Fractional dynamics : recent advances

✍ Scribed by J Klafter; S C Lim; Ralf Metzler

• Publisher: Singapore ; Hackensack, NJ : World Scientific
• Year: 2012
• Tongue: English
• Leaves: 530
• Category: Library

✦ Synopsis

This volume provides the latest developments in the field of fractional dynamics, which covers fractional (anomalous) transport phenomena, fractional statistical mechanics, fractional quantum mechanics and fractional quantum field theory. The contributors are selected based on their active and important contributions to their respective topics. This volume is the first of its kind that covers such a comprehensive range of topics in fractional dynamics. It will point out to advanced undergraduate and graduate students, and young researchers the possible directions of research in this subject.

In addition to those who intend to work in this field and those already in the field, this volume will also be useful for researchers not directly involved in the field, but want to know the current status and trends of development in this subject. This latter group includes theoretical chemists, mathematical biologists and engineers.

Contents:
Classical Systems:
Anomalous Diffusion and Fractional Transport Equations (R Metzler and J-H Jeon)
Stochastic Diffusion and Stable Noise-Induced Phenomena (B Dybiec and E Gudowska-Nowak)
Characteristic Times of Anomalous Diffusion in a Potential (W T Coffey, Y P Kalmykov and S V Titov)
Reactions in Subdiffusive Media and Associated Fractional Equations (S B Yuste, E Abad and K Lindenberg)
Natural and Modified Forms of Distributed-Order Fractional Diffusion Equations (A Chechkin, I M Sokolov and J Klafter)
Anomalous Transport in the Presence of Truncated Lévy Flights (D del-Castillo-Negrete)
Anomalous Diffusion: From Fractional Master Equations to Path Integrals (R Friedrich)
Fractional Feynman–Kac Equation for Anomalous Diffusion Functionals (S Carmi and E Barkai)
Foundations of Fractional Dynamics: A Short Account (R Hilfer)
Parametric Subordination in Fractional Diffusion Processes (R Gorenflo and F Mainardi)
Fractional Calculus, Anomalous Diffusion, and Probability (M M Meerschaert)
Fractional Langevin Equation (E Lutz)
Subdiffusive Dynamics in Washboard Potentials: Two Different Approaches and Different Univesality Classes (I Goychuk and P Hänggi)
Identification and Validation of Fractional Subdiffusion Dynamics (K Burnecki, M Magdziarz and A Weron)
A Class of CTRWs: Compound Fractional Poisson Processes (E Scalas)
Origin of Allometry Hypothesis (B J West and D West)
Quantum Systems:
Principles of Fractional Quantum Mechanics (N Laskin)
Two Examples of Fractional Quantum Dynamics (A Iomin)
Fractional Dynamics of Open Quantum Systems (V E Tarasov)
Casimir Effect Associated with Fractional Klein–Gordon Field (S C Lim and L P Teo)

Readership: Researchers in mathematical physics.

✦ Table of Contents

Contents......Page 8
Preface......Page 6
List of Contributors......Page 10
Classical Systems......Page 16
1. Anomalous Diffusion and Fractional Transport Equations R. Metzler and J.-H. Jeon......Page 18
1. Introduction......Page 19
2. Continuous Time Random Walk and Fractional Diffusion......Page 22
2.1. Physical view of CTRW......Page 24
3. Fractional Fokker–Planck–Smoluchowski Equation......Page 25
3.1. Subdiffusive case......Page 26
3.2. Subordination scheme for the subdiffusive case......Page 29
3.3. Levy flights in external potentials......Page 30
3.4.1. Subdiffusion......Page 31
3.4.2. Levy flights......Page 33
4. Randomness of Long Time Averages in Subdiffusive CTRW Processes......Page 35
Acknowledgments......Page 41
References......Page 42
1. Introduction......Page 48
2. White Non-Gaussian Noises......Page 51
2.1. Escape from finite intervals......Page 52
2.2. Markovian non-Gaussian Kramers problem......Page 56
3.1. Escape from finite intervals......Page 58
3.2. Non-Markovian Kramers problem......Page 61
4. Summary......Page 62
References......Page 63
1. Introduction......Page 66
2. Normal Diffusion......Page 70
3. Anomalous Diffusion......Page 77
4. Fractional Diffusion of a Particle in a Double Well Potential......Page 81
5. Concluding Remarks......Page 87
References......Page 88
4. Reactions in Subdiffusive Media and Associated Fractional Equations S. B. Yuste, E. Abad and K. Lindenberg......Page 92
1. Introduction......Page 93
2. Subdiffusion and Fractional Calculus......Page 95
3. Reactions Occurring at Spatially Fixed Locations......Page 96
3.1. Single-particle target problem......Page 97
3.2. Many-particle target problem......Page 100
3.3. Escape problems......Page 102
4. Reactions Occurring at Random Locations......Page 104
4.1. Mobile particles and traps......Page 105
4.2. Fractional diffusion-reaction equations......Page 109
4.3. Single-particle target problem with a reactivity field......Page 112
4.4. Reaction-subdiffusion equations and morphogen gradient formation......Page 113
4.4.2. Piecewise constant reactivity......Page 114
4.4.3. Exponentially decaying reactivity......Page 116
5. Final Remarks......Page 118
References......Page 119
5. Natural and Modified Forms of Distributed-Order Fractional Diffusion Equations A. Chechkin, I. M. Sokolov and J. Klafter......Page 122
1. Introduction......Page 123
2.1. Riemann–Liouville form......Page 125
3.1. Natural form......Page 126
4. Natural Form of Distributed-Order Time Fractional Diffusion Equation......Page 127
4.2. Generic case of double order equation: Decelerating subdiffusion......Page 128
4.3. Relation to CTRW......Page 129
4.4.1. Probability density function and mean square displacement......Page 130
4.4.2. Distributed-order fractional Fokker– Planck equation for superslow processes......Page 132
4.4.3. Relation to CTRW......Page 133
5. Modified Form of Distributed-Order Time Fractional Diffusion Equation......Page 134
5.2. Generic case of double order equation: Accelerating subdiffusion......Page 135
6. Natural Form of Distributed-Order Space Fractional Diffusion Equation......Page 137
7. Modified Form of Distributed-Order Space Fractional Diffusion Equation......Page 138
7.2. Fractional diffusion equation for a power law truncated Levy process......Page 139
References......Page 140
6. Anomalous Transport in the Presence of Truncated Levy Flights D. del-Castillo-Negrete......Page 144
1. Introduction......Page 145
2. Continuous Time Random Walk for General Levy Processes......Page 147
3. Green’s Function of Tempered Fractional Diffusion Equation......Page 152
3.1. Symmetric solution......Page 154
3.2. Asymmetric solution......Page 156
4. Transition from Super-diffusive to Sub-diffusive Transport......Page 157
5. Truncation Effects in Super-diffusive Front Propagation......Page 160
5.1. Diffusive case......Page 162
5.2. Fractional case......Page 163
5.3. Truncated case......Page 165
5.4. ν > λ......Page 166
5.5. ν < λ......Page 169
6. Conclusions......Page 170
References......Page 171
7. Anomalous Diffusion: From Fractional Master Equations to Path Integrals R. Friedrich......Page 174
1. Introduction......Page 175
2. The Master Equation......Page 176
3. Fractional Master Equations......Page 178
4. Anomalous Diffusion via Subordination......Page 181
5.1. Generalized Ornstein–Uhlenbeck process......Page 182
5.2. Advection of a particle......Page 183
5.4. Ballistic motion in a landscape with random scatterers......Page 184
6. Approximate Solutions of the Master Equation......Page 186
7. Subdiffusive Reaction-Diffusion Equations......Page 187
8. Path-Integral Representation......Page 188
8.1. Generalized Ornstein–Uhlenbeck processes......Page 192
9. Master Equation from Path Integrals......Page 193
10. Summary and Outlook......Page 194
Appendix. Derivation of Master Equation......Page 195
References......Page 197
1. Introduction......Page 200
2.1. Fractional Feynman–Kac equation......Page 203
2.2. A backward equation......Page 209
3.1. Occupation time in half-space......Page 210
3.2. First passage time......Page 212
3.3. The maximal displacement......Page 213
4.1. The fluctuations of the time-averaged position......Page 214
4.2. Simulation of trajectories......Page 218
5. Summary......Page 219
References......Page 220
1. Introduction......Page 224
3. Time Evolution of Observables......Page 226
4. Time Evolution of States......Page 228
6. Statement of the Problem......Page 229
7. Induced Measure Preserving Transformations......Page 230
8. Fractional Time Evolution......Page 231
9. Irreversibility......Page 233
11. Experimental Evidence......Page 235
12. Dissipative Systems......Page 238
References......Page 240
10. Parametric Subordination in Fractional Diffusion Processes R. Gorenflo and F. Mainardi......Page 244
1. Introduction......Page 245
2.1. The Fourier transform......Page 246
2.3. The auxiliary functions of Mittag-Leffer type......Page 247
2.4. The auxiliary functions of Wright type......Page 249
2.5. The Levy stable distributions......Page 252
3. The Space-Time Fractional Di.usion......Page 256
3.1. The Riesz–Feller space-fractional derivative......Page 257
3.3. The fundamental solution of the space-time fractional diffusion equation......Page 258
3.4. Alternative forms of the space-time fractional diffusion equation......Page 260
4.1. The analytical interpretation via operational time......Page 261
4.2. Stochastic interpretation......Page 263
4.3. Evolution equations for the densities of physical and operational time in mutual dependence......Page 265
4.4. The random walks......Page 266
5. Graphical Representations and Conclusions......Page 268
References......Page 275
1. Introduction......Page 280
2. Fractional Derivatives and Probability......Page 282
3. Fractional Derivatives in Time......Page 288
4. Vector Fractional Calculus......Page 290
5. Multi-Scaling Fractional Derivatives......Page 293
6. Simulation......Page 294
7. Tempered Fractional Derivatives......Page 296
References......Page 297
1. Introduction......Page 300
2. Langevin Equation......Page 302
2.1. Solution for the free particle......Page 303
2.3. Mean-square displacement......Page 305
3. Fractional Langevin Equation......Page 306
3.1. Memory effects......Page 307
3.2. Solution for the free particle......Page 309
3.3. Velocity correlation function and mean-square displacement......Page 310
4. Experimental Observation of Fractional Brownian Motion......Page 311
5. Aging......Page 313
6. Overdamped Regime and Critical Exponents......Page 314
7. Ergodicity......Page 317
References......Page 319
1. Introduction......Page 322
2. Free Subdiffusion and Constant Bias......Page 328
4.1. FFPE dynamics......Page 331
4.2. GLE dynamics in periodic potentials......Page 333
5. Summary and Conclusions......Page 339
Appendix A. Continuous Time Random Walk and Random Clock......Page 340
References......Page 342
1. Introduction......Page 346
2.2. Fractional Levy stable motion......Page 348
2.3. Fractional Fokker–Planck equation......Page 349
3.1. Kolmogorov–Smirnov statistic......Page 350
3.2. FIRT estimator......Page 352
4. Sample MSD......Page 354
5. Sample p-variation......Page 355
6. Statistical Validation......Page 358
7. The Case of Confined Systems......Page 361
References......Page 364
1. Introductory Notes......Page 368
2. Compound Poisson Process and Generalizations......Page 369
3. Compound Fractional Poisson Processes......Page 378
4. Limit Theorems......Page 385
References......Page 388
16. Origin of Allometry Hypothesis B. J. West and D. West......Page 390
1.1. Theoretical allometry......Page 391
1.2. Empirical allometry......Page 393
2. Random Allometry Coefficients......Page 395
3. Fractional Diffusion Equation......Page 397
4. Support Origin of Allometry Hypothesis......Page 401
5. Discussion......Page 402
References......Page 404
Quantum Systems......Page 406
17. Principles of Fractional Quantum Mechanics N. Laskin......Page 408
1. Introduction......Page 409
2.1. Quantum Riesz fractional derivative......Page 411
2.2. Hermiticity of the fractional Hamiltonian operator......Page 413
2.3. The parity conservation law for fractional quantum mechanics......Page 414
2.4. Current density......Page 415
2.5. The time-independent fractional Schrodinger equation......Page 417
3. Path Integral......Page 418
3.1.1. Fox H-function representation for a free particle kernel......Page 420
4.1.1. Scaling properties of the 1D fractional Schrodinger equation......Page 424
4.1.2. Exact solution......Page 425
4.1.3. The 3D generalization......Page 426
4.2. The infinite potential well......Page 428
4.3. Fractional Bohr atom......Page 430
4.4.1. Quarkonium and fractional oscillator......Page 432
4.4.2. Spectrum of the 1D fractional oscillator in semiclassical approximation......Page 433
5.1. Bound state in δ-potential well......Page 435
5.2. Linear potential field......Page 436
6.1. Density matrix......Page 437
6.1.1. Motion equation for the density matrix......Page 439
7. Conclusion......Page 440
References......Page 441
1. Introduction......Page 444
2. Mathematical Tools......Page 446
3. Fractional Time Schrodinger Equation......Page 447
4. Diffusion Comb Model......Page 449
5. Quantum Comb Model......Page 450
5.1. Comb FTSE......Page 451
5.2. Green’s function......Page 452
6. Fractional Kicked Rotor with Dissipation......Page 453
6.1. Fractional particle on a circle......Page 454
6.2. Fractional kicked rotor......Page 455
6.3. Wigner representation......Page 457
6.4. Spectrum......Page 459
7. Conclusion......Page 460
Acknowledgments......Page 461
References......Page 462
1. Introduction......Page 464
2. Quantum Operations and Superoperators......Page 466
3. Fractional Powers of Infinitesimal Generators......Page 467
4. Markovian Dynamics for Quantum Observables......Page 470
5. Fractional Quantum Dynamics for Observables......Page 471
6. Fractional Dynamical Semigroup......Page 473
7. Fractional Equation for Quantum States......Page 474
8. Fractional Hamiltonian Dynamics......Page 477
9. Fractional Dynamics of Open Quantum Oscillator......Page 482
10. Quantum Analogs of Derivatives of Integer Orders......Page 485
11. Quantum Analogs of Fractional Derivatives......Page 487
12. Fractional Generalization of Quantum Hamiltonian Systems......Page 490
13. Conclusion......Page 493
References......Page 494
1. Introduction......Page 498
2. Casimir Effect Associated with Fractional Klein–Gordon Field......Page 501
3. Topological Symmetry Breaking of Self-interacting Fractional Klein–Gordon Field......Page 505
4. Casimir Piston Associated with Massless Fractional Klein–Gordon Field......Page 508
5. Casimir Piston Associated with Massive Fractional Klein–Gordon Field......Page 512
6. Conclusion......Page 517
References......Page 518
Index......Page 522

✦ Subjects

Математика;Нелинейная динамика;