Generalized electrodiffusion equation with fractality of space-time

: pp. 163-172
Revised: December 30, 2016
Lviv Polytechnic National University
Lviv Polytechnic National University
Lviv Polytechnic National University
Lviv Polytechnic National University; Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine

The new non-Markovian electrodiffusion equations of ions in spatially heterogeneous environment with fractal structure and generalized Cattaneo-type diffusion equation with taking into account fractality of space-time are obtained. Different models of the frequency dependence of memory functions, which lead to known diffusion equations with fractality of space-time and their generalizations are considered.

  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. 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).
  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, 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, 1–9 (2007).
  8. Hobbie R. K., Roth B. J. Intermediate Physics for Medicine and Biology (Springer, New York, 2007).
  9. Compte A., Metzler R. The generalized Cattaneo equation for the description of anomalous transport processes. Journal of Physics A: Mathematical and General, 30, 7277–7289 (1997).
  10. Metzler, R. Klafter, J. The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Physics Reports 339, 1–77 (2000).
  11. Hilfer R. Fractional dynamics, irreversibility and ergodicity breaking. Chaos, Solitons Fractals, 5, 1475–1484 (1995).
  12. Hilfer R. Fractional diffusion based on Riemann-Liouville fractional derivatives. The Journal of Physical Chemistry B, 104, 3914–3917 (2000).
  13. Hilfer R. Fractional Time Evolution, chap. II, 87–130 (World Scientific, Singapore, New Jersey, London, Hong Kong, 2000).
  14. Kosztołowicz T., Dworecki K., Mrówczyński S. Measuring subdiffusion parameters. Phys. Rev. E 71, 041105 (2005).
  15. Kosztołowicz T. Subdiffusive random walk in a membrane system: the generalized method of images approach. Journal of Statistical Mechanics: Theory and Experiment, 2015, P10021 (2015).
  16. 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 TiO2 nanoporous in aqueous solution. The Journal of Physical Chemistry B, 104, 2287–2298 (2000).
  17. Bisquert J., Compte A. Theory of the electrochemical impedance of anomalous diffusion. Journal of Electroanalytical Chemistry, 499, 112–120 (2001).
  18. Kosztołowicz T., Lewandowska K. D. Hyperbolic subdiffusive impedance. Journal of Physics A: Mathematical and Theoretical, 42, 055004 (2009).
  19. 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, 84–96 (2014).
  20. Berkowitz B., Scher H. Theory of anomalous chemical transport in random fracture networks. Phys. Rev. E, 57, 5858–5869 (1998).
  21. Bouchaud J. P., Georges A. Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications. Physics Reports, 195, 127–293 (1990).
  22. Nigmatullin R. R. To the Theoretical Explanation of the “Universal Response”. Physica Status Solidi (b), 123, 739–745 (1984).
  23. Nigmatullin R. R. On the Theory of Relaxation for Systems with “Remnant” Memory. Physica Status Solidi (b), 124, 389–393 (1984).
  24. Nigmatullin R. R. The realization of the generalized transfer equation in a medium with fractal geometry. Physica Status Solidi (b), 133, 425–430 (1986).
  25. Nigmatullin R. R. Fractional integral and its physical interpretation. Theoretical and Mathematical Physics, 90, 242–251 (1992).
  26. Nigmatullin R. R., Ryabov Y. E. Cole-Davidson dielectric relaxation as a self-similar relaxation process. Physics of the Solid State, 39, 87–90 (1997).
  27. Nigmatullin R. R. Dielectric relaxation phenomenon based on the fractional kinetics: theory and its experimental confirmation. Physica Scripta, 2009, 014001 (2009).
  28. 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, 1604–1619 (2012).
  29. 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, 1–7 (2012).
  30. Gryhorchak I. I., Kostrobiy P. P., Stasyuk I. V., Tokarchuk M. V., Velychko O. V., Ivashchyshyn F. O., Markovych B. M. Fizychni protsesy ta yikh mikroskopichni modeli v periodychnykh neorhanichno/orhanichnykh klatratakh (Rastr-7, L’viv, 2015), (in Ukrainian).
  31. Kostrobij P. P., Grygorchak I. I., Ivaschyshyn F. O., Markovych B. M., Viznovych O., Tokarchuk M. V. Mathematical modeling of subdiffusion impedance in multilayer nanostructures. Mathematical Modeling and Computing, 2, 154–159 (2015).
  32. Balescu, R. Anomalous transport in turbulent plasmas and continuous time random walks. Phys. Rev. E 51, 4807–4822 (1995).
  33. Tribeche M., Shukla P. K. Charging of a dust particle in a plasma with a non extensive electron distribution function. Physics of Plasmas, 18, 103702 (2011).
  34. Gong J., Du J. Dust charging processes in the nonequilibrium dusty plasma with nonextensive power-law distribution. Physics of Plasmas, 19, 023704 (2012).
  35. 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, 5096–5103 (2001).
  36. Tarasov V. E. Electromagnetic field of fractal distribution of charged particles. Physics of Plasmas, 12, 082106 (2005).
  37. Tarasov V. E. Magnetohydrodynamics of fractal media. Physics of Plasmas, 13, 052107 (2006).
  38. Monin A. S. Uravnenija turbulentnoj difuzii. DAN SSSR, ser. geofiz. 2, 256–259 (1955).
  39. Klimontovich J. L. Vvedenie v Fiziku otkrytyh sistem (Moskva Janus, 2002), (in Russian).
  40. Zaslavsky G. Chaos, fractional kinetics, and anomalous transport. Physics Reports, 371, 461–580 (2002).
  41. Zaslavsky G. Fractional kinetic equation for hamiltonian chaos. Physica D: Nonlinear Phenomena, 76, 110–122 (1994).
  42. Saichev A. I., Zaslavsky G. M. Fractional kinetic equations: solutions and applications. Chaos, 7, 753–764 (1997).
  43. Zaslavsky G., Edelman M. Fractional kinetics: from pseudochaotic dynamics to Maxwell’s demon. Physica D: Nonlinear Phenomena, 193, 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, 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, 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, 055201 (2007).
  47. Kosztołowicz T., Lewandowska K. D. Time evolution of the reaction front in a subdiffusive system. Phys. Rev. E, 78, 066103 (2008).
  48. Shkilev V. P. Subdiffusion of mixed origin with chemical reactions. Journal of Experimental and Theoretical Physics, 117, 1066–1070 (2013).
  49. Uchaikin V. V. Fractional phenomenology of cosmic ray anomalous diffusion. Physics-Uspekhi, 56, 1074–1119 (2013).
  50. Stanislavsky A. A. Probability Interpretation of the Integral of Fractional Order. Theoretical and Mathematical Physics, 138, 418–431 (2004).
  51. Tarasov V. E. Fractional generalization of Liouville equations. Chaos, 14, 123–127 (2004).
  52. Tarasov V. E. Fractional Liouville and BBGKI equations. Journal of Physics: Conference Series, 7, 17 (2005).
  53. Tarasov V. E. Fractional systems and fractional Bogoliubov hierarchy equations. Phys. Rev. E, 71, 011102 (2005).
  54. Tarasov V. E. Fractional statistical mechanics. Chaos, 16, 033108 (2006).
  55. Tarasov V. E. Transport equations from Liouville equations for fractional systems. International Journal of Modern Physics B, 20, 341–353 (2006).
  56. Tarasov V. E. Fractional diffusion equations for open quantum system. Nonlinear Dynamics, 71, 663–670 (2013).
  57. Tarasov V. E. Fractional Heisenberg equation. Physics Letters A, 372, 2984–2988 (2008).
  58. Tarasov V. E. Fractional hydrodynamic equations for fractal media. Annals of Physics, 318, 286–307 (2005).
  59. Tarasov V. E. Liouville and Bogoliubov equations with fractional derivatives. Modern Physics Letters B, 21, 237–248 (2007).
  60. Tarasov V. E. The fractional Chapman–Kolmogorov equation. Modern Physics Letters B, 21, 163–174 (2007).
  61. Tarasov V. E. Fractional generalization of the quantum Markovian master equation. Theoretical and Mathematical Physics, 158, 179–195 (2009).
  62. Tarasov V. E. Quantum dissipation from power-law memory. Annals of Physics, 327, 1719–1729 (2012).
  63. Tarasov V. E. Power-law spatial dispersion from fractional Liouville equation. Physics of Plasmas, 20, 102110 (2013).
  64. Tarasov V. E. Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media. Nonlinear Physical Science (Springer Berlin Heidelberg, 2010), 1st edition.
  65. Kobelev L. Y. The Multifractal Time and Irreversibility in Dynamic Systems (2000). arXiv:physics/0002002.
  66. Kobelev Y. L., Kobelev L. Y., Romanov E. P. Kinetic equations for large systems with fractal structures. Doklady Physics, 45, 194–197 (2000).
  67. Kobelev Y. L., Kobelev L. Y., Kobelev V. L., Romanov E. P. Description of diffusion in fractal media on the basis of the klimontovich kinetic equation in fractal space. Doklady Physics, 47, 580–582 (2002).
  68. Lukashchuk S. Y. Time-fractional extensions of the Liouville and Zwanzig equations. Central European Journal of Physics, 11, 740–749 (2013).
  69. Kostrobij P., Markovych B., Viznovych O., Tokarchuk M. Generalized diffusion equation with fractional derivatives within renyi statistics. Journal of Mathematical Physics, 57, 093301 (2016).
  70. Zubarev D. N. Modern methods of the statistical theory of nonequilibrium processes. Journal of Soviet Mathematics, 16, 1509–1571 (1981).
  71. Zubarev D. N., Morozov V. G., Röpke G. Statistical mechanics of nonequilibrium processes, vol. 1 (Fizmatlit, 2002).
  72. Zubarev D. N., Morozov V. G., Röpke G. Statistical mechanics of nonequilibrium processes, vol. 2 (Fizmatlit, 2002).
  73. Markiv B. B., Tokarchuk R. M., Kostrobij P. P., Tokarchuk M. V. Nonequilibrium statistical operator method in Renyi statistics. Physica A: Statistical Mechanics and its Applications, 390, 785–791 (2011).
  74. Cottrill-Shepherd K., Naber M. Fractional differential forms. Journal of Mathematical Physics, 42, 2203–2212 (2001).
  75. Mainardi F. Fractional Calculus, 291–348 (Springer Vienna, Vienna, 1997).
  76. Caputo M., Mainardi F. A new dissipation model based on memory mechanism. Pure and Applied Geophysics, 91, 134–147 (1971).
  77. Qi H., Jiang X. Solutions of the space-time fractional Cattaneo diffusion equation. Physica A: Statistical Mechanics and its Applications, 390, 1876–1883 (2011).
  78. Sun H., Chen W., Li C., Chen Y. Fractional differential models for anomalous diffusion. Physica A: Statistical Mechanics and its Applications, 389, 2719–2724 (2010).
    Math. Model. Comput. Vol. 3, No. 2, pp. 163-172 (2016)