Long-range Interactions, Stochasticity

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