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定 價(jià)：89 元 叢書名：非線性物理科學(xué)
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Fractional Dynamics:Applications of Fractional Calculus to Dynamics of Particles,Fields and Media presents applications of fractional calculus,integral and differential equations of non-integer orders in describing systems with long-time memory,non-local how.asp?id=934204" title="局外人.鼠
Fractional Dynamics:Applications of Fractional Calculus to Dynamics of Particles，F(xiàn)ields and Media presents applications of fractional calculus，integral anddifferential equations of non-integer orders in describing systems with long-timememory，non-local spatial and fractal properties.Mathematical models of fractalmedia and distributions，generalized dynamical systems and discrete maps，non-local statistical mechanics and kinetics，dynamics of open quantum systems，thehydrodynamics and electrodynamics of complex media with non-local propertiesand memory are considered.
This book is intended to meet the needs of scientists and graduate studentsin physics，mechanics and applied mathematics who are interested in electro-dynamics，statistical and condensed matter physics，quantum dynamics，complexmedia theories and kinetics，discrete maps and lattice models，and nonlineardynamics and chaos. Dr.Vasily E.Tarasov is a Senior Research Associate at Nuclear Physics Instituteof Moscow State University and an Associate Professor at Applied Mathematicsand Physics Department of Moscow Aviation Institute.
Fractional calculus is a theory of integrals and derivatives of any arbitrary real（orcomplex）order.It has a long history from 30 September 1695，when the derivativeof order a=1/2 was mentioned by Leibniz.The fractional differentiation and frac-tional integration go back to many great mathematicians such as Leibniz，Liouville，Grfinwald，Letnikov，Riemann，Abel，Riesz and Weyl.The integrals and derivativesof non-integer order，and the fractional integro-differential equations have foundmany applications in recent studies in theoretical physics，mechanics and appliedmathematics.New possibilities in mathematics and theoretical physics appear，when the ordera of the differential operator Dxa or the integral operator Ixa becomes an arbitraryparameter.The fractional calculus is a powerful tool to describe physical systemsthat have long-term memory and long-range spatial interactions.In general，manyusual properties of the ordinary（first-order）derivative Dx are not realized for frac-tional derivative operators Da.For example，a product rule，chain rule and semi-group property have strongly complicated analogs for the operators D~a.Most of the processes associated with complex systems have nonlocal dynamicsand it can be characterized by long-term memory in time.The fractional integrationand fractional differentiation operators allow one to consider some of those charac-teristics.Using fractional calculus，it is possible to obtain useful dynamical mod-els，where fractional integro-differential operators in the time and space variablesdescribe the long-term memory and nonlocal spatial properties of the complex me-dia and processes.We should note that close connections exist between fractionaldifferential and integral equations，and the dynamics of many complex systems，anomalous processes and fractal media.There are many interesting books about fractional calculus，fractional differentialequations，and their physical applications.The first book dedicated specifically tothe theory of fractional integrals and derivatives，is the one by Oldham and Spanierpublished in 1974.There exists the remarkably comprehensive encyclopedic-typemonograph by Samko，Kilbus and Marichev，which was published in Russian in1987 and in English in 1993.
Part I Fractionfil Continuous Models ofFractal 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 Fraetal 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 Electrodynamies 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 Grtinwald-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
……
Part Ⅳ Fractional Temporal Dynamics
Part Ⅴ Fractional Quantum Dynamics
Index
Statistical mechanics is the application of probability theory to study the dynam-ics of systems of arbitrary number of particles（Gibbs，1960;Bogoliubov，1960;Bogolyubov，1970）.Equations with derivatives of non-integer order have many ap-plications in physical kinetics（see，for example，（Zaslavsky，2002，2005;Uchaikin，2008）and（Zaslavsky，1994;Saichev and Zaslavsky，1997;Weitzner and Zaslavsky，2001;Chechkin et al.，2002;Saxena et al.，2002;Zelenyi and Milovanov，2004;Zaslavsky and Edelman，2004;Nigmatullin，2006;Tarasov and Zaslavsky，2008;Rastovic，2008））.Fractional calculus is used to describe anomalous diffusion，andtransport theory（Montroll and Shlesinger，1984;Metzler and Klafter，2000;Za-slavsky，2002;Uchaikin，2003a，b;Metzler and Klafter，2004）.Application of frac-tional integration and differentiation in statistical mechanics was also consideredin（Tarasov，2006a，2007a）and（Tarasov，2004，2005b，a，2006b，2007b）.Fractionalkinetic equations usually appear from some phenomenological models.We suggestfractional generalizations of some basic equations of statistical mechanics.To ob-tain these equations，the probability conservation in a fractional differential volumeelement of the phase space can be used（Tarasov，2006a，2007a）.This element canbe considered as a small part of the phase space set with non-integer-dimension.Wederive the Liouville equation with fractional derivatives with respect to coordinatesand momenta.The fractional Liouville equation（Tarasov，2006a，2007a）is obtainedfrom the conservation of probability to find a system in a fractional volume element.This equation is used to derive fractional Bogolyubov and fractional kinetic equa-tions with fractional derivatives.Statistical mechanics of fractional generalizationof the Hamiltonian systems is discussed.Liouville and Bogolyubov equations withfractional coordinate and momenta derivatives are considered as a basis to derivefractional kinetic equations.The Vlasov equation with derivatives of non-integer or-der is obtained.The Fokker-Planck equation that has fractional phase space deriva-tives is derived from fractional Bogolyubov equation.
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