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 <=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
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