Long-range Interactions, Stochasticity

《长距离相互作用、随机及分数维动力学》编辑推荐：NonlinearPhysicalSciencefocusesontherecentadvancesoffundamentaltheoriesandprinciples,analyticalandsymbolicapproaches,aswellascomputationaltechniquesinnonlinearphysicalscienceandnonlinearmathematicswithengineeringapplications.

In memory of Dr. George Zaslavsky, Long-range Interaction, 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. 本书介绍了连续及离散动力系统的自相似、随机性及分数维性。

编者：罗朝俊（墨西哥）阿弗莱诺维奇（ValentinAfraimovich）丛书主编：（瑞典）伊布拉基莫夫

Dr.AlbertC.J.LuoisaProfessoratSouthernIllinoisUniversityEdwardsville,USA.
Dr.ValentinAfraimovichisaProiessoratSanLuisPotosiUniversity,Mexico.

1FractionalZaslavskyandHenonDiscreteMaps
VasilyE.Tarasov
1.1Introduction
1.2Fractionalderivatives
1.2.1FractionalRiemann-Liouvillederivatives
1.2.2FractionalCaputoderivatives
1.2.3FractionalLiouvillederivatives
1.2.4Interpretationofequationswithfractionalderivatives.
1.2.5Discretemapswithmemory
1.3FractionalZaslavskymaps
1.3.1DiscreteChirikovandZaslavskymaps
1.3.2FractionaluniversalandZaslavskymap
1.3.3Kickeddampedrotatormap
1.3.4FractionalZaslavskymapfromfractionaldifferentialequations
1.4FractionalH6nonmap
1.4.1Henonmap
1.4.2FractionalHenonmap
1.5FractionalderivativeinthekickedtermandZaslavskymap
1.5.1Fractionalequationanddiscretemap
1.5.2Examples
1.6FractionalderivativeinthekickeddampedtermandgeneralizationsofZaslavskyandHenonmaps
1.6.1Fractionalequationanddiscretemap
1.6.2FractionalZaslavskyandHenonmaps
1.7Conclusion
References

2Self-similarity,StochasticityandFractionality
VladimirVUchaikin
2.1Introduction
2.1.1Tenyearsago
2.1.2Twokindsofmotion
2.1.3Dynamicself-similarity
2.1.4Stochasticself-similarity
2.1.5Self-similarityandstationarity
2.2FromBrownianmotiontoLevymotion
2.2.1Brownianmotion
2.2.2Self-similarBrownianmotioninnonstationarynonhomogeneousenvironment
2.2.3Stablelaws
2.2.4DiscretetimeLevymotion
2.2.5ContinuoustimeLevymotion
2.2.6FractionalequationsforcontinuoustimeLevymotion
2.3FractionalBrownianmotion
2.3.1DifferentialBrownianmotionprocess
2.3.2IntegralBrownianmotionprocess
2.3.3FractionalBrownianmotion
2.3.4FractionalGaussiannoises
2.3.5BarnesandAllanmodel
2.3.6FractionalLevymotion
2.4FractionalPoissonmotion
2.4.1Renewalprocesses
2.4.2Self-similarrenewalprocesses
2.4.3Threeformsoffractaldustgenerator
2.4.4ntharrivaltimedistribution
2.4.5FractionalPoissondistribution
2.5FractionalcompoundPoissonprocess
2.5.1CompoundPoissonprocess
2.5.2Levy-Poissonmotion
2.5.3FractionalcompoundPoissonmotion
2.5.4Alinkbetweensolutions
2.5.5FractionalgeneralizationoftheLevymotion
Acknowledgments
Appendix.Fractionaloperators
References

3Long-rangeInteractionsandDilutedNetworks
AntoniaCiani,DuccioFanelliandStefanoRuffo
3.1Long-rangeinteractions
3.1.1Lackofadditivity
3.1.2Equilibriumanomalies:Ensembleinequivalence,negativespecificheatandtemperaturejumps
3.1.3Non-equilibriumdynamicalproperties
3.1.4QuasiStationaryStates
3.1.5Physicalexamples
3.1.6Generalremarksandoutlook
3.2HamiltonianMeanFieldmodel:equilibriumandout-of-equilibriumfeatures
3.2.1Themodel
3.2.2Equilibriumstatisticalmechanics
3.2.3OntheemergenceofQuasiStationaryStates:Non-
equilibriumdynamics
3.3IntroducingdilutionintheHamiltonianMeanFieldmodel
3.3.1HamiltonianMeanFieldmodelonadilutednetwork
3.3.2OnequilibriumsolutionofdilutedHamiltonianMeanField
3.3.3OnQuasiStationaryStatesinpresenceofdilution
3.3.4Phasetransition
3.4Conclusions
Acknowledgments
References

4MetastabilityandTransientsinBrainDynamics:ProblemsandRigorousResults
ValentinS.Afraimovich,MehmetK.Muezzinogluand
MikhailI.Rabinovich
4.1Introduction:whatwediscussandwhynow
4.1.1Dynamicalmodelingofcognition
4.1.2Brainimaging
4.1.3Dynamicsofemotions
4.2Mentalmodes
4.2.1Statespace
4.2.2Functionalnetworks
4.2.3Emotion-cognitiontandem
4.2.4Dynamicalmodelofconsciousness
4.3Competition——robustnessandsensitivity
4.3.1Transientsversusattractorsinbrain
4.3.2Cognitivevariables
4.3.3Emotionalvariables
4.3.4Metastabilityanddynamicalprinciples
4.3.5Winnerlesscompetition——structuralstabilityoftransients
4.3.6Examples:competitivedynamicsinsensorysystems
4.3.7Stableheteroclinicchannels
4.4Basicecologicalmodel
4.4.1TheLotka-Volterrasystem
4.4.2Stressandhysteresis
4.4.3Moodandcognition
4.4.4Intermittentheteroclinicchannel
4.5Conclusion
Acknowledgments
Appendix1
Appendix2
References

5DynamicsofSolitonChains:FromSimpletoComplexandChaoticMotions
KonstantinA.Gorshkov,LevA.OstrovskyandYuryA.Stepanyants
5.1Introduction
5.2Stablesolitonlatticesandahierarchyofenvelopesolitons
5.3ChainsofsolitonswithintheframeworkoftheGardnermodel
5.4Unstablesolitonlatticesandstochastisation
5.5Solitonstochastisationandstrongwaveturbulenceinaresonatorwithexternalsinusoidalpumping
5.6Chainsoftwo-dimensionalsolitonsinpositive-dispersionmedia
5.7Conclusion
FewwordsinmemoryofGeorgeM.Zaslavsky
References

6WhatisControlofTurbulenceinCrossedFields?-Don'tEvenThinkofEliminatingAllVortexes!
DimitriVolchenkov
6.1Introduction
6.2Stochastictheoryofturbulenceincrossedfields:vortexesofallsizesdieout,butone
6.2.1Themethodofrenormalizationgroup
6.2.2Phenomenologyoffullydevelopedisotropicturbulence
6.2.3QuantumfieldtheoryformulationofstochasticNavier-Stokesturbulence
6.2.4AnalyticalpropertiesofFeynmandiagrams
6.2.5UltravioletrenormalizationandRG-equations
6.2.6WhatdotheRGrepresentationssum?
6.2.7Stochasticmagnetichydrodynamics
6.2.8Renormalizationgroupinmagnetichydrodynamics
6.2.9Criticaldimensionsinmagnetichydrodynamics
6.2.10Criticaldimensionsofcompositeoperatorsinmagnetichydrodynamics
6.2.11Operatorsofthecanonicaldimensiond=2
6.2.12Vectoroperatorsofthecanonicaldimensiond=3
6.2.13Instabilityinmagnetichydrodynamics
6.2.14Longlifetoeddiesofapreferablesize
6.3Insearchofloststability
6.3.1Phenomenologyoflong-rangeturbulenttransportinthescrape-offlayer(SOL)ofthermonuclearreactors
6.3.2Stochasticmodelsofturbulenttransportincross-fieldsystems
6.3.3Iterativesolutionsincrossedfields
6.3.4Functionalintegralformulationofcross-fieldturbulenttransport
6.3.5Large-scaleinstabilityofiterativesolutions
6.3.6Turbulencestabilizationbythepoloidalelectricdrift
6.3.7QualitativediscretetimemodelofanomaloustransportintheSOL
6.4Conclusion
References

7EntropyandTransportinBilliards
M.CourbageandS.M.SaberiFathi
7.1Introduction
7.2Entropy
7.2.1EntropyintheLorentzgas
7.2.2Somedynamicalpropertiesofthebarrierbilliardmodel
7.3Transport
7.3.1TransportinLorentzgas
7.3.2Transportinthebarrierbilliard
7.4Concludingremarks
References
Index