar relaxation process.  Physics of the Solid State.39 (1), 87--90 (1997).
  25. Nigmatullin R. R.  Dielectric relaxation phenomenon based on the fractional kinetics: theory and its experimental confirmation.  Physica Scripta. T136, 014001 (2009).
  26. Khamzin A. A., Nigmatullin R. R., Popov I. I.  Microscopic model of a non-Debye dielectric relaxation: The Cole--Cole law and its generalization.  Theoretical and Mathematical Physics. 173 (2), 1604--1619 (2012).
  27. Popov I. I., Nigmatullin R. R., Koroleva E. Y., Nabereznov A. A.  The generalized {Jonscher's} relationship for conductivity and its confirmation for porous structures.  Journal of Non-Crystalline Solids. 358 (1), 1--7 (2012).
  28. Grygorchak I. I., Kostrobij P. P., Stasjuk I. V., Tokarchuk M. V., Velychko O. V., Ivaschyshyn F. O., Markovych B. M.  Fizichni procesy ta ih mikroskopichni modeli v periodychnyh neorganichno/organichnih klatratah.  Rastr-7, Lviv (2015), (in Ukrainian).
  29. Kostrobij P. P., Grygorchak I. I., Ivaschyshyn F. O., Markovych B. M., Viznovych O. V., Tokarchuk M. V.  Mathematical modeling of subdiffusion impedance in multilayer nanostructures.  Mathematical Modeling and Computing. 2 (2), 154--159 (2015).
  30. Kostrobij P., Grygorchak I., Ivashchyshyn F., Markovych B., Viznovych O., Tokarchuk M.  Generalized Electrodiffusion Equation with Fractality of Space–Time: Experiment and Theory.  The Journal of Physical Chemistry A. 122 (16), 4099--4110 (2018).
  31. Balescu R.  Anomalous transport in turbulent plasmas and continuous time random walks.  Phys. Rev. E. 51 (5), 4807--4822 (1995).
  32. Tribeche M., Shukla P. K.  Charging of a dust particle in a plasma with a non extensive electron distribution function.  Physics of Plasmas. 18 (10), 103702 (2011).
  33. Gong J., Du J.  Dust charging processes in the nonequilibrium dusty plasma with nonextensive power-law distribution.  Physics of Plasmas. 19 (2), 023704 (2012).
  34. Carreras B. A., Lynch V. E., Zaslavsky G. M.  Anomalous diffusion and exit time distribution of particle tracers in plasma turbulence model.  Physics of Plasmas. 8 (12), 5096--5103 (2001).
  35. Tarasov V. E.  Electromagnetic field of fractal distribution of charged particles.  Physics of Plasmas. 12 (8), 082106 (2005).
  36. Tarasov V. E.  Magnetohydrodynamics of fractal media.  Physics of Plasmas. 13 (5), 052107 (2006).
  37. Monin A. S.  Uravnenija turbulentnoj difuzii.  DAN SSSR, ser. geofiz. 2, 256--259 (1955), (in Russian).
  38. Klimontovich J. L.  Vvedenie v fiziku otkrytyh sistem.  Moskva, Janus (2002), (in Russian).
  39. Zaslavsky G. M.  Chaos, fractional kinetics, and anomalous transport.  Physics Reports. 371 (6), 461--580 (2002).
  40. Tarasov V. E.  Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media.  Nonlinear Physical Science, Springer Berlin Heidelberg (2010).
  41. Zaslavsk G. M.  Fractional kinetic equation for Hamiltonian chaos.  Physica D: Nonlinear Phenomena. 76 (1), 110--122 (1994).
  42. Saichev A. I., Zaslavsky G. M.  Fractional kinetic equations: solutions and applications.  Chaos. 7 (4), 753--764 (1997).
  43. Zaslavsky G. M., Edelman M. A.  Fractional kinetics: from pseudochaotic dynamics to Maxwell’s Demon.  Physica D: Nonlinear Phenomena. 193 (1–4), 128--147 (2004).
  44. Nigmatullin R.  'Fractional' kinetic equations and ‘universal’ decoupling of a memory function in mesoscale region.  Physica A: Statistical Mechanics and its Applications. 363 (2), 282--298 (2006).
  45. Chechkin A. V., Gonchar V. Y., Szydłowski M.  Fractional kinetics for relaxation and superdiffusion in a magnetic field.  Physics of Plasmas. 9 (1), 78--88 (2002).
  46. Gafiychuk V. V., Datsko B. Y.  Stability analysis and oscillatory structures in time-fractional reaction-diffusion systems.  Phys. Rev. E. 75 (5), 055201 (2007).
  47. Kosztołowicz T., Lewandowska K. D.  Time evolution of the reaction front in a subdiffusive system.  Phys. Rev. E. 78 (6), 066103 (2008).
  48. Shkilev V. P.  Subdiffusion of mixed origin with chemical reactions.  Journal of Experimental and Theoretical Physics. 117 (6),  1066--1070 (2013).
  49. Hobbie R. K., Roth B. J.  Intermediate Physics for Medicine and Biology.  Springer-Verlag, New York (2007).
  50. Jeon J. H., Monne H. M. S., Javanainen M., Metzler R.  Anomalous Diffusion of Phospholipids and Cholesterols in a Lipid Bilayer and its Origins.  Phys. Rev. Lett. 109 (18), 188103 (2012).
  51. Höfling F., Franosch T.  Anomalous transport in the crowded world of biological cells.  Reports on Progress in Physics. 76 (4), 046602 (2013).
  52. Uchaikin V. V.  Fractional phenomenology of cosmic ray anomalous diffusion.  Physics-Uspekhi. 56 (11), 1074--1119 (2013).
  53. Szymanski J., Weiss M.  Elucidating the Origin of Anomalous Diffusion in Crowded Fluids.  Phys. Rev. Lett. 103 (3), 038102 (2009).
  54. Sandev T., Tomovski Z., Dubbeldam J. L. A., Chechkin A.  Generalized diffusion-wave equation with memory kernel.  Journal of Physics A: Mathematical and Theoretical. 52 (1), 015201  (2018).
  55. Sandev T., Metzler R., Chechkin A.  Generalised Diffusion and Wave Equations: Recent Advances.  arXiv:1903.01166 (2019).
  56. Giusti A.  Dispersion relations for the time-fractional Cattaneo--Maxwell heat equation.  Journal of Mathematical Physics. 59 (1), 013506 (2018).
  57. Kostrobij P., Markovych B., Viznovych O., Tokarchuk M.  Generalized diffusion equation with fractional derivatives within Renyi statistics.  Journal of Mathematical Physics. 57 (9), 093301 (2016).
  58. Kostrobij P., Markovych B., Viznovych O., Tokarchuk M.  Generalized electrodiffusion equation with fractality of space--time.  Mathematical Modeling and Computing. 3 (2),  163--172 (2016).
  59. Glushak P. A., Markiv B. B., Tokarchuk M. V.  Zubarev's Nonequilibrium Statistical Operator Method in the Generalized Statistics of Multiparticle Systems.  Theoretical and Mathematical Physics. 194 (1), 57--73 (2018).
  60. Kostrobij P., Markovych B., Viznovych O., Tokarchuk M.  Generalized transport equation with nonlocality of space–-time. Zubarev’s NSO method.  Physica A: Statistical Mechanics and its Applications. 514, 63--70 (2019).
  61. Zubarev D. N.  Modern methods of the statistical theory of nonequilibrium processes.  Journal of Soviet Mathematics. 16 (6), 1509--1571 (1981).
  62. Zubarev D. N., Morozov V. G., Röpke G.  Statistical mechanics of nonequilibrium processes. Vol. 1.  Moscow, Fizmatlit (2002), (in Russian).
  63. Zubarev D. N., Morozov V. G., Röpke G.  Statistical mechanics of nonequilibrium processes. Vol. 2.  Moscow, Fizmatlit (2002), (in Russian).
  64. Markiv B., Tokarchuk R., Kostrobij P., Tokarchuk M.  Nonequilibrium statistical operator method in Renyi statistics.  Physica A: Statistical Mechanics and its Applications. 390 (5), 785--791 (2011).
  65. Cottrill-Shepherd K., Naber M.  Fractional differential forms.  Journal of Mathematical Physics. 42 (5), 2203--2212 (2001).
  66. Mainardi F.  Fractional Calculus.  Springer, Vienna (1997).
  67. Caputo M., Mainardi F.  A new dissipation model based on memory  mechanism.  Pure and Applied Geophysics. 91 (1), 134--147 (1971).