Abstract
We consider anomalous stochastic processes based on the renewal continuous time random walk model with different forms for the probability density of waiting times between individual jumps. In the corresponding continuum limit we derive the generalized diffusion and Fokker-Planck- Smoluchowski equations with the corresponding memory kernels. We calculate the qth order moments in the unbiased and biased cases, and demonstrate that the generalized Einstein relation for the considered dynamics remains valid. The relaxation of modes in the case of an external harmonic potential and the convergence of the mean squared displacement to the thermal plateau are analyzed.
References
[1] K.H. Andersen, P. Castiglione, A. Mazzino and A. Vulpiani, Simple stochastic models showing strong anomalous diffusion. Eur. Phys. J. B 18 (2000), 447-452.Search in Google Scholar
[2] E. Bacry, J. Delour and J.F. Muzy, Multifractal random walk. Phys. Rev. E 64 (2001), 026103.10.1103/PhysRevE.64.026103Search in Google Scholar PubMed
[3] E. Barkai, R. Metzler and J. Klafter, From continuous time random walks to the fractional Fokker-Planck equation. Phys. Rev. E 61, No 1 (2000), 132-138.Search in Google Scholar
[4] F. Bartumeus and S.A. Levin, Fractal reorientation clocks: Linking animal behaviour to statistical patterns of search. Proc. Natl. Acad. Sci. USA 105 (2008), 19072-19077.Search in Google Scholar
[5] C. Berg and G. Forst, Potential Theory on Locally Compact Abelian Groups. Springer, Berlin (1975).10.1007/978-3-642-66128-0Search in Google Scholar
[6] B. Berkowitz, A. Cortis, M. Dentz and H. Scher, Modeling non-Fickian transport in geological formations as a continuous time random walk. Rev. Geophys. 44 (2006), RG2003.10.1029/2005RG000178Search in Google Scholar
[7] S. Burov, R. Metzler and E. Barkai, Aging and nonergodicity beyond the Khinchin theorem. Proc. Natl. Acad. Sci. USA 107 (2010), 13228-13233.10.1073/pnas.1003693107Search in Google Scholar PubMed PubMed Central
[8] J.-P. Bouchaud and A. Georges, Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications. Phys. Rep. 195 (1990), 127-293.Search in Google Scholar
[9] K. Burnecki, E. Kepten, J. Janczura, I. Bronshtein, Y. Garini and A. Weron, Universal algorithm for identification of fractional Brownian motion. A case of telomere subdiffusion. Biophys. J. 103 (2012), 1839-1847.Search in Google Scholar
[10] A. Cairoli and A. Baule, Anomalous processes with general waiting times: functionals and multi-point structure. arXiv:condmat/ 1411.7005.Search in Google Scholar
[11] E. Capelas de Oliveira, F. Mainardi and J. Vaz Jr., Models based on Mittag-Leffler functions for anomalous relaxation in dielectrics. Eur. Phys. J. 193 (2011), 161-171.Search in Google Scholar
[12] 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), 046129.10.1103/PhysRevE.66.046129Search in Google Scholar PubMed
[13] A.V. Chechkin, R. Gorenflo, I.M. Sokolov and V.Yu. Gonchar, Distributed order fractional diffusion equation. Fract. Calc. Appl. Anal. 6, No 3 (2003), 259-279.Search in Google Scholar
[14] A.V. Chechkin, J. Klafter and I.M. Sokolov, Fractional Fokker-Planck equation for ultraslow kinetics. Europhysics Letters 63, No 3 (2003), 326-332.Search in Google Scholar
[15] A. Chechkin, I.M. Sokolov and J. Klafter, Natural and modified forms of distributed order fractional diffusion equations. In: Fractional Dynamics: Recent Advances, World Scientific, Singapore (2011).Search in Google Scholar
[16] A.G. Cherstvy, A.V. Chechkin and R. Metzler, Anomalous diffusion and ergodicity breaking in heterogeneous diffusion processes. New J. Phys. 15 (2013), 083039.10.1088/1367-2630/15/8/083039Search in Google Scholar
[17] A.G. Cherstvy, A.V. Chechkin and R. Metzler, Particle invasion, survival, and non-ergodicity in 2D diffusion processes with spacedependent diffusivity. Soft Matter 10 (2014), 1591-1601.Search in Google Scholar
[18] A.G. Cherstvy and R. Metzler, Nonergodicity, fluctuations, and criticality in heterogeneous diffusion processes. Phys. Rev. E 90 (2014), 012134.10.1103/PhysRevE.90.012134Search in Google Scholar PubMed
[19] A.G. Cherstvy and R. Metzler, Population splitting, trapping, and non-ergodicity in heterogeneous diffusion processes. Phys. Chem. Chem. Phys. 15 (2013), 20220-20235.Search in Google Scholar
[20] J. Dr¨ager and J. Klafter, Strong anomaly in diffusion generated by iterated maps. Phys. Rev. Lett. 84 (2000), 5998-6001.Search in Google Scholar
[21] C.H. Eab and S.C. Lim, Fractional Langevin equations of distributed order. Phys. Rev. E 83 (2011), 031136.10.1103/PhysRevE.83.031136Search in Google Scholar PubMed
[22] A. Erdélyi, W. Magnus, F. Oberhettinger and F.G. Tricomi, Higher Transcedential Functions, Vol. 3, McGraw-Hill, New York (1955).Search in Google Scholar
[23] K.S. Fa and K.G. Wang, Integrodifferential diffusion equation for continuous-time random walk. Phys. Rev. E 81 (2010), 011126.10.1103/PhysRevE.81.011126Search in Google Scholar PubMed
[24] W. Feller, An Introduction to Probability Theory and Its Applications, Vol. II, Wiley, New York (1968).Search in Google Scholar
[25] A. Fuliński, How to generate and measure anomalous diffusion in simple systems. J. Chem. Phys. 138 (2013), 021101.10.1063/1.4775737Search in Google Scholar PubMed
[26] A. Fuliński, Anomalous diffusion and weak nonergodicity. Phys. Rev. E 83 (2011), 061140.10.1103/PhysRevE.83.061140Search in Google Scholar PubMed
[27] R. Fürth, Editor, Albert Einstein: Investigations on the Theory of the Brownian Movement. Dover, New York (1956).Search in Google Scholar
[28] N. Gal and D. Weihs, Experimental evidence of strong anomalous diffusion in living cells. Phys. Rev. E 81 (2010), 020903(R). 10.1103/PhysRevE.81.020903Search in Google Scholar PubMed
[29] A. Godec, A. V. Chechkin, E. Barkai, H. Kantz and R. Metzler, Localisation and universal fluctuations in ultraslow diffusion processes. J. Phys. A: Math. Theor. 47 (2014), 492002.10.1088/1751-8113/47/49/492002Search in Google Scholar
[30] I. Golding and E. C. Cox, Physical nature of bacterial cytoplasm. Phys. Rev. Lett. 96 (2006), 098102.10.1103/PhysRevLett.96.098102Search in Google Scholar PubMed
[31] M.C. González, C.A. Hidalgo and A.-L. Barabási, Understanding individual human mobility patterns. Nature 453 (2008), 779-782.Search in Google Scholar
[32] R. Gorenflo and F. Mainardi, Fractional Calculus: Integral and Differential Equations of Fractional Order. In: Fractals and Fractional Calculus in Continuum Mechanics, Springer, New York (1997), 223-276.10.1007/978-3-7091-2664-6_5Search in Google Scholar
[33] I. Goychuk, Viscoelastic subdiffusion: From anomalous to normal. Phys. Rev. E 80 (2009), 046125.10.1103/PhysRevE.80.046125Search in Google Scholar PubMed
[34] I. Goychuk, Viscoelastic subdiffusion: Generalized Langevin equation approach. Adv. Chem. Phys. 150 (2012), 187-253.Search in Google Scholar
[35] G. Guigas, C. Kalla and M. Weiss, Probing the nanoscale viscoelasticity of intracellular fluids in living cells. Biophys. J. 93 (2007), 316-323.Search in Google Scholar
[36] P. Guo, C.B. Zeng, C.P. Li and Y.Q. Chen, Numerics for the fractional Langevin Equation driven by the fractional Brownian motion. Fract. Cal. Appl. Anal. 16, No 1 (2013), 123-141; DOI: 10.2478/s13540-013-0009-8; http://www.degruyter.com/view/j/fca.2013.16.issue-1/issue-files/fca.2013.16.issue-1.xml.10.2478/s13540-013-0009-8Search in Google Scholar
[37] M. Hahn and S. Umarov, Fractional Fokker-Planck-Kolmogorov type equations and their associated stochastic differential equations. Fract. Calc. Appl. Anal. 14 (2011), 56-79; DOI: 10.2478/s13540-011-0005-9; http://www.degruyter.com/view/j/fca.2011.14.issue-1/issue-files/fca.2011.14.issue-1.xml.10.2478/s13540-011-0005-9Search in Google Scholar
[38] P. H¨anggi, Correlation functions and master equations of generalized non-Markovian Langevin equations. Zeit. Physik B 31 (1978), 407-416.Search in Google Scholar
[39] P. H¨anggi and F. Mojtabai, Thermally activated escape rate in presence of long-time memory. Phys. Rev. E 26 (1982), 1168-1170.Search in Google Scholar
[40] S. Havlin and D. Ben-Avraham, Diffusion in disordered media. Adv. Phys. 51 (2002), 187-292.Search in Google Scholar
[41] S. Havlin and G. H. Weiss, A new class of long-tailed pausing time densities for the CTRW. J. Stat. Phys. 58 (1990), 1267-1273.Search in Google Scholar
[42] Y. He, S. Burov, R. Metzler and E. Barkai, Random time-scale invariant diffusion and transport coefficients. Phys. Rev. Lett. 101 (2008), 058101.10.1103/PhysRevLett.101.058101Search in Google Scholar PubMed
[43] E. Heinsalu, M. Patriarca, I. Goychuk, G. Schmid and P. H¨anggi, Fractional Fokker-Planck dynamics: Numerical algorithm and simulations. Phys. Rev. E 73 (2006), 046133. 10.1103/PhysRevE.73.046133Search in Google Scholar PubMed
[44] R. Hilfer, Exact solutions for a class of fractal time random walks. Fractals 3 (1995), 211-216.Search in Google Scholar
[45] R. Hilfer, Fractional diffusion based on Riemann-Liouville fractional derivatives. J. Phys. Chem. B 104 (2000), 3914-3917.Search in Google Scholar
[46] R. Hilfer, On fractional diffusion and continuous time random walks. Physica A 329 (2003), 35-40.Search in Google Scholar
[47] R. Hilfer and L. Anton, Fractional master equations and fractal time random walks. Phys. Rev. E 51 (1995), R848.10.1103/PhysRevE.51.R848Search in Google Scholar
[48] F. Höfling and T. Franosch, Anomalous transport in the crowded world of biological cells. Rep. Progr. Phys. 76 (2013), 046602.10.1088/0034-4885/76/4/046602Search in Google Scholar PubMed
[49] J.-H. Jeon, A.V. Chechkin and R. Metzler, Scaled Brownian motion: a paradoxical process with a time dependent diffusivity for the description of anomalous diffusion. Phys. Chem. Chem. Phys. 16 (2014), 15811-15817.Search in Google Scholar
[50] J.-H. Jeon, N. Leijnse, L.B. Oddershede and 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), 045011.10.1088/1367-2630/15/4/045011Search in Google Scholar
[51] J.-H. Jeon, V. Tejedor, S. Burov, E. Barkai, C. Selhuber-Unkel, K. Berg-Sorensen, L. Oddershede and R. Metzler, In vivo anomalous diffusion and weak ergodicity breaking of lipid granules. Phys. Rev. Lett. 106 (2011), 048103.10.1103/PhysRevLett.106.048103Search in Google Scholar PubMed
[52] M. Jullien, J. Paret and P. Tabeling, Richardson pair dispersion in two-dimensional turbulence. Phys. Rev. Lett. 82 (1999), 2872.10.1103/PhysRevLett.82.2872Search in Google Scholar
[53] J.W. Kantelhardt, S.A. Zschiegner, E. Koscielny-Bunde, S. Havlin, A. Bunde and H.E. Stanley, Multifractal detrended fluctuation analysis of nonstationary time series. Physica A 316 (2002), 87-114.Search in Google Scholar
[54] J. Klafter, A. Blumen and M.F. Shlesinger, Stochastic pathway to anomalous diffusion. Phys. Rev. A 35 (1987), 3081.10.1103/PhysRevA.35.3081Search in Google Scholar
[55] A. Klemm and R. Kimmich, NMR microscopy of pore-space backbones in rock, sponge, and sand in comparison with random percolation model objects. Phys. Rev. E 55 (1997), 4413.10.1103/PhysRevE.55.4413Search in Google Scholar
[56] A. Klemm, R. Metzler and R. Kimmich, Diffusion on random-site percolation clusters: Theory and NMR microscopy experiments with model objects. Phys. Rev. E 65 (2002), 021112.10.1103/PhysRevE.65.021112Search in Google Scholar PubMed
[57] G. Kneller, A scaling approach to anomalous diffusion. J. Chem Phys. 141 (2014), 041105.10.1063/1.4891357Search in Google Scholar PubMed
[58] A.N. Kolmogorov, Curves in Hilbert spaces invariant relative to oneparametric group of motions. Dokl. Akad. Nauk SSSR 26 (1940), 6-9.Search in Google Scholar
[59] S.C. Kou, Stochastic modeling in nanoscale biophysics: Subdiffusion within proteins. Ann. Appl. Stat. 2 (2008), 501-535. Search in Google Scholar
[60] S.C. Lim and S.V. Muniandy, Self-similar Gaussian processes for modeling anomalous diffusion. Phys. Rev. E 66 (2002), 021114.10.1103/PhysRevE.66.021114Search in Google Scholar PubMed
[61] M.A. Lomholt, T. Koren, R. Metzler and J. Klafter, Lévy strategies in intermittent search processes are advantageous. Proc. Natl. Acad. Sci. USA 105 (2008), 11055-11059.10.1073/pnas.0803117105Search in Google Scholar
[62] M.A. Lomholt, L. Lizana, R. Metzler and T. Ambjörnsson, Microscopic origin of the logarithmic time evolution of aging processes in complex systems. Phys. Rev. Lett. 110 (2013), 208301.10.1103/PhysRevLett.110.208301Search in Google Scholar PubMed
[63] C. Loverdo, O. Bénichou, M. Moreau and R. Voiturierz, Enhanced reaction kinetics in biological cells. Nature Phys. 4 (2008), 134-137.Search in Google Scholar
[64] Y. Luchko and R. Gorenflo, An operational method for solving fractional differential equations with the Caputo derivatives. Acta Math. Vietnamica 24 (1999), 207-233.Search in Google Scholar
[65] E. Lutz, Fractional Langevin equation. Phys. Rev. E 64 (2001), 051106.10.1103/PhysRevE.64.051106Search in Google Scholar PubMed
[66] M. Magdziarz, Langevin picture of subdiffusion with infinitely divisible waiting times. J. Stat. Phys. 135 (2009), 763-772.Search in Google Scholar
[67] B.B. Mandelbrot, Multifractals and 1/f Noise: Wild Self-Affinity in Physics. Springer, Berlin (1999).10.1007/978-1-4612-2150-0Search in Google Scholar
[68] B.B. Mandelbrot and J.W. van Ness, Fractional Brownian motions, fractional noises and applications. SIAM Rev. 10 (1968), 422-437.Search in Google Scholar
[69] P. Massignan, C. Manzo, J. A. Torreno-Pina, M. F. García-Parako, M. Lewenstein and G. L. Lapeyre, Jr., Nonergodic subdiffusion from Brownian motion in an inhomogeneous medium. Phys. Rev. Lett. 112 (2014), 150603.10.1103/PhysRevLett.112.150603Search in Google Scholar PubMed
[70] Y. Meroz, I.M. Sokolov and J. Klafter, Subdiffusion of mixed origins: When ergodicity and nonergodicity coexist. Phys. Rev. E 81 (2010), 010101(R).10.1103/PhysRevE.81.010101Search in Google Scholar PubMed
[71] R. Metzler, Generalized Chapman-Kolmogorov equation: A unifying approach to the description of anomalous transport in external fields. Phys. Rev. E 62 (2000), 6233.10.1103/PhysRevE.62.6233Search in Google Scholar PubMed
[72] R. Metzler, E. Barkai and J. Klafter, Anomalous diffusion and relaxation close to thermal equilibrium: A fractional Fokker-Planck equation approach. Phys. Rev. Lett. 82 (1999), 3563.10.1103/PhysRevLett.82.3563Search in Google Scholar
[73] R. Metzler, E. Barkai and J. Klafter, Deriving fractional Fokker-Planck equations from a generalised master equation. Europhys. Lett. 46, No 4 (1999), 431-436.Search in Google Scholar
[74] R. Metzler and J. Klafter, Subdiffusive transport close to thermal equilibrium: From the Langevin equation to fractional diffusion. Phys. Rev. E 61 (2000), 6308. 10.1103/PhysRevE.61.6308Search in Google Scholar PubMed
[75] R. Metzler, J.-H. Jeon, A.G. Cherstvy and 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.Search in Google Scholar
[76] R. Metzler and J. Klafter, The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339 (2000), 1-77.Search in Google Scholar
[77] R. Metzler and J. Klafter, 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-R208.10.1088/0305-4470/37/31/R01Search in Google Scholar
[78] E.W. Montroll, Random walks on lattices. III. Calculation of firstpassage times with application to exciton trapping on photosynthetic units. J. Math. Phys. 10 (1969), 753.10.1063/1.1664902Search in Google Scholar
[79] S. Ott and J. Mann, An experimental investigation of the relative diffusion of particle pairs in three-dimensional turbulent flow. J. Fluid Mech. 422 (2000), 207-223.Search in Google Scholar
[80] V.V. Palyulin, A.V. Chechkin and R. Metzler, Lévy flights do not always optimize random blind search for sparse targets. Proc. Natl. Acad. Sci. USA 111 (2014), 2931-2936.Search in Google Scholar
[81] W. Pan, L. Filobelo, N.D.Q. Pham, O. Galkin, V.V. Uzunova and P.G. Vekilov, Viscoelasticity in homogeneous protein solutions. Phys. Rev. Lett. 102 (2009), 058101.10.1103/PhysRevLett.102.058101Search in Google Scholar PubMed
[82] J. Paneva-Konovska, Convergence of series in three parametric Mittag- Leffler functions. Math. Slovaca 64, No 1 (2014), 73-84; DOI: 10.2478/s12175-013-0188-0.10.2478/s12175-013-0188-0Search in Google Scholar
[83] I. Podlubny, Fractional Differential Equations. Academic Press, San Diego (1999).Search in Google Scholar
[84] T.R. Prabhakar, A singular equation with a generalized Mittag-Leffler function in the kernel. Yokohama Math. J. 19 (1971), 7-15.Search in Google Scholar
[85] A. Rebenshtok, S. Denisov, P. H¨anggi and E. Barkai, Non-normalizable densities in strong anomalous diffusion: Beyond the central limit theorem. Phys. Rev. Lett. 112 (2014), 110601.10.1103/PhysRevLett.112.110601Search in Google Scholar PubMed
[86] L.F. Richardson, Atmospheric diffusion shown on a distance-neighbour graph. Proc. Roy. Soc. A 110 (1926), 709-737.Search in Google Scholar
[87] D. Robert, T.H. Nguyen, F. Gallet and C. Wilhelm, In vivo determination of fluctuating forces during endosome trafficking using a combination of active and passive microrheology. PLoS ONE 4 (2010), e10046.10.1371/journal.pone.0010046Search in Google Scholar PubMed PubMed Central
[88] L.P. Sanders, M.A. Lomholt, L. Lizana, K. Fogelmark, R. Metzler and 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. 10.1088/1367-2630/16/11/113050Search in Google Scholar
[89] T. Sandev, A. Chechkin, N. Korabel, H. Kantz, I.M. Sokolov and R. Metzler, Distributed order diffusion equations and multifractality: models and solutions. Submitted.Search in Google Scholar
[90] T. Sandev, R. Metzler and Ž . Tomovski, Fractional diffusion equation with a generalized Riemann-Liouville time fractional derivative. J. Phys. A: Math. Theor. 44 (2011), 255203.10.1088/1751-8113/44/25/255203Search in Google Scholar
[91] T. Sandev, R. Metzler and Ž. Tomovski, Correlation functions for the fractional generalized Langevin equation in the presence of internal and external noise. J. Math. Phys. 55 (2014), 023301.10.1063/1.4863478Search in Google Scholar
[92] T. Sandev, I. Petreska and E.K. Lenzi, Time-dependent Schrödingerlike equation with nonlocal term. J. Math. Phys. 55 (2014), 092105.10.1063/1.4894059Search in Google Scholar
[93] T. Sandev and Ž. Tomovski, Langevin equation for a free particle driven by power law type of noises. Phys. Lett. A 378 (2014), 1-9.Search in Google Scholar
[94] T. Sandev, Ž. Tomovski and J.L.A. Dubbeldam, Generalized Langevin equation with a three parameter Mittag-Leffler noise. Physica A 390 (2011), 3627-3636.Search in Google Scholar
[95] R.K. Saxena, A.M. Mathai and H.J. Haubold, Unified fractional kinetic equation and a fractional diffusion equation. Astrophys. Space Sci. 290 (2004), 299-310.Search in Google Scholar
[96] M. Saxton, Wanted: A positive control for anomalous subdiffusion. Biophys. J. 103 (2012), 2411-2422.Search in Google Scholar
[97] M. Saxton, Single-particle tracking: the distribution of diffusion coefficients. Biophys. J. 72 (1997), 1744-1753.Search in Google Scholar
[98] H. Scher, G. Margolin, R. Metzler, J. Klafter and B. Berkowitz, The dynamical foundation of fractal stream chemistry: The origin of extremely long retention times. Geophys. Res. Lett. 29 (2002), 1061.10.1029/2001GL014123Search in Google Scholar
[99] H. Scher and E.W. Montroll, Anomalous transit-time dispersion in amorphous solids. Phys. Rev. B 12 (1975), 2455.10.1103/PhysRevB.12.2455Search in Google Scholar
[100] R. Schilling, R. Song and Z. Vondracek, Bernstein Functions. De Gruyter, Berlin (2010).Search in Google Scholar
[101] F.G. Schmitt and L. Seuront, Multifractal random walk in copepod behaviour. Physica A 301 (2001), 375-396.Search in Google Scholar
[102] M. Schubert, E. Preis, J.C. Blakesley, P. Pingel, U. Scherf and D. Neher, Mobility relaxation and electron trapping in a donor/acceptor copolymer. Phys. Rev. B 87 (2013), 024203.10.1103/PhysRevB.87.024203Search in Google Scholar
[103] L. Seuront, F.G. Schmitt, M.C. Brewer, J.R. Strickler and S. Souissi, From random walk to multifractal random walk in zooplankton swimming behaviour. Zoological Studies 43 (2004), 498-510.Search in Google Scholar
[104] I.M. Sokolov, Thermodynamics and fractional Fokker-Planck equations. Phys. Rev. E 63 (2001), 056111. 10.1103/PhysRevE.63.056111Search in Google Scholar PubMed
[105] I. M. Sokolov, A. V. Chechkin and J. Klafter, Distributed-order fractional kinetics. Acta Phys. Polon. B 35 (2004), 1323-1341.Search in Google Scholar
[106] I.M. Sokolov and J. Klafter, From diffusion to anomalous diffusion: A century after Einstein’s Brownian motion. Chaos 15 (2005), 26103.10.1063/1.1860472Search in Google Scholar PubMed
[107] T.H. Solomon, E.R. Weeks and H.L. Swinney, Observation of anomalous diffusion and Lévy flights in a two-dimensional rotating flow. Phys. Rev. Lett. 71 (1993), 3975.10.1103/PhysRevLett.71.3975Search in Google Scholar PubMed
[108] C.M. Song, T. Koren, P. Wang and A.-L. Barabási, Modelling the scaling properties of human mobility. Nature Phys. 6 (2010), 818-823.Search in Google Scholar
[109] M. Spanner, F. Höfling, G.E. Schröder-Turk, K. Mecke and T. Franosch, Anomalous transport of a tracer on percolating clusters. J. Phys. Cond. Mat. 23 (2011), 234120.10.1088/0953-8984/23/23/234120Search in Google Scholar PubMed
[110] A. Stanislavsky and K. Weron, Numerical scheme for calculating of the fractional two-power relaxation laws in time-domain of measurements. Comput. Phys. Commun. 183 (2012), 320-323.Search in Google Scholar
[111] A. Stanislavsky, K. Weron and J. Trzmiel, Subordination model of anomalous diffusion leading to the two-power-law relaxation responses. EPL 91 (2010), 40003.10.1209/0295-5075/91/40003Search in Google Scholar
[112] A. Stanislavsky, K. Weron and A. Weron, Diffusion and relaxation controlled by tempered α-stable processes. Phys. Rev. E 78 (2008), 051106.10.1103/PhysRevE.78.051106Search in Google Scholar PubMed
[113] A. Stanislavsky, K. Weron and A. Weron, Anomalous diffusion with transient subordinators: A link to compound relaxation laws. J. Chem. Phys. 140 (2014), 054113.10.1063/1.4863995Search in Google Scholar PubMed
[114] J. Szymanski and M. Weiss, Elucidating the origin of anomalous diffusion in crowded fluids. Phys. Rev. Lett. 103 (2009), 038102.10.1103/PhysRevLett.103.038102Search in Google Scholar PubMed
[115] S.M.A. Tabei, S. Burov, H.Y. Kim, A. Kuznetsov, T. Huynh, J. Jureller, L.H. Philipson, A.R. Dinner and N.F. Scherer, Intracellular transport of insulin granules is a subordinated random walk. Proc. Natl. Acad. Sci. USA 110 (2013), 4911-4916.Search in Google Scholar
[116] A.A. Tateishi, E.K. Lenzi, L.R. da Silva, H.V. Ribeiro, S. Picoli Jr. and R.S. Mendes, Different diffusive regimes, generalized Langevin and diffusion equations. Phys. Rev. E 85 (2012), 011147.10.1103/PhysRevE.85.011147Search in Google Scholar PubMed
[117] F. Thiel and I.M. Sokolov, Scaled Brownian motion as a mean-field model for continuous-time random walks. Phys. Rev. E 89 (2014), 012115.10.1103/PhysRevE.89.012115Search in Google Scholar PubMed
[118] Ž. Tomovski, T. Sandev, R. Metzler and J. Dubbeldam, Generalized space-time fractional diffusion equation with composite fractional time derivative. Physica A 391 (2012), 2527-2542. Search in Google Scholar
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