Ultraslow diffusion is characterized by a logarithmic growth of the mean squared displacement (MSD) as a function of time. It occurs in complex arrangements of molecules, microbes, and many-body systems. This paper reviews mechanical models for ultraslow diffusion in heterogeneous media from both macroscopic and microscopic perspectives. Macroscopic models are typically formulated in terms of a diffusion equation that employs noninteger order derivatives (distributed order, structural, and comb models (CM)) or employs a diffusion coefficient that is a function of space or time. Microscopic models are usually based on the continuous time random walk (CTRW) theory, but use a weighted logarithmic function as the limiting formula of the waiting time density. The similarities and differences between these models are analyzed and compared with each other. The corresponding MSD in each case is tabulated and discussed from the perspectives of the underlying assumptions and of real-world applications in heterogeneous materials. It is noted that the CMs can be considered as a type of two-dimensional distributed order fractional derivative model (DFDM), and that the structural derivative models (SDMs) generalize the DFDMs. The heterogeneous diffusion process model (HDPM) with time-dependent diffusivity can be rewritten to a local structural derivative diffusion model mathematically. The ergodic properties, aging effect, and velocity autocorrelation for the ultraslow diffusion models are also briefly discussed.
References
1.
Metzler
, R.
, and Klafter
, J.
, 2000
, “The Random Walk's Guide to Anomalous Diffusion: A Fractional Dynamics Approach
,” Phys. Rep.
, 339
(1
), pp. 1
–77
.2.
Schumer
, R.
, Meerschaert
, M.
, and Baeumer
, B.
, 2009
, “Fractional Advection Dispersion Equations for Modeling Transport at the Earth Surface
,” J. Geophys. Res: Earth Surf.
, 114
(F4
), p. F00A07
.3.
Klages
, R.
, Radons
, G.
, and Sokolov
, I. M.
, 2008
, Anomalous Transport: Foundations and Applications
, Wiley
, New York
.4.
Bénichou
, O.
, and Oshanin
, G.
, 2002
, “Ultraslow Vacancy-Mediated Tracer Diffusion in Two Dimensions: The Einstein Relation Verified
,” Phys. Rev. E.
, 66
(3
), p. 031101
.5.
Rosenau
, P.
, 1995
, “Fast and Superfast Diffusion Processes
,” Phys. Rev. Lett.
, 74
(7
), pp. 1056
–1059
.6.
Kuipers
, N.
, and Beenackers
, A.
, 1993
, “Non-Fickian Diffusion With Chemical Reaction in Glassy Polymers With Swelling Induced by the Penetrant: A Mathematical Model
,” Chem. Eng. Sci.
, 48
(16
), pp. 2957
–2971
.7.
Metzler
, R.
, Barkai
, E.
, and Klafter
, J.
, 1999
, “Anomalous Diffusion and Relaxation Close to Thermal Equilibrium: A Fractional Fokker-Planck Equation Approach
,” Phys. Rev. Lett.
, 82
(18
), pp. 3563
–3567
.8.
Metzler
, R.
, Jeon
, J. H.
, Cherstvy
, A. G.
, and Barkai
, E.
, 2014
, “Anomalous Diffusion Models and Their Properties: Non-Stationarity, Nonergodicity, and Ageing at the Centenary of Single Particle Tracking
,” Phys. Chem. Chem. Phys.
, 16
(44
), pp. 24128
–24164
.9.
Sikora
, G.
, Burnecki
, K.
, and Wyłomańska
, A.
, 2017
, “Mean Squared Displacement Statistical Test for Fractional Brownian Motion
,” Phys. Rev E.
, 95
(3
), p. 032110
.10.
Uneyama
, T.
, Miyaguchi
, T.
, and Akimoto
, T.
, 2015
, “Fluctuation Analysis of Time-Averaged Mean-Square Displacement for the Langevin Equation With Time-Dependent and Fluctuating Diffusivity
,” Phys. Rev. E.
, 92
(3
), p. 032140
.11.
Trotta
, E.
, and Zimbardo
, G.
, 2015
, “A Numerical Study of Lévy Random Walks: Mean Square Displacement and Power-Law Propagators
,” J. Plasma. Phys.
, 81
, p. 325810108
.12.
Godec
, A.
, Chechkin
, A. V.
, Barkai
, E.
, Kantz
, H.
, and Metzler
, R.
, 2014
, “Localisation and Universal Fluctuations in Ultraslow Diffusion Processes
,” J. Phys. A-Math. Theor.
, 47
(49
), p. 492002
.13.
Einstein
, A.
, 1905
, “On the Movement of Small Particles Suspended in Stationary Liquids Required by Molecular-Kinetic Theory of Heat
,” Ann. Phys.
, 17
(8
), pp. 549
–560
.14.
El Masri
, D.
, Berthier
, L.
, and Cipelletti
, L.
, 2010
, “Subdiffusion and Intermittent Dynamic Fluctuations in the Aging Regime of Concentrated Hard Spheres
,” Phys. Rev. E.
, 82
(3 Pt 1
), p. 031503
.15.
Ingo
, C.
, Magin
, R.
, Colon-Perez
, L.
, Triplett
, W.
, and Mareci
, T.
, 2014
, “On Random Walks and Entropy in Diffusion-Weighted Magnetic Resonance Imaging Studies of Neural Tissue
,” Magnet. Reson. Med.
, 71
(2
), pp. 617
–627
.16.
Zhang
, Y.
, 2010
, “Moments for Tempered Fractional Advection-Diffusion Equations
,” J. Stat. Phys.
, 139
(5
), pp. 915
–939
.17.
Zhang
, Y.
, Benson
, D. A.
, Meerschaert
, M.
, LaBolle
, E.
, and Scheffler
, H.
, 2006
, “Random Walk Approximation of Fractional-Order Multiscaling Anomalous Diffusion
,” Phys. Rev. E.
, 74
(2
), p. 026706
.18.
Lomholt
, M.
, Lizana
, L.
, Metzler
, R.
, and Ambjornsson
, T.
, 2013
, “Microscopic Origin of the Logarithmic Time Evolution of Aging Processes in Complex Systems
,” Phys. Rev. Lett.
, 110
(20
), p. 208301
.19.
Sperl
, M.
, 2005
, “Nearly Logarithmic Decay in the Colloidal Hard-Sphere System
,” Phys. Rev. E.
, 71
(6 Pt 1
), p. 060401
.20.
Limpert
, E.
, Stahel
, W. A.
, and Abbt
, M.
, 2001
, “Log-Normal Distributions Across the Sciences: Keys and Clues
,” Bioscience
, 51
(5
), pp. 341
–352
.21.
Sandev
, T.
, Chechkin
, A. V.
, Korabel
, N.
, Kantz
, H.
, Sokolov
, I. M.
, and Metzler
, R.
, 2015
, “Distributed-Order Diffusion Equations and Multifractality: Models and Solutions
,” Phys. Rev. E.
, 92
(4
), p. 042117
.22.
Sinai
, Y. G.
, 1982
, “The Limit Behavior of a One-Dimensional Random Walk in a Random Medium
,” Theory Pobab. Appl.
, 27
(2
), pp. 247
–258
.23.
Stanley
, H. E.
, and Havlin
, S.
, 1987
, “Generalisation of the Sinai Anomalous Diffusion Law
,” J. Phys. A Gen. Phys.
, 20
(9
), p. L615
.24.
Sanders
, L. P.
, Lomholt
, M. A.
, Lizana
, L.
, Fogelmark
, K.
, Metzler
, R.
, and Ambjörnsson
, T.
, 2014
, “Severe Slowing-Down and Universality of the Dynamics in Disordered Interacting Many-Body Systems: Ageing and Ultraslow Diffusion
,” New J. Phys.
, 16
(11
), p. 113050
.25.
Brilliantov
, N.
, and Pöschel
, T.
, 2005
, “Kinetic Theory of Granular Gases
,” Phys. Today
, 58
(10
), pp. 84
–86
.26.
Drager
, J.
, and Klafter
, J.
, 2000
, “Strong Anomaly in Diffusion Generated by Iterated Maps
,” Phys. Rev. Lett.
, 84
(26 Pt 1
), pp. 5998
–6001
.27.
Cassi
, D.
, and Regina
, S.
, 1996
, “Random Walks on Bounded Structures
,” Phys. Rev. Lett.
, 76
(16
), pp. 2914
–2917
.28.
Vaknin
, A.
, Ovadyahu
, Z.
, and Pollak
, M.
, 2000
, “Aging Effects in an Anderson Insulator
,” Phys. Rev. Lett.
, 84
(15
), pp. 3402
–3405
.29.
Brauns
, E.
, Madaras
, M.
, Coleman
, R.
, Murphy
, C.
, and Berg
, M.
, 2002
, “Complex Local Dynamics in DNA on the Picosecond and Nanosecond Time Scales
,” Phys. Rev. Lett.
, 88
(15
), p. 158101
.30.
Matan
, K.
, Williams
, R.
, Witten
, T.
, and Nagel
, S.
, 2002
, “Crumpling a Thin Sheet
,” Phys. Rev. Lett.
, 88
(7
), p. 076101
.31.
Ben-David
, O.
, Rubinstein
, S.
, and Fineberg
, J.
, 2010
, “Slip-Stick and the Evolution of Frictional Strength
,” Nature
, 463
(7277
), p. 76
.32.
Richard
, P.
, Nicodemi
, M.
, Delannay
, R.
, Ribière
, P.
, and Bideau
, D.
, 2005
, “Slow Relaxation and Compaction of Granular Systems
,” Nat. Mater.
, 4
(2
), pp. 121
–128
.33.
Boettcher
, S.
, and Sibani
, P.
, 2011
, “Aging in Dense Colloids as Diffusion in the Logarithm of Time
,” J. Phys. Cond. Matt.
, 23
(6
), p. 065103
.34.
Watamabe
, H.
, 2018
, “Empirical Observations of Ultraslow Diffusion Driven by the Fractional Dynamics in Languages
,” Phys. Rev. E.
, 98
(1
), p. 012308
.35.
Song
, C.
, Koren
, T.
, Wang
, P.
, and Barabasi
, A.
, 2010
, “Modelling the Scaling Properties of Human Mobility
,” Nat. Phys.
, 6
(10
), pp. 818
–823
.36.
Chen
, W.
, Liang
, Y.
, and Hei
, X.
, 2016
, “Structural Derivative Based on Inverse Mittag-Leffler Function for Modeling Ultraslow Diffusion
,” Fract. Calc. Appl. Anal.
, 19
(5
), pp. 1316
–1346
.37.
Hilfer
, R.
, and Seybold
, H.
, 2006
, “Computation of the Generalized Mittag-Leffler Function and Its Inverse in the Complex Plane
,” Integr. Transforms Spec. Funct.
, 17
(9
), pp. 637
–652
.38.
Hardy
, G.
, 1928
, “Gösta Mittag-Leffler
,” J. Lond. Math. Soc.
, 3
(2
), pp. 156
–160
.39.
Hiramoto-Yamaki
, N.
, Tanaka
, K. A. K.
, Suzuki
, K. G. N.
, Hirosawa
, K. M.
, Miyahara
, M. S. H.
, Kalay
, Z.
, Tanaka
, K.
, Kasai
, R. S.
, Kusumi
, A.
, and Fujiwara
, T. K.
, 2014
, “Ultrafast Diffusion of a Fluorescent Cholesterol Analog in Compartmentalized Plasma Membranes
,” Traffic
, 15
(6
), pp. 583
–612
.40.
Bouchaud
, J. P.
, and Georges
, A.
, 1990
, “Anomalous Diffusion in Disordered Media: Statistical Mechanisms, Models and Physical Applications
,” Phys. Rep.
, 195
(4–5
), pp. 127
–293
.41.
Piryatinska
, A.
, Saichev
, A.
, and Woyczynski
, W.
, 2005
, “Models of Anomalous Diffusion: The Subdiffusive Case
,” Phys. A
, 349
(3–4
), pp. 375
–420
.42.
Meroz
, Y.
, and Sokolov
, I. M.
, 2015
, “A Toolbox for Determining Subdiffusive Mechanisms
,” Phys. Rep.
, 573
, pp. 1
–29
.43.
Eliazar
, I. I.
, and Shlesinger
, M. F.
, 2013
, “Fractional Motions
,” Phys. Rep.
, 527
(2
), pp. 101
–129
.44.
Sokolov
, I. M.
, 2012
, “Models of Anomalous Diffusion in Crowded Environments
,” Soft Matter
, 8
(35
), pp. 9043
–9052
.45.
Zoia
, A.
, 2008
, “Continuous-Time Random-Walk Approach to Normal and Anomalous Reaction-Diffusion Processes
,” Phys. Rev. E.
, 77
(4
), p. 041115
.46.
Cortis
, A.
, and Knudby
, C.
, 2006
, “A Continuous Time Random Walk Approach to Transient Flow in Heterogeneous Porous Media
,” Water Resour. Res.
, 42
(10
), p. W10201
.47.
Le Vot
, F.
, Abad
, E.
, and Yuste
, S.
, 2017
, “Continuous-Time Random-Walk Model for Anomalous Diffusion in Expanding Media
,” Phys. Rev. E.
, 96
(3–1
), p. 032117
.48.
Chen
, W.
, Sun
, H.
, Zhang
, X.
, and Korošak
, D.
, 2010
, “Anomalous Diffusion Modeling by Fractal and Fractional Derivatives
,” Comput. Math. Appl.
, 59
(5
), pp. 1754
–1758
.49.
Benson
, D.
, Tadjeran
, C.
, Meerschaert
, M.
, Farnham
, I.
, and Pohll
, G.
, 2004
, “Radial Fractional-Order Dispersion Through Fractured Rock
,” Water Resour. Res.
, 40
(12
), pp. 87
–87
.50.
Liang
, Y.
, Chen
, W.
, Akpa
, B.
, Neuberger
, T.
, Webb
, A.
, and Magin
, R.
, 2017
, “Using Spectral and Cumulative Spectral Entropy to Classify Anomalous Diffusion in Sephadex™ Gels
,” Comput. Math. Appl.
, 73
(5
), pp. 765
–774
.51.
Chen
, W.
, 2006
, “Time-Space Fabric Underlying Anomalous Diffusion
,” Chaos Soliton. Fract.
, 28
(4
), pp. 923
–929
.52.
Liang
, Y.
, Ye
, A.
, Chen
, W.
, Gatto
, R.
, Colon-Perez
, L.
, Mareci
, T.
, and Magin
, R.
, 2016
, “A Fractal Derivative Model for the Characterization of Anomalous Diffusion in Magnetic Resonance Imaging
,” Commun. Nonlinear Sci.
, 39
, pp. 529
–537
.53.
Chen
, W.
, Wang
, F.
, Zheng
, B.
, and Cai
, W.
, 2017
, “Non-Euclidean Distance Fundamental Solution of Hausdorff Derivative Partial Differential Equations
,” Eng. Anal. Bound. Elem.
, 84
, pp. 213
–219
.54.
Wei
, S.
, Chen
, W.
, and Hon
, Y.
, 2016
, “Characterizing Time Dependent Anomalous Diffusion Process: A Survey on Fractional Derivative and Nonlinear Models
,” Physica A.
, 462
, pp. 1244
–1251
.55.
Sun
, Y.
, Liang
, M.
, and Chang
, T.
, 2012
, “Time/Depth Dependent Diffusion and Chemical Reaction Model of Chloride Transportation in Concrete
,” Appl. Math. Model.
, 36
(3
), pp. 1114
–1122
.56.
Su
, N.
, Sander
, G. C.
, Liu
, F.
, Anh
, V.
, and Barry
, D. A.
, 2005
, “Similarity Solutions for Solute Transport in Fractal Porous Media Using a Time-and-Scale-Dependent Dispersivity
,” Appl. Math. Model.
, 29
(9
), pp. 852
–870
.57.
Chechkin
, A. V.
, Gorenflo
, R.
, and Sokolov
, I. M.
, 2002
, “Retarding Subdiffusion and Accelerating Superdiffusion Governed by Distributed-Order Fractional Diffusion Equations
,” Phys. Rev. E.
, 66
(4
), p. 046129
.58.
Eab
, C.
, and Lim
, S.
, 2011
, “Fractional Langevin Equations of Distributed Order
,” Phys. Rev. E.
, 83
(3 Pt 1
), p. 031136
.59.
Sokolov
, I. M.
, Chechkin
, A. V.
, and Klafter
, J.
, 2004
, “Distributed-Order Fractional Kinetics
,” Acta Phys. Pol. B
, 35
(4
), pp. 1323
–1341
.http://www.actaphys.uj.edu.pl/fulltext?series=Reg&vol=35&page=132360.
Kochubei
, A.
, 2008
, “Distributed Order Calculus and Equations of Ultraslow Diffusion
,” J. Math. Anal. Appl.
, 340
(1
), pp. 252
–281
.61.
Mainardi
, F.
, Mura
, A.
, Pagnini
, G.
, and Gorenflo
, R.
, 2008
, “Time-Fractional Diffusion of Distributed Order
,” J. Vib. Control.
, 14
(9–10
), pp. 1267
–1290
.62.
Chechkin
, A. V.
, Gorenflo
, R.
, and Sokolov
, I. M.
, 2003
, “Distributed Order Time Fractional Diffusion Equation
,” Fract. Calc. Appl. Anal.
, 6
(3
), pp. 259
–279
.http://www.diogenes.bg/fcaa/volume6/fcaa63/achechkin.pdf63.
Su
, N.
, Nelson
, P.
, and Connor
, S.
, 2015
, “The Distributed-Order Fractional Diffusion-Wave Equation of Groundwater Flow: Theory and Application to Pumping and Slug Tests
,” J. Hydrol.
, 529
(3
), pp. 1262
–1273
.64.
Chechkin
, A. V.
, Klafter
, J.
, and Sokolov
, I. M.
, 2003
, “Fractional Fokker-Planck Equation for Ultraslow Kinetics
,” EPL.
, 63
(3
), pp. 326
–332
.65.
Yang
, X.
, Liang
, Y.
, and Chen
, W.
, 2018
, “A Local Structural Derivative PDE Model for Ultraslow Creep
,” Comput. Math. Appl.
, 76
(7
), pp. 1713
–1718
.66.
Liang
, Y.
, and Chen
, W.
, 2018
, “A Non-Local Structural Derivative Model for Characterization of Ultraslow Diffusion in Dense Colloids
,” Commun. Nonlinear Sci.
, 56
, pp. 131
–137
.67.
Chen
, W.
, and Liang
, Y.
, 2017
, “New Methodologies in Fractional and Fractal Derivatives Modeling
,” Chaos Soliton. Fract.
, 102
, pp. 72
–77
.68.
Xu
, W.
, Chen
, W.
, Liang
, Y.
, and Weberszpil
, J.
, 2017
, “A Spatial Structural Derivative Model for Ultraslow Diffusion
,” Therm. Sci.
, 21
(Suppl. 1
), pp. 121
–127
.69.
Liang
, Y.
, 2018
, “Diffusion Entropy Method for Ultraslow Diffusion Using Inverse Mittag-Leffler Function
,” Fract. Calc. Appl. Anal.
, 21
(1
), pp. 104
–117
.70.
Dzhanoev
, A.
, and Sokolov
, I. M.
, 2018
, “The Effect of the Junction Model on the Anomalous Diffusion in the 3D Comb Structure
,” Chaos Solitons. Fract.
, 106
, pp. 330
–336
.71.
Sandev
, T.
, Iomin
, A.
, Kantz
, H.
, Metzler
, R.
, and Chechkin
, A. V.
, 2016
, “Comb Model With Slow and Ultraslow Diffusion
,” Math. Model. Nat. Phenom
, 11
(3
), pp. 18
–33
.72.
Havlin
, S.
, Kiefer
, J.
, and Weiss
, G. H.
, 1987
, “Anomalous Diffusion on a Random Comblike Structure
,” Phys. Rev. A.
, 36
(3
), pp. 1403
–1408
.73.
Iomin
, A.
, and Méndez
, V.
, 2016
, “Does Ultra-Slow Diffusion Survive in a Three Dimensional Cylindrical Comb
,” Chaos Solitons Fractals
, 82
, pp. 142
–147
.74.
Cherstvy
, A. G.
, and Metzler
, R.
, 2013
, “Population Splitting, Trapping, and Non-Ergodicity in Heterogeneous Diffusion Processes
,” Phys. Chem. Chem. Phys.
, 15
(46
), pp. 20220
–20235
.75.
Cherstvy
, A. G.
, and Metzler
, R.
, 2014
, “Non-Ergodicity, Fluctuations, and Criticality in Heterogeneous Diffusion Processes
,” Phys. Rev. E.
, 90
(1
), p. 012134
.76.
Cherstvy
, A. G.
, Chechkin
, A. V.
, and Metzler
, R.
, 2014
, “Particle Invasion, Survival, and Non-Ergodicity in 2D Diffusion, Processes With Space-Dependent Diffusivity
,” Soft Matter
, 10
(10
), pp. 1591
–1601
.77.
Bodrova
, A.
, Chechkin
, A. V.
, Cherstvy
, A. G.
, Safdari
, H.
, Sokolov
, I. M.
, and Metzler
, R.
, 2016
, “Underdamped Scaled Brownian Motion: (Non-)Existence of the Overdamped Limit in Anomalous Diffusion
,” Sci. Rep.
, 6
, p. 30520
.78.
Jeon
, J. H.
, Chechkin
, A. V.
, and Metzler
, R.
, 2014
, “Scaled Brownian Motion: A Paradoxical Process With a Time Dependent Diffusivity for the Description of Anomalous Diffusion
,” Phys. Chem. Chem. Phys.
, 16
(30
), pp. 15811
–15817
.79.
Bodrova
, A.
, Chechkin
, A. V.
, Cherstvy
, A. G.
, and Metzler
, R.
, 2015
, “Ultraslow Scaled Brownian Motion
,” New J. Phys.
, 17
(6
), p. 063038
.80.
Havlin
, S.
, and Weiss
, G. H.
, 1990
, “A New Class of Long-Tailed Pausing Time Densities for the CTRW
,” J. Stat. Phys.
, 58
(5–6
), pp. 1267
–1273
.81.
Meerschaert
, M.
, and Scheffler
, H.
, 2005
, “Limit Theorems for Continuous Time Random Walks With Slowly Varying Waiting Times
,” Stat. Probab. Lett.
, 71
(1
), pp. 15
–22
.82.
Meerschaert
, M.
, and Scheffler
, H.
, 2006
, “Stochastic Model for Ultraslow Diffusion
,” Stoch. Proc. Appl.
, 116
(9
), pp. 1215
–1235
.83.
Denisov
, S. I.
, and Kantz
, H.
, 2011
, “Continuous-Time Random Walk With a Super-heavy-Tailed Distribution of Waiting Times
,” Phys. Rev. E.
, 83
(4 Pt 1
), p. 041132
.84.
Denisov
, S.
, Bystrik
, Y.
, and Kantz
, H.
, 2013
, “Limiting Distributions of Continuous-Time Random Walks With Superheavy-Tailed Waiting Times
,” Phys. Rev. E.
, 87
(2
), p. 022117
.85.
Denisov
, S.
, and Kantz
, H.
, 2010
, “Continuous-Time Random Walk Theory of Superslow Diffusion
,” EPL
, 92
(3
), p. 30001
.86.
Denisov
, S.
, Yuste
, S.
, Bystrik
, Y.
, Kantz
, H.
, and Lindenberg
, K.
, 2011
, “Asymptotic Solutions of Decoupled Continuous-Time Random Walks With Superheavy-Tailed Waiting Time and Heavy-Tailed Jump Length Distributions
,” Phys. Rev. E.
, 84
(6
), p. 061143
.87.
Liang
, Y.
, and Chen
, W.
, 2018
, “Continuous Time Random Walk Model With Asymptotical Probability Density of Waiting Times Via Inverse Mittag-Leffler Function
,” Commun. Nonlinear Sci.
, 57
, pp. 439
–448
.88.
Huang
, J.
, and Zhao
, H.
, 2017
, “Ultraslow Diffusion and Weak Ergodicity Breaking in Right Triangular Billiards
,” Phys. Rev. E.
, 95
(3
), p. 032209
.89.
Burov
, S.
, Jeon
, J. H.
, Metzler
, R.
, and Barkai
, E.
, 2011
, “Single Particle Tracking in Systems Showing Anomalous Diffusion: The Role of Weak Ergodicity Breaking
,” Phys. Chem. Chem. Phys.
, 13
(5
), pp. 1800
–1812
.90.
Weber
, S.
, Spakowitz
, A.
, and Theriot
, J.
, 2010
, “Bacterial Chromosomal Loci Move Subdiffusively Through a Viscoelastic Cytoplasm
,” Phys. Rev. Lett.
, 104
(23
), p. 238102
.91.
Safdari
, H.
, Cherstvy
, A. G.
, Chechkin
, A. V.
, Bodrova
, A.
, and Metzler
, R.
, 2017
, “Aging Underdamped Scaled Brownian Motion: Ensemble- and Time-Averaged Particle Displacements, Nonergodicity, and the Failure of the Overdamping Approximation
,” Phys. Rev. E.
, 95
(1
), p. 012120
.92.
Chechkin
, A. V.
, Kantz
, H.
, and Metzler
, R.
, 2017
, “Ageing Effects in Ultraslow Continuous Time Random Walks
,” Eur. Phys. J. B.
, 90
(11
), p. 205
.93.
Lutz
, E.
, 2001
, “Fractional Langevin Equation
,” Phys. Rev. E
, 64
(5 Pt 1
), p. 051106
.94.
Heymans
, N.
, and Podlubny
, I.
, 2006
, “Physical Interpretation of Initial Conditions for Fractional Differential Equations With Riemann-Liouville Fractional Derivatives
,” Rheol. Acta.
, 45
(5
), pp. 765
–771
.95.
Caputo
, M.
, 1959
, “Conformal Projection of an Ellipsoid of Revolution When the Scale Factor and Its Normal Derivative Are Assigned on a Geodesic Line of the Ellipsoid
,” J. Geophys. Res.
, 64
(11
), pp. 1867
–1873
.96.
Abramowitz
, M.
, and Stegun
, I.
, 1970
, Handbook of Mathematical Functions
, Dover
, New York
.97.
Mainardi
, F.
, and Pagnini
, G.
, 2003
, “The Wright Functions as Solutions of the Time-Fractional Diffusion Equation
,” Appl. Math. Comput.
, 141
(1
), pp. 51
–62
.98.
99.
Liang
, Y.
, Chen
, W.
, and Cai
, W.
, 2019
, Hausdorff Calculus: Applications to Fractal Systems
, De Gruyter
, Berlin
.100.
Zeng
, C.
, and Chen
, Y.
, 2015
, “Global Padé Approximations of the Generalized Mittag-Leffler Function and Its Inverse
,” Fract. Calc. Appl. Anal.
, 18
(6
), pp. 1492
–1506
.101.
Ingo
, C.
, Barrick
, T.
, Webb
, A.
, and Ronen
, I.
, 2017
, “Accurate Padé Global Approximations for the Mittag-Leffler Function, Its Inverse, and Its Partial Derivatives to Efficiently Compute Convergent Power Series
,” Int. J. Appl. Comput. Math.
, 3
(2
), pp. 347
–362
.102.
Gorenflo
, R.
, Kilbas
, A.
, Mainardi
, F.
, and Rogosin
, S.
, 2014
, Mittag-Leffler Functions, Related Topics and Applications
, Springer-Verlag
, Berlin
.103.
Su
, X.
, Chen
, W.
, Xu
, W.
, and Liang
, Y.
, 2018
, “Non-Local Structural Derivative Maxwell Model for Characterizing Ultra-Slow Rheology in Concrete
,” Constr. Build. Mater.
, 190
, pp. 342
–348
.104.
Kirane
, M.
, and Torebek
, B.
, 2018
, “Extremum Principle for the Hadamard Derivatives and Its Application to Nonlinear Fractional Partial Differential Equations
,” arXiv: 1806.04886.105.
Sandev
, T.
, Chechkin
, A. V.
, Kantz
, H.
, and Metzler
, R.
, 2015
, “Diffusion and Fokker-Planck-Smoluchowski Equations With Generalized Memory Kernel
,” Fract. Calc. Appl. Anal.
, 18
(4
), pp. 1006
–1038
.106.
Kochubei
, A.
, 2011
, “General Fractional Calculus, Evolution Equations, and Renewal Processes
,” Integr. Equ. Oper. Theory
, 71
(4
), pp. 583
–600
.107.
Cherstvy
, A. G.
, Chechkin
, A. V.
, and Metzler
, R.
, 2013
, “Anomalous Diffusion and Ergodicity Breaking in Heterogeneous Diffusion Processes
,” New J. Phys.
, 15
(8
), p. 083039
.108.
Safdari
, H.
, Cherstvy
, A. G.
, Chechkin
, A. V.
, Thiel
, F.
, Sokolov
, I. M.
, and Metzler
, R.
, 2015
, “Quantifying the Non-Ergodicity of Scaled Brownian Motion
,” J. Phys. A.
, 48
(37
), p. 375002
.109.
Safdari
, H.
, Chechkin
, A. V.
, Jafari
, G.
, and Metzler
, R.
, 2015
, “Aging Scaled Brownian Motion
,” Phys. Rev. E.
, 91
(4
), p. 042107
.110.
Montroll
, E.
, and Weiss
, G. H.
, 1965
, “Random Walks on Lattices II
,” J. Math. Phys.
, 6
(2
), pp. 167
–181
.111.
Leonenko
, N.
, and Olenko
, A.
, 2013
, “Tauberian and Abelian Theorems for Long-Range Dependent Random Fields
,” Comput. Appl.
, 15
(4
), pp. 715
–742
.112.
Kesten
, H.
, 1986
, “The Limit Distribution of Sinai's Random Walk in Random Environment
,” Phys. A
, 138
(1–2
), pp. 299
–309
.113.
Klemm
, A.
, Müller
, H.
, and Kimmich
, R.
, 1999
, “Evaluation of Fractal Parameters of percolation model Objects and Natural Porous Media by Means of NMR Microscopy
,” Phys. A
, 266
(1–4
), pp. 242
–246
.114.
Weeks
, E.
, and Weitz
, D.
, 2002
, “Subdiffusion and the Cage Effect Studied Near the Colloidal Glass Transition
,” Chem. Phys.
, 284
(1–2
), pp. 361
–367
.115.
Courtland
, R.
, and Weeks
, E.
, 2003
, “Direct Visualization of Aging in Colloidal Glasses
,” J. Phys. Condens. Matter
, 15
(1
), p. S359
.116.
Bel
, G.
, and Barkai
, E.
, 2006
, “Random Walk to a Nonergodic Equilibrium Concept
,” Phys. Rev. E.
, 73
(1 Pt 2
), p. 016125
.117.
Evangelista
, L. R.
, and Lensi
, E. K.
, 2018
, Fractional Diffusion Equations and Anomalous Diffusion
, Cambridge University Press
, Cambridge, UK
.Copyright © 2019 by ASME
You do not currently have access to this content.
