Anomalous transport: foundations and applications

Anomalous Transport......Page 4
Contents......Page 8
Preface......Page 18
List of Contributors......Page 22
1 In Memoriam: Radu Balescu......Page 28
1.1 Radu Balescu’s Abstract for the Conference on Anomalous Transport in Bad Honnef......Page 29
1.2 The Scientific Career of Radu Balescu by Boris Weyssow......Page 30
1.3 My Memory of Radu Balescu by Angelo Vulpiani......Page 32
1.4 My Memory of Radu Balescu by Francesco Mainardi......Page 33
1.5 In Memoriam: Radu Balescu by Raul Sánchez......Page 35
1.6 Remembering Radu Balescu by Diego del-Castillo-Negrete......Page 37
References......Page 38
Part I Fractional Calculus and Stochastic Theory......Page 40
Introduction to Part I......Page 42
2.1.1 Leibniz......Page 44
2.1.3 Paradoxa and Problems......Page 45
2.1.4 Liouville......Page 47
2.1.7 Riemann......Page 48
2.2.1 Fractional Integrals......Page 49
2.2.2 Fractional Derivatives......Page 58
2.2.3 Eigenfunctions......Page 71
2.3.2 Fractional Space......Page 73
2.3.3 Fractional Time......Page 75
2.3.4 Identification of α from Models......Page 80
Appendix A: Tables......Page 86
Appendix B: Function Spaces......Page 88
Appendix C: Distributions......Page 92
References......Page 96
3.1 St. Petersburg Paradox......Page 102
3.2 Holtsmark Distribution......Page 103
3.3 Activated Hopping......Page 104
3.4 Deterministic Examples of Long Tail Distributions......Page 105
3.5 RandomWalks and Master Equations......Page 107
3.6 RandomWalks and Upper Critical Dimensions......Page 109
3.7 Weierstrass Random Walk......Page 110
3.8 Fractal Time Random Walk......Page 112
3.9 Coupled Memory Random Walks: Diffusion or Telegraph Equation......Page 113
3.10 Random Walks: Coupled Memory Lévy Walks: Turbulent and Relativistic......Page 116
References......Page 118
4.1 Introduction......Page 120
4.2 An Outline of the Gnedenko-\rKovalenko Theory of Thinning......Page 122
4.3 The Continuous Time Random Walk (CTRW)......Page 125
4.4 Manipulations: Rescaling and Respeeding......Page 128
4.5 Power Laws and Asymptotic Universality of the Mittag-Leffler Waiting-Time Density......Page 130
4.6 Passage to the Diffusion Limit in Space......Page 131
4.7 The Time-Fractional Drift Process......Page 137
4.8 Conclusions......Page 140
Appendix A: The Time-Fractional Derivatives......Page 144
Appendix B: The Space-Fractional Derivatives......Page 146
Appendix C: The Mittag-Leffler Function......Page 150
References......Page 152
5 Introduction to the Theory of Lévy Flights......Page 156
5.1 Lévy Stable Distributions......Page 157
5.2 Underlying Random Walk Processes......Page 160
5.3 Space Fractional Fokker–Planck Equation......Page 162
5.4.1 First Passage Time and Leapover Properties......Page 164
5.4.2 Lévy Flights and the Method of Images......Page 167
5.5.2 Lévy Flights in an Harmonic Potential......Page 168
5.5.3 Lévy Flights in a Quartic Potential, 1 < \r\\alpha < 2......Page 169
5.5.4 Lévy Flights in a More General Potential Well......Page 171
5.5.5 Kramers Problem for Lévy Flights......Page 172
5.6.1 Langevin Description......Page 174
5.6.3 Space-Homogeneous Relaxation in Absence of External Field......Page 175
5.6.5 Relaxation of the Linear Oscillator [84]......Page 176
5.6.6 Relaxation in a Magnetized Plasma......Page 177
5.6.7 Damped Lévy Flights [89]......Page 179
5.7 Power-Law Truncated Lévy Flights......Page 180
5.8 Summary......Page 185
References......Page 186
6.1 Introduction......Page 190
6.2.1 Anomalous Transport of Tracers in Flows with Coherent Structures......Page 192
6.2.2 Continuous Time Random Walk Model and Fractional Diffusion......Page 197
6.3 Fractional Diffusion......Page 199
6.3.1 Fractional Derivatives......Page 200
6.3.2 Fractional Diffusion Equation: Green’s Function and Self-Similar Scaling......Page 204
6.3.3 Interplay of Regular and Fractional Diffusion......Page 206
6.3.4 Fractional Diffusion Equation for Truncated Lévy Processes......Page 208
6.3.5 Nonlocality and Uphill Transport......Page 212
6.3.6 Fractional Diffusion Models of Turbulent Transport......Page 217
6.4 Fractional Diffusion in Finite-Size Domains......Page 220
6.4.1 Finite-Size Domain Model......Page 221
6.4.2 Finite-Difference Numerical Integration Method......Page 222
6.4.3 Examples......Page 227
6.5 Summary and Conclusions......Page 235
References......Page 238
7.1 Introduction......Page 240
7.2 Normal Occupation Time Statistics......Page 242
7.2.1 Ergodicity for Bounded Normal Diffusion......Page 246
7.2.3 Occupation Times for Unbounded Normal Diffusion......Page 247
7.3 Anomalous Diffusion......Page 248
7.3.1 Derivation of Fractional Equation for Occupation Times......Page 249
7.3.2 Weak Ergodicity Breaking......Page 252
7.3.3 Example: Anomalous Diffusion in an Interval......Page 254
7.4 DeterministicWeak Ergodicity Breaking......Page 255
7.5 Discussion......Page 261
Appendix A......Page 264
References......Page 266
Part II Dynamical Systems and Deterministic Transport......Page 268
Introduction to Part II......Page 270
8.1 Introduction......Page 272
8.2 Transport and Thermodynamic Formalism......Page 273
8.3 The Periodic Orbits Approach......Page 276
8.4 One-Dimensional Transport: Kneading Determinant......Page 280
8.5 An Anomalous Example......Page 283
8.6 Probabilistic Approximations......Page 290
8.7 Conclusions......Page 292
References......Page 293
9.1 Introduction......Page 296
9.2 Continuous Time Random Walk Model......Page 297
9.3.1 Phase Space of Hamiltonian Systems......Page 299
9.3.2 Main Properties of Hamiltonian Chaos......Page 300
9.3.3 Connection to CTRW......Page 303
9.4.1 Incompressible Fluid Flows as Hamiltonian Systems......Page 305
9.4.2 Specific Model......Page 306
9.4.3 Transport Properties of Passive Tracers......Page 308
9.4.4 Effect of Noise Perturbation......Page 309
9.5 Discussion......Page 315
References......Page 316
10.1 Introduction......Page 320
10.2.1 Fourier and the Problem of Heat Transport in the Earth......Page 321
10.2.2 From the Old Kinetic Theory to Boltzmann......Page 322
10.2.3 The Harmonic Pathology and the Phonons......Page 324
10.2.4 The Fermi-Pasta-Ulam (FPU) Numerical Experiment\r......Page 325
10.3.1 Homogeneous Chain......Page 328
10.3.2 Disordered Chain......Page 331
10.4.1 Early Studies......Page 337
10.4.2 Green-Kubo\r Formalism......Page 338
10.4.4 The Strong-Chaos Regime......Page 339
10.4.5 Integrable Nonlinear Systems......Page 341
10.4.6 Universality......Page 342
10.4.7 Anomalous Diffusion......Page 344
References......Page 347
Part III Anomalous Transport in Disordered Systems......Page 350
Introduction to Part III......Page 352
11.1 Introduction......Page 354
11.2.1 Trapping-Induced Subdiffusion......Page 355
11.2.2 Two-Step Relaxation......Page 359
11.2.3 Superposition of Relaxation Times and Stretched Exponentials......Page 362
11.3.1 Superdiffusion and Lévy Flights......Page 365
11.3.2 Rearrangement-Induced Stress Fields in Elastic Media......Page 367
11.4 Conclusion......Page 370
References......Page 371
12.1 Introduction......Page 374
12.2 Model......Page 379
12.3 Results......Page 380
12.4 Summary......Page 391
References......Page 392
13.1 Introduction......Page 394
13.2 Subdiffusion Contexts and Modeling Approaches......Page 396
13.3 Target and Trapping Problem......Page 400
13.3.1 Target Problem......Page 401
13.3.2 Trapping Problem......Page 403
13.4.1 Annihilation and Coalescence Reactions......Page 405
13.4.2 Annihilation of Two Species, A + B ->\r0......Page 407
13.5 Reactions with Nonhomogeneous Distribution of Reactants......Page 408
13.5.1 Reaction-\rSubdiffusion Equations......Page 409
13.5.2 Reaction Fronts......Page 412
13.6 Reactants with Different (Sub)diffusion Exponents......Page 414
13.7 Finale......Page 419
References......Page 420
14.1 Motivation......Page 424
14.2 Anomalous Diffusion on Regular Sierpinski Carpets......Page 425
14.2.1 Regular Sierpinski Carpets......Page 426
14.2.2 Random Walks on Fractals......Page 428
14.2.3 Resistance Scaling Method......Page 430
14.2.4 Master Equation Approach......Page 432
14.2.5 Diffusion Simulation......Page 433
14.3 Anomalous Diffusion on Disordered Sierpinski Carpets......Page 434
14.3.1 Disordered Sierpinski Carpets......Page 435
14.3.2 Modeling Carpets and Calculating Diffusion Exponents......Page 436
14.3.3 Results and Discussion......Page 439
14.4 Fractional Differential Equation......Page 443
14.4.1 Sierpinski Gasket......Page 445
14.4.2 Koch Curve......Page 448
14.5 Summary......Page 451
References......Page 453
Part IV Applications to Complex Systems and Experimental Results......Page 456
Introduction to Part IV......Page 458
15.1 Introduction......Page 460
15.2 The Basic Idea......Page 461
15.3 Typical Distributions f(\\beta\r)......Page 462
15.4 Asymptotic Behavior for Large Energies......Page 464
15.5 Anomalous Diffusion in Superstatistical Systems......Page 466
15.6 From Time Series to Superstatistics......Page 469
15.7 Overview of Applications......Page 472
15.8 Lagrangian Turbulence......Page 473
15.9 Defect Turbulence......Page 475
15.10 Statistics of Cosmic Rays......Page 477
15.11 Statistics of Train Delays......Page 480
15.12 Conclusion and Outlook......Page 482
References......Page 483
16 Money Circulation Science \r-  Fractional Dynamics in Human Mobility......Page 486
16.2 Diffusive Dispersal and Reaction Kinetics......Page 487
16.2.2 A Word of Caution......Page 488
16.3 Long Distance Dispersal and Lévy Flights......Page 489
16.3.1 Lévy Flights......Page 490
16.4 Human Travel in the 21st Century......Page 491
16.5.1 The Lack of Scale in Money Movements......Page 493
16.6 Is That All?......Page 496
16.7 Scaling Analysis......Page 497
16.8 Scale Free Waiting Times......Page 498
16.9 Ambivalent Processes......Page 500
16.9.1 Scaling Relation......Page 501
16.9.2 The Limiting Function for Amibivalent Processes......Page 503
References......Page 509
17.1 General Introduction......Page 512
17.2 From Nano- to Milliseconds: Segment Diffusion in Polymer Melts......Page 514
17.2.1 Evaluation Theory for the Field-Cycling NMR Relaxometry Technique......Page 515
17.2.2 Results for Polymer Melts......Page 518
17.2.3 Discussion of the Results for Polymer Melts......Page 519
17.3.1 The Measuring Principles of Field-Gradient NMR Diffusometry......Page 522
17.3.2 Sub- and Superdiffusive Transport in Porous Glasses......Page 526
17.3.3 Discussion of the Results for Porous Glasses......Page 531
17.4.1 NMR Microscopy Diffusometry Techniques......Page 532
17.4.2 Diffusion of Solvents in Swollen Polymers......Page 536
17.4.3 Diffusion on Random-Site Percolation Clusters......Page 539
17.4.4 Discussion of the Results for Percolation Clusters......Page 541
17.5 General Discussion and Outlook......Page 542
References......Page 544
18.1 Introduction......Page 546
18.2.1 Zeolites......Page 547
18.2.2 Mesoporous Glass and Silica......Page 548
18.3.1 The Phenomenon and Measurement of Concentration......Page 549
18.3.2 Diffusion Measurement......Page 551
18.4.1 Single-File Diffusion......Page 553
18.4.2 Molecular Transport in Intersecting Channels......Page 554
18.5.1 Dynamics in Random Pore Spaces......Page 557
18.5.2 Molecular Dynamics in Channel Pores......Page 562
18.6 Conclusions......Page 566
References......Page 567
19.1 Introduction......Page 572
19.2 Thirty-Year Old Enigma about the Diffusion Rate of Membrane Molecules in the Plasma Membrane......Page 576
19.3 Macroscopic Diffusion Coefficients for Transmembrane Proteins are Suppressed by the Presence of the Membrane Skeleton......Page 578
19.4 Single-Molecule Tracking Revealed That Transmembrane Proteins Undergo Hop Diffusion......Page 579
19.5 Corralling Effects of the Membrane Skeleton for Transmembrane Proteins (the Membrane-Skeleton Fence Model)......Page 585
19.6 Phospholipids Also Undergo Hop Diffusion in the Plasma Membrane......Page 588
19.7 The Biological Significance of Oligomerization-Induced Trapping Based on the Membrane-Skeleton Fences and Pickets......Page 595
19.8 A Paradigm Shift of the Plasma Membrane Structure Concept is Necessary: From the Simple Two-dimensional Continuum Fluid Model to the Compartmentalized Fluid Model......Page 597
References......Page 599
Index......Page 602