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I. Math Framework for Non-Equilibrium Hierarchies Multiscale Analysis II

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I. Math Framework for Non-Equilibrium Hierarchies Multiscale Analysis II Fusion Fedorova, Zeitlin Outline LOCALIZATION AND PATTERN FORMATION IN Multiphysics Hopes MODELS OF FUSION/ENERGY CONFINEMENT IN Introduction PLASMA PHYSICS: BBGKY hierarchy I. Math Framework for Non-Equilibrium Hierarchies Multiscale analysis II. BBGKY hierarchy and Reductions Variational approach TOWARDS WAVELETONS IN (PLASMA) PHYSICS Modeling of patterns Vlasov-Poisson system Multiscale representations Antonina N. Fedorova Michael G. Zeitlin RATE/RMS models RATE equations IPME RAS, V.O. Bolshoj pr., 61, 199178, St. Petersburg, Russia Multiscale http://www.ipme.ru/zeitlin.html, http:/mp.ipme.ru/zeitlin.html representations Conclusions IPAM Meeting, 2012 Outline Fusion Fedorova, Zeitlin Multiphysics Outline Multiphysics Hopes Hopes Introduction Introduction BBGKY hierarchy BBGKY hierarchy Multiscale analysis Variational approach Multiscale analysis Modeling of patterns Vlasov-Poisson system Variational approach Multiscale representations Modeling of patterns RATE/RMS models Vlasov-Poisson system RATE equations Multiscale Multiscale representations representations Conclusions RATE/RMS models RATE equations Multiscale representations Conclusions 1. The only important result we found (after Martin Kruskal) Fusion Fedorova, Zeitlin A MAGNETICALLY CONFINED PLASMA CANNOT BE IN Outline THERMODYNAMICAL EQUILIBRIUM... Multiphysics Folklor Hopes Introduction BBGKY hierarchy 2. Current state of the Problem Multiscale analysis And I walk out of space Variational approach Into an overgrown garden of values, Modeling of patterns And tear up seeming stability Vlasov-Poisson system And self-comprehension of causes. Multiscale And your, infinity, textbook representations I read by myself, without people - RATE/RMS models Leafless, savage mathematical book, RATE equations Multiscale A problem book of gigantic radicals.. representations Conclusions Osip Mandelstam 3. Just for beauty Fusion Fedorova, Zeitlin I learnt to distrust all physical concepts as the basis for a theory. Instead one should put one’s trust in a mathematical scheme, even Outline if the scheme does not appear at first sight to be connected with Multiphysics physics. One should concentrate on getting interesting Hopes mathematics. Introduction G–d used beautiful mathematics in creating the world. BBGKY hierarchy Multiscale analysis I consider that I understand an equation when I can predict the Variational approach properties of its solutions, without actually solving it. Modeling of patterns Vlasov-Poisson system PAM Dirac Multiscale representations RATE/RMS models 4. Just for drive RATE equations Multiscale Wir m¨ussen wissen. Wir werden wissen. representations We must know. We will know. Conclusions David Hilbert Fusion Fedorova, Zeitlin Key words: Outline Localization, Multiphysics Localized modes, Hopes Patterns, Introduction Pattern formation, BBGKY hierarchy Controllable patterns, Multiscale analysis Waveletons, Variational approach (Non-linear) Pseudo-differential dynamics (ΨDOD), Modeling of patterns Multiscales, Vlasov-Poisson system Multiscale Multiresolution, representations Local/non-linear harmonic analysis (wavelet, Weyl-Heisenberg,...), RATE/RMS models SYMMETRY (Hidden, etc), RATE equations Functional spaces, Multiscale Topology of configuration space (tokamaks vs.stellarators vs.N-kamaks) representations Orbits, Conclusions Variational methods, Minimal complexity/Effectiveness of numerics Non-equilibrium ensembles, hierarchy of kinetics equations (BBGKY) and reasonable reductions/truncations {waveleton}:={soliton} F {wavelet} Fusion Fedorova, Zeitlin waveleton ≈ (meta) stable localized (controllable) Outline pattern Multiphysics Hopes Fusion state = (meta) stable state (long-living (meta) Introduction stable fluctuation) BBGKY hierarchy Multiscale analysis Variational approach in which most of energy of the system is concentrated Modeling of patterns in the relative small area of the whole phase space Vlasov-Poisson system during time which is enough to take it ouside for Multiscale possible usage (zero measure, min entropy). representations RATE/RMS models RATE equations Multiscale representations Conclusions Kindergarten ”Multiphysics” Fusion Fedorova, Zeitlin Outline KINDERGARTEN Multiphysics Hopes Base things: States, Observables, Measures, Measurement, ... Introduction What we are looking for? BBGKY hierarchy Multiscale analysis Pre-fusion, Fusion, Post-fusion state (KMS, SRB, ... ?) Variational approach Modeling of patterns States: Functional spaces Vlasov-Poisson system Observables: Operator algebras Multiscale Key point: Symmetry representations RATE/RMS models Filtrations: Tower of Scales RATE equations Multiscale It is result of Underlying Hidden Symmetry representations Conclusions ... Vc ⊂ ... Vn ⊂ ... Vn+1 ⊂ ... ∩ Vn = ∅ ∪ Vn = H After that: operators (FWT), measures, ... Fusion Fedorova, Zeitlin U(V ) − W ∗{(ψ(f ),ψ∗(f ), supp f ⊂ V } Outline ∗ U = ∪U(V ), u − C (closure in norm topology) Multiphysics ∗ 3 ∗ 3 Hopes ∃NV = ψ (x)ψ(x)d x, [ψ(x),ψ (y)]± = δ (x − y) V Introduction BBGKY hierarchy ∗ 3 ∗ ψ(f ) = ψ(x)f (x)d x, [ψ(g),ψ (f )]± =(g, f ) Multiscale analysis ∗ Variational approach |1 >= ψ (f )Ω0, ψ(f )Ω0 =0, Ω0 − cyclic, Modeling of patterns R(U) - representation, ω -state, ω(A) - expectation value of A ∈ U in Vlasov-Poisson system Multiscale state ω representations GNS: representation Rω(U) in Hω and Ω ∈ Hω such that: RATE/RMS models RATE equations ω(A)=(Ω, R(A)Ω), ∀A Multiscale representations Conclusions Gibbs states, equlibrium states Fusion Fedorova, Zeitlin ′ ′ iH (t) β,µ, V , H = H − µN Uv (t) = e v Outline Multiphysics ′ ′ −βH −1 −βH Hopes Gibbs: ωv (A) = Trv (ρv A) ρv (Trv e v ) e v Introduction BBGKY hierarchy Well defined expectation functional, ∀A ∈ U(V ) Multiscale analysis Variational approach RN Modeling of patterns ∃ limn→∞ ωvn (A) = ω(A), Vn+1 ⊃ Vn, ∪Vn = Vlasov-Poisson system Multiscale KMS, Tomita-Takesaki,... representations RATE/RMS models RATE equations Multiscale representations Conclusions Fusion Fedorova, Zeitlin Outline Multiphysics Hopes Introduction BBGKY hierarchy Multiscale analysis Variational approach Modeling of patterns Vlasov-Poisson system Multiscale representations RATE/RMS models Figure 1: Localized modes. RATE equations Multiscale representations Conclusions Fusion 2500 Fedorova, Zeitlin 2000 Outline 1500 Multiphysics Hopes 1000 Introduction 500 BBGKY hierarchy 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Multiscale analysis Variational approach Figure 2: Kick Modeling of patterns Vlasov-Poisson system Multiscale −1 representations −2 RATE/RMS models −3 RATE equations −4 −5 Multiscale −6 representations −7 Conclusions −8 −9 −10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure 3: MRA for Kick Fusion 6 Fedorova, Zeitlin 5 Outline 4 Multiphysics 3 Hopes 2 Introduction 1 BBGKY hierarchy 0 Multiscale analysis 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Variational approach Figure 4: Multi-Kicks Modeling of patterns Vlasov-Poisson system Multiscale −1 representations −2 RATE/RMS models −3 RATE equations −4 −5 Multiscale representations −6 −7 Conclusions −8 −9 −10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure 5: MRA for Multi-Kicks Fusion −3 x 10 4 Fedorova, Zeitlin 3.5 3 Outline 2.5 Multiphysics 2 1.5 Hopes 1 Introduction 0.5 BBGKY hierarchy 0 −0.5 Multiscale analysis 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Variational approach Figure 6: RW-fractal Modeling of patterns Vlasov-Poisson system Multiscale −1 representations −2 RATE/RMS models −3 RATE equations −4 −5 Multiscale representations −6 −7 Conclusions −8 −9 −10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure 7: MRA for RW-fractal Fusion Fedorova, Zeitlin Outline Multiphysics Hopes Introduction BBGKY hierarchy Multiscale analysis Variational approach Modeling of patterns Figure 8: Localized modes. Vlasov-Poisson system Multiscale representations RATE/RMS models RATE equations 0.8 Multiscale 0.6 representations 0.4 Conclusions 0.2 distribution function F(q,p) 0 30 30 20 20 10 10 0 0 coordinate (q) momentum (p) Figure 9: Localized mode contribution to distribution function. Fusion Fedorova, Zeitlin 1 Outline 0.8 0.6 Multiphysics 0.4 Hopes 0.2 distribution function F(q,p) 0 Introduction 60 60 40 40 BBGKY hierarchy 20 20 0 0 coordinate (q) momentum (p) Multiscale analysis Variational approach Figure 10: Chaotic-like pattern. Modeling of patterns Vlasov-Poisson system Multiscale representations 3 RATE/RMS models 2 RATE equations 1 0 W(q,p) Multiscale −1 representations −2 150 Conclusions 150 100 100 50 50 0 0 coordinate (q) momentum (p) Figure 11: Entangled-like pattern. Fusion Fedorova, Zeitlin 8 Outline 6 4 Multiphysics W(q,p) 2 Hopes 0 −2 Introduction 30 30 20 25 BBGKY hierarchy 20 10 15 10 5 0 0 coordinate (q) momentum (p) Multiscale analysis Variational approach Figure 12: Localized pattern: waveleton. Modeling of patterns Vlasov-Poisson system Multiscale representations 1 RATE/RMS models 0.8 RATE equations 0.6 0.4 Multiscale 0.2 representations distribution function F(q,p) 0 60 Conclusions 60 40 40 20 20 0 0 coordinate (q) momentum (p) Figure 13: Localized pattern: waveleton. Fusion Fedorova, Zeitlin 5 Outline 4 3 Multiphysics 2 1 Hopes 0 Introduction −1 150 120 100 100 BBGKY hierarchy 80 50 60 40 20 0 0 Multiscale analysis Variational approach Figure 14: Waveleton Pattern Modeling of patterns Vlasov-Poisson system Multiscale representations 5 RATE/RMS models 4 3 RATE equations 2 Multiscale 1 representations 0 −1 150 Conclusions 120 100 100 80 50 60 40 20 0 0 Figure 15: Waveleton Pattern Fusion Fedorova, Zeitlin 0.8 Outline 0.6 Multiphysics 0.4 Hopes 0.2 distribution
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