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Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particl

样章
作者：
Vasily E. Tarasov
定价：
89.00元
ISBN：
978-7-04-029473-6
版面字数：
780.000千字
开本：
16开
全书页数：
505页
装帧形式：
精装
重点项目：
暂无
出版时间：
2010-08-20
读者对象：
学术著作
一级分类：
自然科学
二级分类：
数学与统计
三级分类：
动力系统

In memory of Dr. George Zaslavsky, "Long-range Interactions, Stochasticity and Fractional Dynamics" covers the recent developments of long-range interaction, fractional dynamics, brain dynamics and stochastic theory of turbulence, each chapter was written by established scientists in the field. The book is dedicated to Dr. George Zaslavsky, who was one of three founders of the theory of Hamiltonian chaos. The book discusses self-similarity and stochasticity and fractionality for discrete and continuous dynamical systems, as well as long-range interactions and diluted networks. A comprehensive theory for brain dynamics is also presented. In addition, the complexity and stochasticity for soliton chains and turbulence are addressed. The book is intended for researchers in the field of nonlinear dynamics in mathematics, physics and engineering. Dr. Albert C.J. Luo is a Professor at Southern Illinois University Edwardsville, USA. Dr. Valentin Afraimovich is a Professor at San Luis Potosi University, Mexico.

  • Front Matter
  • Part I Fractional Continuous Models of Fractal Distributions
  • 1 Fractional Integration and Fractals
    • 1.1 Riemann-Liouville fractional integrals
    • 1.2 Liouville fractional integrals
    • 1.3 Riesz fractional integrals
    • 1.4 Metric and measure spaces
    • 1.5 Hausdorff measure
    • 1.6 Hausdorff dimension and fractals
    • 1.7 Box-counting dimension
    • 1.8 Mass dimension of fractal systems
    • 1.9 Elementary models of fractal distributions
    • 1.10 Functions and integrals on fractals
    • 1.11 Properties of integrals on fractals
    • 1.12 Integration over non-integer-dimensional space
    • 1.13 Multi-variable integration on fractals
    • 1.14 Mass distribution on fractals
    • 1.15 Density of states in Euclidean space
    • 1.16 Fractional integral and measure on the real axis
    • 1.17 Fractional integral and mass on the real axis
    • 1.18 Mass of fractal media
    • 1.19 Electric charge of fractal distribution
    • 1.20 Probability on fractals
    • 1.21 Fractal distribution of particles
    • References
  • 2 Hydrodynamics of Fractal Media
    • 2.1 Introduction
    • 2.2 Equation of balance of mass
    • 2.3 Total time derivative of fractional integral
    • 2.4 Equation of continuity for fractal media
    • 2.5 Fractional integral equation of balance of momentum
    • 2.6 Differential equations of balance of momentum
    • 2.7 Fractional integral equation of balance of energy
    • 2.8 Differential equation of balance of energy
    • 2.9 Euler’s equations for fractal media
    • 2.10 Navier-Stokes equations for fractal media
    • 2.11 Equilibrium equation for fractal media
    • 2.12 Bernoulli integral for fractal media
    • 2.13 Sound waves in fractal media
    • 2.14 One-dimensional wave equation in fractal media
    • 2.15 Conclusion
    • References
  • 3 Fractal Rigid Body Dynamics
    • 3.1 Introduction
    • 3.2 Fractional equation for moment of inertia
    • 3.3 Moment of inertia of fractal rigid body ball
    • 3.4 Moment of inertia for fractal rigid body cylinder
    • 3.5 Equations of motion for fractal rigid body
    • 3.6 Pendulum with fractal rigid body
    • 3.7 Fractal rigid body rolling down an inclined plane
    • 3.8 Conclusion
    • References
  • 4 Electrodynamics of Fractal Distributions of Charges and Fields
    • 4.1 Introduction
    • 4.2 Electric charge of fractal distribution
    • 4.3 Electric current for fractal distribution
    • 4.4 Gauss’ theorem for fractal distribution
    • 4.5 Stokes’ theorem for fractal distribution
    • 4.6 Charge conservation for fractal distribution
    • 4.7 Coulomb’s and Biot-Savart laws for fractal distribution
    • 4.8 Gauss’ law for fractal distribution
    • 4.9 Ampere’s law for fractal distribution
    • 4.10 Integral Maxwell equations for fractal distribution
    • 4.11 Fractal distribution as an effective medium
    • 4.12 Electric multipole expansion for fractal distribution
    • 4.13 Electric dipole moment of fractal distribution
    • 4.14 Electric quadrupole moment of fractal distribution
    • 4.15 Magnetohydrodynamics of fractal distribution
    • 4.16 Stationary states in magnetohydrodynamics of fractal distributions
    • 4.17 Conclusion
    • References
  • 5 Ginzburg-Landau Equation for Fractal Media
    • 5.1 Introduction
    • 5.2 Fractional generalization of free energy functional
    • 5.3 Ginzburg-Landau equation from free energy functional
    • 5.4 Fractional equations from variational equation
    • 5.5 Conclusion
    • References
  • 6 Fokker-Planck Equation for Fractal Distributions of Probability
    • 6.1 Introduction
    • 6.2 Fractional equation for average values
    • 6.3 Fractional Chapman-Kolmogorov equation
    • 6.4 Fokker-Planck equation for fractal distribution
    • 6.5 Stationary solutions of generalized Fokker-Planck equation
    • 6.6 Conclusion
    • References
  • 7 Statistical Mechanics of Fractal Phase Space Distributions
    • 7.1 Introduction
    • 7.2 Fractal distribution in phase space
    • 7.3 Fractional phase volume for configuration space
    • 7.4 Fractional phase volume for phase space
    • 7.5 Fractional generalization of normalization condition
    • 7.6 Continuity equation for fractal distribution in configuration space
    • 7.7 Continuity equation for fractal distribution in phase space
    • 7.8 Fractional average values for configuration space
    • 7.9 Fractional average values for phase space
    • 7.10 Generalized Liouville equation
    • 7.11 Reduced distribution functions
    • 7.12 Conclusion
    • References
  • Part II Fractional Dynamics and Long-Range Interactions
  • 8 Fractional Dynamics of Media with Long-Range Interaction
    • 8.1 Introduction
    • 8.2 Equations of lattice vibrations and dispersion law
    • 8.3 Equations of motion for interacting particles
    • 8.4 Transform operation for discrete models
    • 8.5 Fourier series transform of equations of motion
    • 8.6 Alpha-interaction of particles
    • 8.7 Fractional spatial derivatives
    • 8.8 Riesz fractional derivatives and integrals
    • 8.9 Continuous limits of discrete equations
    • 8.10 Linear nearest-neighbor interaction
    • 8.11 Linear integer long-range alpha-interaction
    • 8.12 Linear fractional long-range alpha-interaction
    • 8.13 Fractional reaction-diffusion equation
    • 8.14 Nonlinear long-range alpha-interaction
    • 8.15 Fractional 3-dimensional lattice equation
    • 8.16 Fractional derivatives from dispersion law
    • 8.17 Fractal long-range interaction
    • 8.18 Fractal dispersion law
    • 8.19 Gr¨unwald-Letnikov-Riesz long-range interaction
    • 8.20 Conclusion
    • References
  • 9 Fractional Ginzburg-Landau Equation
    • 9.1 Introduction
    • 9.2 Particular solution of fractional Ginzburg-Landau equation
    • 9.3 Stability of plane-wave solution
    • 9.4 Forced fractional equation
    • 9.5 Conclusion
    • References
  • 10 Psi-Series Approach to Fractional Equations
    • 10.1 Introduction
    • 10.2 Singular behavior of fractional equation
    • 10.3 Resonance terms of fractional equation
    • 10.4 Psi-series for fractional equation of rational order
    • 10.5 Next to singular behavior
    • 10.6 Conclusion
    • References
  • Part III Fractional Spatial Dynamics
  • 11 Fractional Vector Calculus
    • 11.1 Introduction
    • 11.2 Generalization of vector calculus
    • 11.3 Fundamental theorem of fractional calculus
    • 11.4 Fractional differential vector operators
    • 11.5 Fractional integral vector operations
    • 11.6 Fractional Green’s formula
    • 11.7 Fractional Stokes’ formula
    • 11.8 Fractional Gauss’ formula
    • 11.9 Conclusion
    • References
  • 12 Fractional Exterior Calculus and Fractional Differential Forms
    • 12.1 Introduction
    • 12.2 Differential forms of integer order
    • 12.3 Fractional exterior derivative
    • 12.4 Fractional differential forms
    • 12.5 Hodge star operator
    • 12.6 Vector operations by differential forms
    • 12.7 Fractional Maxwell’s equations in terms of fractional forms
    • 12.8 Caputo derivative in electrodynamics
    • 12.9 Fractional nonlocal Maxwell’s equations
    • 12.10 Fractional waves
    • 12.11 Conclusion
    • References
  • 13 Fractional Dynamical Systems
    • 13.1 Introduction
    • 13.2 Fractional generalization of gradient systems
    • 13.3 Examples of fractional gradient systems
    • 13.4 Hamiltonian dynamical systems
    • 13.5 Fractional generalization of Hamiltonian systems
    • 13.6 Conclusion
    • References
  • 14 Fractional Calculus of Variations in Dynamics
    • 14.1 Introduction
    • 14.2 Hamilton’s equations and variations of integer order
    • 14.3 Fractional variations and Hamilton’s equations
    • 14.4 Lagrange’s equations and variations of integer order
    • 14.5 Fractional variations and Lagrange’s equations
    • 14.6 Helmholtz conditions and non-Lagrangian equations
    • 14.7 Fractional variations and non-Hamiltonian systems
    • 14.8 Fractional stability
    • 14.9 Conclusion
    • References
  • 15 Fractional Statistical Mechanics
    • 15.1 Introduction
    • 15.2 Liouville equation with fractional derivatives
    • 15.3 Bogolyubov equation with fractional derivatives
    • 15.4 Vlasov equation with fractional derivatives
    • 15.5 Fokker-Planck equation with fractional derivatives
    • 15.6 Conclusion
    • xiv Contents
    • References
  • Part IV Fractional Temporal Dynamics
  • 16 Fractional Temporal Electrodynamics
    • 16.1 Introduction
    • 16.2 Universal response laws
    • 16.3 Linear electrodynamics of medium
    • 16.4 Fractional equations for laws of universal response
    • 16.5 Fractional equations of the Curie-von Schweidler law
    • 16.6 Fractional Gauss’ laws for electric field
    • 16.7 Universal fractional equation for electric field
    • 16.8 Universal fractional equation for magnetic field
    • 16.9 Fractional damping of magnetic field
    • 16.10 Conclusion
    • References
  • 17 Fractional Nonholonomic Dynamics
    • 17.1 Introduction
    • 17.2 Nonholonomic dynamics
    • 17.3 Fractional temporal derivatives
    • 17.4 Fractional dynamics with nonholonomic constraints
    • 17.5 Constraints with fractional derivatives
    • 17.6 Equations of motion with fractional nonholonomic constraints
    • 17.7 Example of fractional nonholonomic constraints
    • 17.8 Fractional conditional extremum
    • 17.9 Hamilton’s approach to fractional nonholonomic constraints
    • 17.10 Conclusion
    • References
  • 18 Fractional Dynamics and Discrete Maps with Memory
    • 18.1 Introduction
    • 18.2 Discrete maps without memory
    • 18.3 Caputo and Riemann-Liouville fractional derivatives
    • 18.4 Fractional derivative in the kicked term and discrete maps
    • 18.5 Fractional derivative in the kicked term and dissipative discrete maps
    • 18.6 Fractional equation with higher order derivatives and discrete map
    • 18.7 Fractional generalization of universal map for 1 <a<=2
    • 18.8 Fractional universal map for a > 2
    • 18.9 Riemann-Liouville derivative and universal map with memory
    • 18.10 Caputo fractional derivative and universal map with memory
    • 18.11 Fractional kicked damped rotator map
    • 18.12 Fractional dissipative standard map
    • 18.13 Fractional H´enon map
    • 18.14 Conclusion
    • References
  • Part V Fractional Quantum Dynamics
  • 19 Fractional Dynamics of Hamiltonian Quantum Systems
    • 19.1 Introduction
    • 19.2 Fractional power of derivative and Heisenberg equation
    • 19.3 Properties of fractional dynamics
    • 19.4 Fractional quantum dynamics of free particle
    • 19.5 Fractional quantum dynamics of harmonic oscillator
    • 19.6 Conclusion
    • References
  • 20 Fractional Dynamics of Open Quantum Systems
    • 20.1 Introduction
    • 20.2 Fractional power of superoperator
    • 20.3 Fractional equation for quantum observables
    • 20.4 Fractional dynamical semigroup
    • 20.5 Fractional equation for quantum states
    • 20.6 Fractional non-Markovian quantum dynamics
    • 20.7 Fractional equations for quantum oscillator with friction
    • 20.8 Quantum self-reproducing and self-cloning
    • 20.9 Conclusion
    • References
  • 21 Quantum Analogs of Fractional Derivatives
    • 21.1 Introduction
    • 21.2 Weyl quantization of differential operators
    • 21.3 Quantization of Riemann-Liouville fractional derivatives
    • 21.4 Quantization of Liouville fractional derivative
    • 21.5 Quantization of nondifferentiable functions
    • 21.6 Conclusion
    • References
  • Index
  • Nonlinear Physical Science
  • 版权

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