分数维动力学 国内英文版PDF|Epub|txt|kindle电子书版本网盘下载

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Part Ⅰ Fraction？l Continuous Models of Fractal Distributions1 Fractional Integration and Fractals3

1.1 Riemann-Liouville fractional integrals4

1.2 Liouville fractional integrals6

1.3 Riesz fractional integrals7

1.4 Metric and measure spaces9

1.5 Hausdorff measure10

1.6 Hausdorff dimension and fractals14

1.7 Box-counting dimension16

1.8 Mass dimension of fractal systems19

1.9 Elementary models of fractal distributions20

1.10 Functions and integrals on fractals22

1.11 Properties of integrals on fractals25

1.12 Integration over non-integer-dimensional space26

1.13 Multi-variable integration on fractals28

1.14 Mass distribution on fractals29

1.15 Density of states in Euclidean space31

1.16 Fractional integral and measure on the real axis32

1.17 Fractional integral and mass on the real axis34

1.18 Mass of fractal media36

1.19 Electric charge of fractal distribution38

1.20 Probability on fractals39

1.21 Fractal distribution of particles41

References44

2 Hydrodynamics of Fractal Media49

2.1 Introduction49

2.2 Equation of balance of mass50

2.3 Total time derivative of fractional integral51

2.4 Equation of continuity for fractal media54

2.5 Fractional integral equation of balance of momentum55

2.6 Differential equations of balance of momentum56

2.7 Fractional integral equation of balance of energy57

2.8 Differential equation of balance of energy58

2.9 Euler's equations for fractal media60

2.10 Navier-Stokes equations for fractal media62

2.11 Equilibrium equation for fractal media63

2.12 Bernoulli integral for fractal media64

2.13 Sound waves in fractal media66

2.14 One-dimensional wave equation in fractal media67

2.15 Conclusion69

References69

3 Fractal Rigid Body Dynamics73

3.1 Introduction73

3.2 Fractional equation for moment of inertia74

3.3 Moment of inertia of fractal rigid body ball76

3.4 Moment of inertia for fractal rigid body cylinder78

3.5 Equations of motion for fractal rigid body81

3.6 Pendulum with fractal rigid body82

3.7 Fractal rigid body rolling down an inclined plane84

3.8 Conclusion85

References86

4 Electrodynamics of Fractal Distributions of Charges and Fields89

4.1 Introduction89

4.2 Electric charge of fractal distribution90

4.3 Electric current for fractal distribution92

4.4 Gauss'theorem for fractal distribution93

4.5 Stokes'theorem for fractal distribution93

4.6 Charge conservation for fractal distribution94

4.7 Coulomb's and Biot-Savart laws for fractal distribution95

4.8 Gauss'law for fractal distribution96

4.9 Ampere's law for fractal distribution97

4.10 Integral Maxwell equations for fractal distribution98

4.11 Fractal distribution as an effective medium100

4.12 Electric multipole expansion for fractal distribution101

4.13 Electric dipole moment of fractal distribution103

4.14 Electric quadrupole moment of fractal distribution104

4.15 Magnetohydrodynamics of fractal distribution107

4.16 Stationary states in magnetohydrodynamics of fractal distributions110

4.17 Conclusion111

References112

5 Ginzburg-Landau Equation for Fractal Media115

5.1 Introduction115

5.2 Fractional generalization of free energy functional116

5.3 Ginzburg-Landau equation from free energy functional117

5.4 Fractional equations from variational equation118

5.5 Conclusion121

References121

6 Fokker-Planck Equation for Fractal Distributions of Probability123

6.1 Introduction123

6.2 Fractional equation for average values124

6.3 Fractional Chapman-Kolmogorov equation125

6.4 Fokker-Planck equation for fractal distribution127

6.5 Stationary solutions of generalized Fokker-Planck equation130

6.6 Conclusion132

References132

7 Statistical Mechanics of Fractal Phase Space Distributions135

7.1 Introduction135

7.2 Fractal distribution in phase space136

7.3 Fractional phase volume for configuration space136

7.4 Fractional phase volume for phase space139

7.5 Fractional generalization of normalization condition139

7.6 Continuity equation for fractal distribution in configuration space141

7.7 Continuity equation for fractal distribution in phase space142

7.8 Fractional average values for configuration space144

7.9 Fractional average values for phase space145

7.10 Generalized Liouville equation146

7.11 Reduced distribution functions147

7.12 Conclusion148

References150

Part Ⅱ Fractional Dynamics and Long-Range Interactions150

8 Fractional Dynamics of Media with Long-Range Interaction153

8.1 Introduction153

8.2 Equations of lattice vibrations and dispersion law155

8.3 Equations of motion for interacting particles160

8.4 Transform operation for discrete models162

8.5 Fourier series transform of equations of motion163

8.6 Alpha-interaction of particles166

8.7 Fractional spatial derivatives170

8.8 Riesz fractional derivatives and integrals174

8.9 Continuous limits of discrete equations177

8.10 Linear nearest-neighbor interaction180

8.11 Linear integer long-range alpha-interaction181

8.12 Linear fractional long-range alpha-interaction184

8.13 Fractional reaction-diffusion equation187

8.14 Nonlinear long-range alpha-interaction190

8.15 Fractional 3-dimensional lattice equation194

8.16 Fractional derivatives from dispersion law195

8.17 Fractal long-range interaction198

8.18 Fractal dispersion law203

8.19 Grünwald-Letnikov-Riesz long-range interaction206

8.20 Conclusion208

References209

9 Fractional Ginzburg-Landau Equation215

9.1 Introduction215

9.2 Particular solution of fractional Ginzburg-Landau equation216

9.3 Stability of plane-wave solution220

9.4 Forced fractional equation221

9.5 Conclusion222

References223

10 Psi-Series Approach to Fractional Equations227

10.1 Introduction227

10.2 Singular behavior of fractional equation228

10.3 Resonance terms of fractional equation229

10.4 Psi-series for fractional equation of rational order230

10.5 Next to singular behavior233

10.6 Conclusion235

References236

Part Ⅲ Fractional Spatial Dynamics241

11 Fractional Vector Calculus241

11.1 Introduction241

11.2 Generalization of vector calculus242

11.3 Fundamental theorem of fractional calculus247

11.4 Fractional differential vector operators250

11.5 Fractional integral vector operations253

11.6 Fractional Green's formula254

11.7 Fractional Stokes'formula257

11.8 Fractional Gauss'formula259

11.9 Conclusion261

References262

12 Fractional Exterior Calculus and Fractional Differential Forms265

12.1 Introduction265

12.2 Differential forms of integer order266

12.3 Fractional exterior derivative269

12.4 Fractional differential forms274

12.5 Hodge star operator279

12.6 Vector operations by differential forms281

12.7 Fractional Maxwell's equations in terms of fractional forms282

12.8 Caputo derivative in electrodynamics284

12.9 Fractional nonlocal Maxwell's equations285

12.10 Fractional waves287

12.11 Conclusion288

References289

13 Fractional Dynamical Systems293

13.1 Introduction293

13.2 Fractional generalization of gradient systems294

13.3 Examples of fractional gradient systems301

13.4 Hamiltonian dynamical systems305

13.5 Fractional generalization of Hamiltonian systems307

13.6 Conclusion311

References312

14 Fractional Calculus of Variations in Dynamics315

14.1 Introduction315

14.2 Hamilton's equations and variations of integer order315

14.3 Fractional variations and Hamilton's equations317

14.4 Lagrange's equations and variations of integer order319

14.5 Fractional variations and Lagrange's equations321

14.6 Helmholtz conditions and non-Lagrangian equations323

14.7 Fractional variations and non-Hamiltonian systems326

14.8 Fractional stability328

14.9 Conclusion330

References331

15 Fractional Statistical Mechanics335

15.1 Introduction335

15.2 Liouville equation with fractional derivatives336

15.3 Bogolyubov equation with fractional derivatives340

15.4 Vlasov equation with fractional derivatives343

15.5 Fokker-Planck equation with fractional derivatives345

15.6 Conclusion349

References350

Part Ⅳ Fractional Temporal Dynamics357

16 Fractional Temporal Electrodynamics357

16.1 Introduction357

16.2 Universal response laws358

16.3 Linear electrodynamics of medium360

16.4 Fractional equations for laws of universal response362

16.5 Fractional equations of the Curie-von Schweidler law364

16.6 Fractional Gauss'laws for electric field366

16.7 Universal fractional equation for electric field369

16.8 Universal fractional equation for magnetic field370

16.9 Fractional damping of magnetic field372

16.10 Conclusion373

References374

17 Fractional Nonholonomic Dynamics377

17.1 Introduction377

17.2 Nonholonomic dynamics378

17.3 Fractional temporal derivatives385

17.4 Fractional dynamics with nonholonomic constraints388

17.5 Constraints with fractional derivatives394

17.6 Equations of motion with fractional nonholonomic constraints396

17.7 Example of fractional nonholonomic constraints398

17.8 Fractional conditional extremum401

17.9 Hamilton's approach to fractional nonholonomic constraints403

17.10 Conclusion405

References406

18 Fractional Dynamics and Discrete Maps with Memory409

18.1 Introduction409

18.2 Discrete maps without memory410

18.3 Caputo and Riemann-Liouville fractional derivatives415

18.4 Fractional derivative in the kicked term and discrete maps418

18.5 Fractional derivative in the kicked term and dissipative discrete maps422

18.6 Fractional equation with higher order derivatives and discrete map425

18.7 Fractional generalization of universal map for 1＜α≤2429

18.8 Fractional universal map for α＞2434

18.9 Riemann-Liouville derivative and universal map with memory436

18.10 Caputo fractional derivative and universal map with memory441

18.11 Fractional kicked damped rotator map445

18.12 Fractional dissipative standard map447

18.13 Fractional Hénon map449

18.14 Conclusion450

References451

Part Ⅴ Fractional Quantum Dynamics457

19 Fractional Dynamics of Hamiltonian Quantum Systems457

19.1 Introduction457

19.2 Fractional power of derivative and Heisenberg equation458

19.3 Properties of fractional dynamics460

19.4 Fractional quantum dynamics of free particle462

19.5 Fractional quantum dynamics of harmonic oscillator463

19.6 Conclusion464

References465

20 Fractional Dynamics of Open Quantum Systems467

20.1 Introduction467

20.2 Fractional power of superoperator468

20.3 Fractional equation for quantum observables471

20.4 Fractional dynamical semigroup473

20.5 Fractional equation for quantum states475

20.6 Fractional non-Markovian quantum dynamics477

20.7 Fractional equations for quantum oscillator with friction478

20.8 Quantum self-reproducing and self-cloning482

20.9 Conclusion486

References487

21 Quantum Analogs of Fractional Derivatives491

21.1 Introduction491

21.2 Weyl quantization of differential operators492

21.3 Quantization of Riemann-Liouville fractional derivatives494

21.4 Quantization of Liouville fractional derivative496

21.5 Quantization of nondifferentiable functions497

21.6 Conclusion500

References501

Index503