Part Ⅰ Fraction？l Continuous Models of Fractal Distributions1 Fractional Integration and Fractals 3
1.1 Riemann-Liouville fractional integrals 4
1.2 Liouville fractional integrals 6
1.3 Riesz fractional integrals 7
1.4 Metric and measure spaces 9
1.5 Hausdorff measure 10
1.6 Hausdorff dimension and fractals 14
1.7 Box-counting dimension 16
1.8 Mass dimension of fractal systems 19
1.9 Elementary models of fractal distributions 20
1.10 Functions and integrals on fractals 22
1.11 Properties of integrals on fractals 25
1.12 Integration over non-integer-dimensional space 26
1.13 Multi-variable integration on fractals 28
1.14 Mass distribution on fractals 29
1.15 Density of states in Euclidean space 31
1.16 Fractional integral and measure on the real axis 32
1.17 Fractional integral and mass on the real axis 34
1.18 Mass of fractal media 36
1.19 Electric charge of fractal distribution 38
1.20 Probability on fractals 39
1.21 Fractal distribution of particles 41
References 44
2 Hydrodynamics of Fractal Media 49
2.1 Introduction 49
2.2 Equation of balance of mass 50
2.3 Total time derivative of fractional integral 51
2.4 Equation of continuity for fractal media 54
2.5 Fractional integral equation of balance of momentum 55
2.6 Differential equations of balance of momentum 56
2.7 Fractional integral equation of balance of energy 57
2.8 Differential equation of balance of energy 58
2.9 Euler's equations for fractal media 60
2.10 Navier-Stokes equations for fractal media 62
2.11 Equilibrium equation for fractal media 63
2.12 Bernoulli integral for fractal media 64
2.13 Sound waves in fractal media 66
2.14 One-dimensional wave equation in fractal media 67
2.15 Conclusion 69
References 69
3 Fractal Rigid Body Dynamics 73
3.1 Introduction 73
3.2 Fractional equation for moment of inertia 74
3.3 Moment of inertia of fractal rigid body ball 76
3.4 Moment of inertia for fractal rigid body cylinder 78
3.5 Equations of motion for fractal rigid body 81
3.6 Pendulum with fractal rigid body 82
3.7 Fractal rigid body rolling down an inclined plane 84
3.8 Conclusion 85
References 86
4 Electrodynamics of Fractal Distributions of Charges and Fields 89
4.1 Introduction 89
4.2 Electric charge of fractal distribution 90
4.3 Electric current for fractal distribution 92
4.4 Gauss'theorem for fractal distribution 93
4.5 Stokes'theorem for fractal distribution 93
4.6 Charge conservation for fractal distribution 94
4.7 Coulomb's and Biot-Savart laws for fractal distribution 95
4.8 Gauss'law for fractal distribution 96
4.9 Ampere's law for fractal distribution 97
4.10 Integral Maxwell equations for fractal distribution 98
4.11 Fractal distribution as an effective medium 100
4.12 Electric multipole expansion for fractal distribution 101
4.13 Electric dipole moment of fractal distribution 103
4.14 Electric quadrupole moment of fractal distribution 104
4.15 Magnetohydrodynamics of fractal distribution 107
4.16 Stationary states in magnetohydrodynamics of fractal distributions 110
4.17 Conclusion 111
References 112
5 Ginzburg-Landau Equation for Fractal Media 115
5.1 Introduction 115
5.2 Fractional generalization of free energy functional 116
5.3 Ginzburg-Landau equation from free energy functional 117
5.4 Fractional equations from variational equation 118
5.5 Conclusion 121
References 121
6 Fokker-Planck Equation for Fractal Distributions of Probability 123
6.1 Introduction 123
6.2 Fractional equation for average values 124
6.3 Fractional Chapman-Kolmogorov equation 125
6.4 Fokker-Planck equation for fractal distribution 127
6.5 Stationary solutions of generalized Fokker-Planck equation 130
6.6 Conclusion 132
References 132
7 Statistical Mechanics of Fractal Phase Space Distributions 135
7.1 Introduction 135
7.2 Fractal distribution in phase space 136
7.3 Fractional phase volume for configuration space 136
7.4 Fractional phase volume for phase space 139
7.5 Fractional generalization of normalization condition 139
7.6 Continuity equation for fractal distribution in configuration space 141
7.7 Continuity equation for fractal distribution in phase space 142
7.8 Fractional average values for configuration space 144
7.9 Fractional average values for phase space 145
7.10 Generalized Liouville equation 146
7.11 Reduced distribution functions 147
7.12 Conclusion 148
References 150
Part Ⅱ Fractional Dynamics and Long-Range Interactions 150
8 Fractional Dynamics of Media with Long-Range Interaction 153
8.1 Introduction 153
8.2 Equations of lattice vibrations and dispersion law 155
8.3 Equations of motion for interacting particles 160
8.4 Transform operation for discrete models 162
8.5 Fourier series transform of equations of motion 163
8.6 Alpha-interaction of particles 166
8.7 Fractional spatial derivatives 170
8.8 Riesz fractional derivatives and integrals 174
8.9 Continuous limits of discrete equations 177
8.10 Linear nearest-neighbor interaction 180
8.11 Linear integer long-range alpha-interaction 181
8.12 Linear fractional long-range alpha-interaction 184
8.13 Fractional reaction-diffusion equation 187
8.14 Nonlinear long-range alpha-interaction 190
8.15 Fractional 3-dimensional lattice equation 194
8.16 Fractional derivatives from dispersion law 195
8.17 Fractal long-range interaction 198
8.18 Fractal dispersion law 203
8.19 Grünwald-Letnikov-Riesz long-range interaction 206
8.20 Conclusion 208
References 209
9 Fractional Ginzburg-Landau Equation 215
9.1 Introduction 215
9.2 Particular solution of fractional Ginzburg-Landau equation 216
9.3 Stability of plane-wave solution 220
9.4 Forced fractional equation 221
9.5 Conclusion 222
References 223
10 Psi-Series Approach to Fractional Equations 227
10.1 Introduction 227
10.2 Singular behavior of fractional equation 228
10.3 Resonance terms of fractional equation 229
10.4 Psi-series for fractional equation of rational order 230
10.5 Next to singular behavior 233
10.6 Conclusion 235
References 236
Part Ⅲ Fractional Spatial Dynamics 241
11 Fractional Vector Calculus 241
11.1 Introduction 241
11.2 Generalization of vector calculus 242
11.3 Fundamental theorem of fractional calculus 247
11.4 Fractional differential vector operators 250
11.5 Fractional integral vector operations 253
11.6 Fractional Green's formula 254
11.7 Fractional Stokes'formula 257
11.8 Fractional Gauss'formula 259
11.9 Conclusion 261
References 262
12 Fractional Exterior Calculus and Fractional Differential Forms 265
12.1 Introduction 265
12.2 Differential forms of integer order 266
12.3 Fractional exterior derivative 269
12.4 Fractional differential forms 274
12.5 Hodge star operator 279
12.6 Vector operations by differential forms 281
12.7 Fractional Maxwell's equations in terms of fractional forms 282
12.8 Caputo derivative in electrodynamics 284
12.9 Fractional nonlocal Maxwell's equations 285
12.10 Fractional waves 287
12.11 Conclusion 288
References 289
13 Fractional Dynamical Systems 293
13.1 Introduction 293
13.2 Fractional generalization of gradient systems 294
13.3 Examples of fractional gradient systems 301
13.4 Hamiltonian dynamical systems 305
13.5 Fractional generalization of Hamiltonian systems 307
13.6 Conclusion 311
References 312
14 Fractional Calculus of Variations in Dynamics 315
14.1 Introduction 315
14.2 Hamilton's equations and variations of integer order 315
14.3 Fractional variations and Hamilton's equations 317
14.4 Lagrange's equations and variations of integer order 319
14.5 Fractional variations and Lagrange's equations 321
14.6 Helmholtz conditions and non-Lagrangian equations 323
14.7 Fractional variations and non-Hamiltonian systems 326
14.8 Fractional stability 328
14.9 Conclusion 330
References 331
15 Fractional Statistical Mechanics 335
15.1 Introduction 335
15.2 Liouville equation with fractional derivatives 336
15.3 Bogolyubov equation with fractional derivatives 340
15.4 Vlasov equation with fractional derivatives 343
15.5 Fokker-Planck equation with fractional derivatives 345
15.6 Conclusion 349
References 350
Part Ⅳ Fractional Temporal Dynamics 357
16 Fractional Temporal Electrodynamics 357
16.1 Introduction 357
16.2 Universal response laws 358
16.3 Linear electrodynamics of medium 360
16.4 Fractional equations for laws of universal response 362
16.5 Fractional equations of the Curie-von Schweidler law 364
16.6 Fractional Gauss'laws for electric field 366
16.7 Universal fractional equation for electric field 369
16.8 Universal fractional equation for magnetic field 370
16.9 Fractional damping of magnetic field 372
16.10 Conclusion 373
References 374
17 Fractional Nonholonomic Dynamics 377
17.1 Introduction 377
17.2 Nonholonomic dynamics 378
17.3 Fractional temporal derivatives 385
17.4 Fractional dynamics with nonholonomic constraints 388
17.5 Constraints with fractional derivatives 394
17.6 Equations of motion with fractional nonholonomic constraints 396
17.7 Example of fractional nonholonomic constraints 398
17.8 Fractional conditional extremum 401
17.9 Hamilton's approach to fractional nonholonomic constraints 403
17.10 Conclusion 405
References 406
18 Fractional Dynamics and Discrete Maps with Memory 409
18.1 Introduction 409
18.2 Discrete maps without memory 410
18.3 Caputo and Riemann-Liouville fractional derivatives 415
18.4 Fractional derivative in the kicked term and discrete maps 418
18.5 Fractional derivative in the kicked term and dissipative discrete maps 422
18.6 Fractional equation with higher order derivatives and discrete map 425
18.7 Fractional generalization of universal map for 1＜α≤2 429
18.8 Fractional universal map for α＞2 434
18.9 Riemann-Liouville derivative and universal map with memory 436
18.10 Caputo fractional derivative and universal map with memory 441
18.11 Fractional kicked damped rotator map 445
18.12 Fractional dissipative standard map 447
18.13 Fractional Hénon map 449
18.14 Conclusion 450
References 451
Part Ⅴ Fractional Quantum Dynamics 457
19 Fractional Dynamics of Hamiltonian Quantum Systems 457
19.1 Introduction 457
19.2 Fractional power of derivative and Heisenberg equation 458
19.3 Properties of fractional dynamics 460
19.4 Fractional quantum dynamics of free particle 462
19.5 Fractional quantum dynamics of harmonic oscillator 463
19.6 Conclusion 464
References 465
20 Fractional Dynamics of Open Quantum Systems 467
20.1 Introduction 467
20.2 Fractional power of superoperator 468
20.3 Fractional equation for quantum observables 471
20.4 Fractional dynamical semigroup 473
20.5 Fractional equation for quantum states 475
20.6 Fractional non-Markovian quantum dynamics 477
20.7 Fractional equations for quantum oscillator with friction 478
20.8 Quantum self-reproducing and self-cloning 482
20.9 Conclusion 486
References 487
21 Quantum Analogs of Fractional Derivatives 491
21.1 Introduction 491
21.2 Weyl quantization of differential operators 492
21.3 Quantization of Riemann-Liouville fractional derivatives 494
21.4 Quantization of Liouville fractional derivative 496
21.5 Quantization of nondifferentiable functions 497
21.6 Conclusion 500
References 501
Index 503