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All issues Volume 11 / No 3 (2016) Math. Model. Nat. Phenom., 11 3 (2016) 18-33 References
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Issue
Math. Model. Nat. Phenom.
Volume 11, Number 3, 2016
Anomalous diffusion
Page(s) 18 - 33
DOI https://doi.org/10.1051/mmnp/201611302
Published online 21 June 2016
  1. J.-P. Bouchaud, A. Georges. Anomalous diffusion in disordered media: statistical mechanisms, models and physical applications. Phys. Rep., 195 (1990) 127–293. [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]
  2. R. Metzler, J. Klafter. The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep., 339 (2000), 1–77 [NASA ADS] [CrossRef] [Google Scholar]
  3. . The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics. J. Phys. A: Math. Gen., 37 (2004), R161–R208. [Google Scholar]
  4. R. Metzler, J.-H. Jeon, A. G. Cherstvy, E. Barkai. Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking. Phys. Chem. Chem. Phys., 16 (2014), 24128–24164. [CrossRef] [PubMed] [Google Scholar]
  5. H. Scher, E. W. Montroll. Anomalous transit-time dispersion in amorphous solids. Phys. Rev. B, 12 (1975), 2455. [CrossRef] [Google Scholar]
  6. J.-H. Jeon, V. Tejedor, S. Burov, E. Barkai, C. Selhuber-Unkel, K. Berg-Sørensen, L. Oddershede, R. Metzler. In vivo anomalous diffusion and weak ergodicity breaking of lipid granules. Phys. Rev. Lett., 106 (2011), 048103. [CrossRef] [PubMed] [Google Scholar]
  7. I. Golding, E. C. Cox. Physical Nature of Bacterial Cytoplasm. Phys. Rev. Lett., 96 (2006), 098102. [CrossRef] [PubMed] [Google Scholar]
  8. J. Szymanski, M. Weiss. Elucidating the origin of anomalous diffusion in crowded fluids. Phys. Rev. Lett., 103 (2009), 038102. [CrossRef] [PubMed] [Google Scholar]
  9. J.-H. Jeon, N. Leijnse, L. B. Oddershede, R. Metzler. Anomalous diffusion and power-law relaxation of the time averaged mean squared displacement in worm-like micellar solutions. New J. Phys., 15 (2013), 045011. [CrossRef] [Google Scholar]
  10. J. F. Reverey, J.-H. Jeon, H. Bao, M. Leippe, R. Metzler, C. Selhuber-Unkel. Superdiffusion dominates intracellular particle motion in the supercrowded cytoplasm of pathogenic Acanthamoeba castellanii. Sci. Rep., 5 (2015), 11690. [CrossRef] [PubMed] [Google Scholar]
  11. A. Caspi, R. Granek, M. Elbaum. Enhanced diffusion in active intracellular transport. Phys. Rev. Lett., 85 (2000), 5655. [CrossRef] [PubMed] [Google Scholar]
  12. A. Godec, M. Bauer, R. Metzler. Collective dynamics effect transient subdiffusion of inert tracers in gel networks, New J. Phys., 16 (2014), 092002. [CrossRef] [Google Scholar]
  13. F. Trovato, V. Tozzini. Diffusion within the cytoplasm: a mesoscale model of interacting macromolecules. Biophys. J., 107 (2014), 2579–2591. [CrossRef] [PubMed] [Google Scholar]
  14. G. R. Kneller, K. Baczynski, M. Pasenkiewicz-Gierula. Molecular dynamics simulation and exact results. J. Chem. Phys., 135 (2011), 141105. [CrossRef] [PubMed] [Google Scholar]
  15. J.-H. Jeon, H. M. Monne, M. Javanainen, R. Metzler. Anomalous diffusion of phospholipids and cholesterols in a lipid bilayer and its origins. Phys. Rev. Lett., 109 (2012), 188103. [CrossRef] [PubMed] [Google Scholar]
  16. S.R. White, M. Barma, Field-induced drift and trapping in percolation networks. J. Phys. A: Math. Gen., 17 (1984), 2995. [CrossRef] [Google Scholar]
  17. G.H. Weiss, S. Havlin, Some properties of a random walk on a comb structure. Physica A, 134 (1986), 474–482. [CrossRef] [Google Scholar]
  18. S. Havlin, J.E. Kiefer, G.H. Weiss. Anomalous diffusion on a random comblike structure. Phys. Rev. A, 36 (1987), 1403–1408. [CrossRef] [Google Scholar]
  19. O. Matan, S. Havlin, D. Staufler. Scaling properties of diffusion on comb-like structures. J. Phys. A: Math. Gen., 22 (1989), 2867. [CrossRef] [Google Scholar]
  20. V.E. Arkhincheev, E.M. Baskin. Anomalous diffusion and drift in a comb model of percolation clusters. Sov. Phys. JETP, 73 (1991), 161–165. [Google Scholar]
  21. I.A. Lubashevski, A.A. Zemlyanov. Continuum description of anomalous diffusion on a comb structure. J. Exper. Theor. Phys., 87 (1998), 700–713. [CrossRef] [Google Scholar]
  22. V.E. Arkhincheev. Anomalous diffusion and charge relaxation on comb model: exact solutions. Physica A, 280 (2000), 304–314; Diffusion on random comb structure: effective medium approximation. Physica A, 307 (2002), 131–141; Unified continuum description for sub-diffusion random walks on multi-dimensional comb model. Physica A, 389 (2010), 1–6. [CrossRef] [Google Scholar]
  23. E. Baskin, A. Iomin. Superdiffusion on a comb structure. Phys. Rev. Lett., 93 (2004), 120603. [CrossRef] [PubMed] [Google Scholar]
  24. A. Iomin, E. Baskin. Negative superdiffusion due to inhomogeneous convection. Phys. Rev. E, 71 (2005), 061101. [CrossRef] [Google Scholar]
  25. L.R. da Silva, A.A. Tateishi, M.K. Lenzi, E.K. Lenzi, P.C. da Silva. Green function for a non-Markovian Fokker-Planck equation: comb-model and anomalous diffusion, Brazilian J. Phys., 39 (2009), 483–487. [Google Scholar]
  26. O.A. Dvoretskaya, P.S. Kondratenko. Anomalous transport regimes and asymptotic concentration distributions in the presence of advection and diffusion on a comb structure. Phys. Rev. E 79 (2009), 041128. [CrossRef] [Google Scholar]
  27. I. Podlubny. Fractional Differential Equations. Acad. Press, San Diego etc., 1999. [Google Scholar]
  28. V. Mendez, A. Iomin. Comb-like models for transport along spiny dendrites. Chaos Solitons Fractals, 53 (2013), 46–51. [CrossRef] [Google Scholar]
  29. A. Iomin, V. Mendez. Reaction-subdiffusion front propagation in a comblike model of spiny dendrites. Phys. Rev. E, 88 (2013), 012706. [CrossRef] [Google Scholar]
  30. A. Iomin. Subdiffusion on a fractal comb. Phys. Rev. E, 83 (2011), 052106. [CrossRef] [Google Scholar]
  31. A. Iomin. Superdiffusive comb: Application to experimental observation of anomalous diffusion in one dimension. Phys. Rev. E, 86 (2012), 032101. [CrossRef] [Google Scholar]
  32. E.K. Lenzi, L.R. da Silva, A.A. Tateishi, M.K. Lenzi, H.V. Ribeiro. Diffusive process on a backbone structure with drift terms. Phys. Rev. E, 87 (2013), 012121. [CrossRef] [Google Scholar]
  33. D. Shamiryan, M.R. Baklanov, P. Lyons, S. Beckx, W. Boullart, K. Maex. Diffusion of solvents in thin porous films. Colloids and Surfaces A: Physicochem. Eng. Aspects, 300 (2007), 111–116. [CrossRef] [Google Scholar]
  34. R.T. Sibatov, E.V. Morozova. Multiple trapping on a comb structure as a model of electron transport in disordered nanostructured semiconductors. J. Exper. Theor. Phys., 120 (2015), 860–870. [CrossRef] [Google Scholar]
  35. L.C.Y. Chu, D. Guha, Y.M.M. Antar. Comb-shaped circularly polarised dielectric resonator antenna. IEEE Electron. Lett., 42 (2006), 785–787. [CrossRef] [Google Scholar]
  36. M. Thiriet. Tissue Functioning and Remodeling in the Circulatory and Ventilatory Systems. Springer, New York, 2013. [Google Scholar]
  37. D. Ben-Avraham, S. Havlin. Diffusion and Reactions in Fractals and Disordered System. Cambridge University Press, Cambridge, 2000. [Google Scholar]
  38. A. Rebenshtok, E. Barkai. Occupation times on a comb with ramified teeth. Phys. Rev. E, 88 (2013), 052126. [CrossRef] [Google Scholar]
  39. V.Yu. Zaburdaev, P.V. Popov, A.S. Romanov, K.V. Chukbar. Stochastic transport through complex comb structures. J. Exper. Theor. Phys, 106 (2008), 999–1005. [CrossRef] [Google Scholar]
  40. 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 (2014), 093050. [CrossRef] [Google Scholar]
  41. T. Sandev, A. Iomin, H. Kantz. Fractional diffusion on a fractal grid comb. Phys. Rev. E, 91 (2015), 032108. [CrossRef] [Google Scholar]
  42. Y. He, S. Burov, R. Metzler, E. Barkai. Random time-scale invariant diffusion and transport coefficients. Phys. Rev. Lett., 101 (2008), 058101. [NASA ADS] [CrossRef] [PubMed] [Google Scholar]
  43. J. H. P. Schulz, E. Barkai, R. Metzler. Aging effects and population splitting in single-particle trajectory averages. Phys. Rev. Lett., 110 (2013), 020602. [CrossRef] [PubMed] [Google Scholar]
  44. Y. Meroz, I. M. Sokolov, J. Klafter. Subdiffusion of mixed origins: When ergodicity and nonergodicity coexist. Phys. Rev. E, 81 (2010), 010101(R). [CrossRef] [Google Scholar]
  45. Y. Mardoukhi, J.-H. Jeon, R. Metzler. Geometry controlled anomalous diffusion in random fractal geometries: looking beyond the infinite cluster. Phys. Chem. Chem. Phys., 17 (2015), 30134. [CrossRef] [PubMed] [Google Scholar]
  46. T. Sandev, A. Chechkin, H. Kantz, R. Metzler. Diffusion and Fokker-Planck-Smoluchowski equations with generalized memory kernel. Fract. Calc. Appl. Anal. 18 (2015), 1006–1038. [CrossRef] [Google Scholar]
  47. A.V. Chechkin, M. Hofmann, I.M. Sokolov. Continuous-time random walk with correlated waiting times. Phys. Rev. E, 80 (2009), 031112. [CrossRef] [Google Scholar]
  48. E. Barkai. Fractional Fokker-Planck equation, solution, and application. Phys. Rev. E, 63 (2001), 046118. [CrossRef] [Google Scholar]
  49. M.M. Meerschaert, P. Straka. Inverse stable subordinators. Math. Model. Nat. Phenom., 8 (2013), 1–16. [Google Scholar]
  50. A.N. Kochubei. General fractional calculus, evolution equations, and renewal processes. Integr. Equ. Oper. Theory, 71 (2011), 583–600. [CrossRef] [Google Scholar]
  51. A.V. Chechkin, R. Gorenflo, I.M. Sokolov. Retarding subdiffusion and accelerating superdiffusion governed by distributed-order fractional diffusion equations. Phys. Rev. E, 66 (2002), 046129. [CrossRef] [Google Scholar]
  52. A.V. Chechkin, J. Klafter, I.M. Sokolov. Fractional Fokker-Planck equation for ultraslow kinetics. EPL, 63 (2003), 326. [CrossRef] [EDP Sciences] [Google Scholar]
  53. A. Chechkin, I.M. Sokolov, J. Klafter. Natural and Modified Forms of Distributed Order Fractional Diffusion Equations, in Fractional Dynamics: Recent Advances, Eds. J. Klafter, S.C. Lim and R. Metzler. World Scientific Publishing Company, Singapore, 2011. [Google Scholar]
  54. F. Mainardi. Fractional Calculus and Waves in Linear Viscoelesticity: An introduction to Mathematical Models. Imperial College Press, London, 2010. [Google Scholar]
  55. T. Sandev, Ž. Tomovski. Langevin equation for a free particle driven by power law type of noises. Phys. Lett. A, 378 (2014), 1–9. [CrossRef] [Google Scholar]
  56. A. Erdelyi, W. Magnus, F. Oberhettinger, F.G. Tricomi. Higher Transcedential Functions. Vol. 3, McGraw-Hill, New York, 1955. [Google Scholar]
  57. Y.G. Sinai. The limiting behavior of a one-dimensional random walk in a random medium. Theor. Probab. Appl., 27 (1982), 256–268. [CrossRef] [Google Scholar]
  58. A. Godec, A. V. Chechkin, E. Barkai, H. Kantz, R. Metzler. Localization and universal fluctuations in ultraslow diffusion processes. J. Phys. A: Math. Theor., 47 (2014), 492002. [CrossRef] [Google Scholar]
  59. A. Bodrova, A. V. Chechkin, A. G. Cherstvy, R. Metzler. Quantifying non-ergodic dynamics of force-free granular gases. Phys. Chem. Chem. Phys., 17 (2015), 21791–21798. [CrossRef] [PubMed] [Google Scholar]
  60. L. P. Sanders, M. A. Lomholt, L. Lizana, K. Fogelmark, R. Metzler, T. Ambjörnsson. Severe slowing-down and universality of the dynamics in disordered interacting many-body systems: ageing and ultraslow diffusion. New J. Phys., 16 (2014), 113050. [CrossRef] [Google Scholar]
  61. J. Dräger, J. Klafter. Strong anomaly in diffusion generated by iterated maps. Phys. Rev. Lett., 84 (2000), 5998. [CrossRef] [PubMed] [Google Scholar]
  62. A. G. Cherstvy, R. Metzler. Population splitting, trapping, and non-ergodicity in heterogeneous diffusion processes. Phys. Chem. Chem. Phys., 15 (2013), 20220–20235. [CrossRef] [PubMed] [Google Scholar]
  63. M.A. Lomholt, L. Lizana, R. Metzler, T. Ambjörnsson, Microscopic origin of the logarithmic time evolution of aging processes in complex systems. Phys. Rev. Lett., 110 (2013), 208301. [CrossRef] [PubMed] [Google Scholar]
  64. A. Bodrova, A. V. Chechkin, A. G. Cherstvy, R. Metzler. Ultraslow scaled Brownian motion. New J. Phys., 17 (2015), 063038. [CrossRef] [Google Scholar]
  65. A.M. Mathai, R.K. Saxena, H.J. Haubold. The H-function: Theory and Applications. New York Dordrecht Heidelberg London, Springer, 2010. [Google Scholar]
  66. R. Schilling, R. Song, Z. Vondracek. Bernstein Functions. De Gruyter, Berlin, 2010. [Google Scholar]
  67. C. Berg, G. Forst. Potential Theory on Locally Compact Abelian Groups. Berlin, Springer, 1975. [Google Scholar]
  68. T.R. Prabhakar. A singular integral equation with a generalized Mittag-Leffler function in the kernel. Yokohama Math. J., 19 (1971), 7–15. [MathSciNet] [Google Scholar]
  69. R.K. Saxena, A.M. Mathai, H.J. Haubold, Unified fractional kinetic equation and a fractional diffusion equation. Astrophys. Space Sci., 209 (2004), 299–310. [CrossRef] [Google Scholar]
  70. T. Sandev, R. Metzler, Z. Tomovski. Correlation functions for the fractional generalized Langevin equation in the presence of internal and external noise. J. Math. Phys., 55 (2014), 023301. [CrossRef] [Google Scholar]
  71. H. Seybold, R. Hilfer. Numerical algorithm for calculating the generalized Mittag-Leffler function. SIAM J. Numer. Anal., 47 (2008), 69–88. [Google Scholar]
  72. Z.L. Huang, X.L. Jin, C.W. Lim, Y. Wang. Statistical analysis for stochastic systems including fractional derivatives. Nonlin. Dyn., 59 (2010), 339–349. [CrossRef] [Google Scholar]
  73. W. Feller. An Introduction to Probability Theory and Its Applications. Vol. II, Wiley, New York, 1968. [Google Scholar]

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