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, , V , H = H − N 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
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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
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−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
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−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 function F(q,p) 0 Introduction 30 30 20 20 BBGKY hierarchy 10 10 0 0 coordinate (q) momentum (p) Multiscale analysis
Variational approach
Figure 16: Localized mode contribution to distribution function. 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 17: Chaotic-like pattern. 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 18: Localized waveleton pattern. Modeling of patterns
Vlasov-Poisson system
Multiscale representations
0.08 RATE/RMS models 0.06 0.04 RATE equations 0.02 0 Multiscale −0.02 representations −0.04
−0.06 60 Conclusions 60 40 50 40 20 30 20 10 0 0
Figure 19: Eigenmode of level 1. Fusion
Fedorova, Zeitlin
8 Outline 6
4 Multiphysics
2 Hopes
0 Introduction −2 30 30 20 25 BBGKY hierarchy 20 10 15 10 5 Multiscale analysis 0 0 Variational approach
Figure 20: Stable waveleton pattern. Modeling of patterns
Vlasov-Poisson system
Multiscale representations
2 RATE/RMS models
1.5
1 RATE equations
0.5
0 Multiscale −0.5 representations
−1
−1.5 Conclusions 60 60 40 50 40 20 30 20 10 0 0
Figure 21: Chaotic-like behaviour. Fusion
Fedorova, Zeitlin −3 x 10 10 Outline
5 Multiphysics
0 Hopes
−5 Introduction 30 30 20 25 20 BBGKY hierarchy 10 15 10 5 0 0 Multiscale analysis
Variational approach
Figure 22: Waveleton-like distribution. Modeling of patterns
Vlasov-Poisson system
Multiscale −1 representations
−2 RATE/RMS models −3
−4 RATE equations
−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 23: Multiscale decomposition. Fusion
Fedorova, Zeitlin
Outline
Multiphysics
Hopes
Introduction
BBGKY hierarchy
Multiscale analysis
Variational approach
Modeling of patterns
Vlasov-Poisson system
Multiscale representations
RATE/RMS models
RATE equations
Multiscale representations
Conclusions
Figure 24: Localized modes. Fusion
MATH Fedorova, Zeitlin How much is it needed? Outline
Multiphysics 0 % (subset of measure zero) ≫ 100 % Hopes
Introduction CPUIntelPentium1-5 SuperStringPhysics BBGKY hierarchy DVD Sub-PlanckianPhysics, Multiscale analysis CellPhone QuantumGravity,Higgs Variational approach
Modeling of patterns ? ←− Plasma Fusion −→ ? Vlasov-Poisson system Multiscale Is MATH really needful thing to describe Plasma World? — representations Or it is a place for engineers only :-) :-( RATE/RMS models RATE equations MOTIVES: Multiscale representations Folklor: a magnetically confined plasma cannot be in thermodynamical Conclusions equilibrium... Instabilities...(meta) stability (long living fluctuations) VERY complex system: lineariazation*, perturbation methods, U(1) Fourier analysis (SIN, COS, EXP i(kx − ωt)) are not proper *) wave eq. vs. sine - Gordon (or KdV or KP etc): solitons, breathers, finite-gap solutions differ from solutions of linear equations. Fusion
HOPES, ILLUSIONS (lost), ... , PARADIGMS, Fedorova, Zeitlin
LESSONS (after Martin Kruskal) Outline
Multiphysics
Hopes Tori Introduction BBGKY hierarchy FPU −→ Liouvillean −→ KAM −→ Cantori −→ ... Multiscale analysis Variational approach
L-Tori Windings (Soliton Era 1967-1984/9) Modeling of patterns KP example Vlasov-Poisson system Multiscale 3 1 representations uy = wx , wy = ut − (6uux + uxxx ) RATE/RMS models 4 4 3 ∂ 1 RATE equations uyy = [ut − (6uux + uxxx )] Multiscale 4 ∂x 4 representations ∂L ∂A Conclusions − =[A, L] ∂t ∂y
2 3 3 L = ∂x + u, A = ∂ + (u∂x + ∂x u) + w x 4 Zero - curvature representation, Lax pair Fusion
Fedorova, Zeitlin 2 u(x, y, t) = 2∂x log Θ(xU + yV + tW + z0) + C Outline
3 Multiphysics w(x, y, t) = ∂x ∂y log Θ(xU + yV + tW + z0) + C1 2 Hopes Θ = Θ(z) - theta function of Riemann surface Γ wrt a basis of cycles Introduction BBGKY hierarchy a1, b1,... bg , Ui , Vi , Wi - b-periods Multiscale analysis
Variational approach Ui = Ω1, Vi = Ω2, Wi = Ω3, Modeling of patterns bi bi bi Vlasov-Poisson system
Ω1, Ω2, Ω3 - normalized second kind diff. Multiscale representations
RATE/RMS models
RATE equations
Multiscale representations
Conclusions Fusion
KP Hierarchy Fedorova, Zeitlin
Γ - Riemann surface, BA function Outline Multiphysics
Hopes ∞ 2 3 ζi kx+k t2+k t3+... Introduction Ψ(x, t2, t3,..., p) = 1 + e ki BBGKY hierarchy i=1 Multiscale analysis KP(t2 = y, t3 = t, t1 = x) Variational approach
∂Ψ Modeling of patterns = AnΨ ∂tn Vlasov-Poisson system n n n n−i Multiscale A1 = ∂x , A2 = L, A3 = A, An = ∂x + ui ∂x representations i=2 RATE/RMS models n n RATE equations u2 ,..., un via ζ1,...,ζn−1 recursively and satisfy infinite system of Multiscale differential equations, KP hierarchy: representations Conclusions ∂ ∂ ∂An ∂Am − + An, − + Am = − +[An, Am]=0, ∂tn ∂tm ∂tm ∂tn n, m =1, 2,... Fusion
Zero-curvature representation: Fµν = 0, Fedorova, Zeitlin
−1 Outline Plane connection A ∼ ∂ gg Moduli space (Hitchin) Multiphysics Casimirs, BT, Kac-Moody/Virasoro, Lie-Poisson, Drinfeld-Sokolov, Hopes Coalgebras, Hopf, Quantum Groups Introduction BBGKY hierarchy
Multiscale analysis KAM, etc. Variational approach Modeling of patterns
Vlasov-Poisson system
Multiscale Hilbert 16; Homoclinic; Heteroclinic; Stable/Unstable; Bifurcations; representations
RATE/RMS models Heteroclinic bifurcations: resonance, transverse RATE equations Multiscale representations Resonance: birth and death of periodic orbit Conclusions (Magnetic-reconection);
Transverse: (un)stability of heteroclinic cycle; Fusion
Arnold diffusion: Fedorova, Zeitlin
Outline
Multiphysics
H(I , p, q, Φ, t), I (0) < δ, I (T ) > K, T > 0 Hopes SRB - New Ergodic Theory Introduction BBGKY hierarchy
Multiscale analysis
Hyperbolic dynamics: Anosov Flows Variational approach
Modeling of patterns Attractor(s) - repeller(s) coexistence Vlasov-Poisson system Multiscale representations
Attractor: Fixed point, Limit cycle, limit tori, Strange attractor (dim RATE/RMS models
∈/ N) RATE equations
Multiscale representations Sensitivity to initial data (Chaos) Conclusions
Hidden symmetries (Fuchsian, Kleinian, PSL(2,Z), Modular)
Thuston-Milnor-Sullivan: Quasiconformal Dynamics Fusion
J. Leray’s way: Fedorova, Zeitlin
Outline Algebraical Analysis (Sato) Multiphysics Hopes
Microlocalizations( Kashiwara-Shapira) ΨDO Introduction
BBGKY hierarchy
Representation Theory of Hidden Symmetries of Functional Spaces Multiscale analysis Local Nonlinear Harmonic Analysis on the Orbits Variational approach Modeling of patterns Frames, Atomic Decomposition, Basis Pursuit, Modulation Spaces/ Vlasov-Poisson system Multiscale Feichtinger’s algebra representations
RATE/RMS models SLE, Dimers (Kenyon), (Random) Matrix models RATE equations
Multiscale Classical/Quantum representations Conclusions
Moyal/Deformation Quantization
Wigner-Weyl (ΨDO)
Sheaves as states: quantum points arXiv:1109.5035, 1109.5042 Fusion
Fedorova, Zeitlin INTRODUCTION Outline Multiphysics
Hopes
Introduction I. Class of Models BBGKY hierarchy a). Individual cM/qM (linear/nonlinear; {cM}⊂{qM}), Multiscale analysis (∗ - Quantization of) Polynomial/Rational Hamiltonians: Variational approach
i j Modeling of patterns H(p, q, t) = aij (t)p q Vlasov-Poisson system i.j Multiscale Important example: Orbital motion (in Storage Rings). representations The magnetic vector potential of a magnet with 2n poles in Cartesian RATE/RMS models RATE equations coordinates is Multiscale A = Knfn(x, y), representations
n Conclusions where fn is a homogeneous function of x and y of order n. The cases n =2 to n = 5 correspond to low-order multipoles: quadrupole, sextupole, octupole, decupole. Fusion
Fedorova, Zeitlin
Outline The corresponding Hamiltonian is: Multiphysics
2 2 Hopes px + py H(x, px , y, py , s) = + Introduction 2 BBGKY hierarchy 1 x 2 y 2 − k1(s) + k1(s) Multiscale analysis ρ2(s) 2 2 Variational approach
Modeling of patterns kn(s) + ijn(s) (n+1) −Re (x + iy) Vlasov-Poisson system (n + 1)! n≥2 Multiscale representations Terms corresponding to kick type contributions of rf-cavity: RATE/RMS models RATE equations L 2π Aτ = − V0 cos k τ δ(s − s0) Multiscale 2πk L representations Conclusions or localized cavity V (s) = V0 δp(s − s0) with n=+∞ δp(s − s0) = n=−∞ δ(s − (s0 + n L)) at position s0. Fusion
Fedorova, Zeitlin
The second example: using Serret–Frenet parametrization, we have Outline after truncation of power series expansion of square root the following Multiphysics approximated (up to octupoles) Hamiltonian for orbital motion in Hopes machine coordinates: Introduction
2 2 BBGKY hierarchy 1 [px + H z] +[pz − H x] H = Multiscale analysis 2 [1 + f (pσ)] Variational approach
+pσ − [1 + Kx x + Kz z] f (pσ) Modeling of patterns
1 2 2 1 2 2 Vlasov-Poisson system + [Kx + g] x + [Kz − g] z − N xz 2 2 Multiscale representations λ 3 2 4 2 2 4 + (x − 3xz ) + (z − 6x z + x ) RATE/RMS models 6 24 RATE equations 1 L eV (s) 2π + cos h σ + ϕ Multiscale 2 representations β0 2π h E0 L Conclusions Then we use series expansion of function f (pσ): ′ ′′ 2 2 2 f (pσ) = f (0) + f (0)pσ + f (0)pσ/2 + ... = pσ − pσ/(2γ0 ) + ... and the corresponding expansion of RHS of equations. Fusion
Fedorova, Zeitlin
Outline
Multiphysics
Hopes
Introduction b). QFT (e.g. second quantization/Fock) BBGKY hierarchy c). BBGKY Hierarchy (with reductions to different forms VMP, Multiscale analysis rms/envelope dynamics (belong to a).) and all that) Variational approach d). Wignerization of a): Wigner-Moyal-Weyl-von Neumann-Lindblad Modeling of patterns e). Wignerization of c): Quantum (Non) Equilibrium Ensembles Vlasov-Poisson system Multiscale Important remark: Symbols of Op instead of Op: ΨDO picture. representations
(Non)Linear ΨDO Dynamics (all qM ⊂ ΨDOD) RATE/RMS models
RATE equations
Multiscale representations
Conclusions Fusion
Fedorova, Zeitlin
Outline II. Effects (what we are interested in) Multiphysics i).Hierarchy of (internal) scales (time, space, phase space). Multiscales Hopes (non-perturbative!): from slow to fast, from coarses to finest level of Introduction resolution/decomposition. Coexistence of hierarchy of multiscale BBGKY hierarchy dynamics (with transitions between scales ...for rmns: cascades in Multiscale analysis hydrodynamics). Variational approach If that, then we need (at least) Modeling of patterns ii). Localized (max!) modes/”harmonics” – Vlasov-Poisson system Nonlinear LOCALIZED Harmonics Multiscale representations instead of plane waves/(sin/cos) – farewell to ’stupid’ Fourier U(1) RATE/RMS models analysis, coherent states (any)/gaussians)... RATE equations What is a MODE/qubit? We take it from our zoo/library! What is Multiscale (simple) PATTERN/STATE? IT’S THE SOLUTION OF ΨDOD representations DECOMPOSED VIA THE BASES OF LOCALIZED (eigen) MODES! Conclusions (first, naive def.). Fusion
Fedorova, Zeitlin
Outline
Multiphysics
Hopes
Introduction
BBGKY hierarchy
Multiscale analysis
Variational approach
Modeling of patterns
Vlasov-Poisson system
Multiscale representations
RATE/RMS models
RATE equations
Multiscale representations
Conclusions
Figure 25: Localized modes. iii).Ensembles of states/qubits: Zoo of Patterns (controllable by Fusion construction) Fedorova, Zeitlin
Outline
Multiphysics ALL THAT inside the Class of Models (ΨDO Dynamics) −→ localized, Hopes entangled/chaotic, decoherent/quasiclassical. Introduction
BBGKY hierarchy
Multiscale analysis III. How? Methods? SYMMETRY! (HIDDEN symmetry) Variational approach Representation theory of internal/hidden/underlying symmetry. Modeling of patterns Kinematical, Dynamical, Hidden. Vlasov-Poisson system Arena (space of representation): (Hilbert) SPACE of STATES. Multiscale representations Symmetry inside Functional spaces! RATE/RMS models METHODS: RATE equations Harmonic analysis on (non)abelian Group Multiscale (kinematical/internal symmetry). Instead of U(1) Fourier analysis: representations Local/Nonlinear (non-abelian) Harmonic Analysis! Conclusions Dynamics on proper orbits/scales (MULTISCALES) in Functional spaces. Remark. ∃ much more powerful technique because of existence of HIDDEN Symmetry∗ *({QMF} −→ Loop groups −→ Cuntz op. alg. −→ Quantum Group structure (∃ natural Fock-like structure also). The key objects are: MULTIRESOLUTION (Multiscale) representation Fusion
(e.g, wavelet/gabor etc. analysis) Fedorova, Zeitlin −→ appearance of MULTISCALES (ORBITS) and localized (NATURAL eigen-) modes. Outline Multiphysics
2500 Hopes
2000 Introduction
1500 BBGKY hierarchy
1000 Multiscale analysis
Variational approach 500 Modeling of patterns 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Vlasov-Poisson system Figure 26: Kick Multiscale representations
RATE/RMS models
RATE equations −1
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−4 Conclusions −5
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Figure 27: MRA for Kick 6 Fusion
5 Fedorova, Zeitlin
4 Outline
Multiphysics 3 Hopes
2 Introduction
1 BBGKY hierarchy
Multiscale analysis 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Variational approach Figure 28: Multi-Kicks Modeling of patterns Vlasov-Poisson system
Multiscale representations −1 RATE/RMS models −2
−3 RATE equations
−4 Multiscale representations −5
−6 Conclusions
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Figure 29: MRA for Multi-Kicks Variational formulation (CMP standard: control of convergence, Fusion reductions to algebraic systems, control of type of behaviour). Fedorova, Zeitlin j i IV. Set-up/Problems (ΨDOD): L {Op }Ψ = 0 Outline Objects inside: Multiphysics 2 i). SPACE of STATES. H = {Ψ} (Hilbert) space of states: L , Sobolev, Hopes 0 k ∞ Schwartz, C , C , ... C , ...; Introduction 2 2 2 2 2 1 1 2 1 1 L (R ) vs. L (S ) vs. L (S × S ) vs. L (S × S ⋉ Zn): BBGKY hierarchy Tokamaks vs. Stellarators. Multiscale analysis Class of smoothness: Variational approach Dynamics with/without Chaos/Fractality. Modeling of patterns i ii). DECOMPOSITIONS. Ψ ≈ i ai e (bases, frames, atomic Vlasov-Poisson system decomposition): (exp) control of convergence, max(!) rate of Multiscale convergence for any Ψ in any H. representations RATE/RMS models iii). OBSERVABLES/OPERATORS (ODO, PDO, ΨDO, SIO,..., RATE equations Microlocal analysis of Kashiwara-Shapira): i Multiscale < Ψ|Op |Ψ > – Max sparse! (from functions to sheafs) representations Almost (max!) diagonal according to FWT (BCR Th.): Conclusions
D11 0 0 ... 0 D22 0 ... 0 0 D33 ... . . . . . . . .. iv). MEASURES: multifractal wavelet measures { i } Fusion
−3 x 10 Fedorova, Zeitlin 4
3.5
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2.5 Multiphysics 2
1.5 Hopes
1 Introduction 0.5
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−0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Multiscale analysis Figure 30: RW-fractal Variational approach Modeling of patterns
Vlasov-Poisson system
−1 Multiscale −2 representations
−3 RATE/RMS models −4
−5 RATE equations
−6
−7 Multiscale representations −8
−9 Conclusions
−10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Figure 31: MRA for RW-fractal v). VARIATIONAL/PROJECTION methods (from Galerkin to Floer/Rabinowitz, symplectic AW in coadjoint orbital dynamics). Preservation of Poisson/symplectic structures! Fusion
Fedorova, Zeitlin
Outline vi). MULTIRESOLUTION or wavelet microscope Multiphysics Hopes 1). (internal) symmetry (kinematic): Introduction affine group = {translations, dilations} or others... BBGKY hierarchy
2). representation/action of this symmetry on H = {Ψ} Multiscale analysis ⇓ as a result (coherence!) Variational approach a). localized bases b). multiscales Modeling of patterns vii). Effectiveness of proper numerics: CPU-time, HDD-space, minimal Vlasov-Poisson system complexity of algorithms and (Shannon) entropy of calculations. Multiscale Finally: LOCALIZES MODES, ZOO of PATTERNS: LOCALIZED, representations CHAOTIC/ENTANGLED,... RATE/RMS models LOCALIZED PATTERN: CONFINEMENT of ENERGY (FUSION RATE equations Multiscale STATE = WAVELETON). representations
V. Quantization: ∗ Star Product, Conclusions Deformation, Quantum Group, ...Renormalization,... Fusion
Fedorova, Zeitlin
Outline
Multiphysics
Hopes
Introduction
BBGKY hierarchy
Multiscale analysis
Variational approach
Modeling of patterns
Figure 32: Localized modes. Vlasov-Poisson system
Multiscale representations
RATE/RMS models
0.8 RATE equations
0.6 Multiscale 0.4 representations 0.2
distribution function F(q,p) 0 Conclusions 30 30 20 20 10 10 0 0 coordinate (q) momentum (p)
Figure 33: Localized mode contribution to distribution function. Fusion
Fedorova, Zeitlin
Outline 1 0.8 Multiphysics 0.6
0.4 Hopes
0.2 Introduction distribution function F(q,p) 0 60 60 40 40 BBGKY hierarchy 20 20 0 0 coordinate (q) momentum (p) Multiscale analysis
Variational approach Figure 34: Chaotic-like pattern. Modeling of patterns
Vlasov-Poisson system
Multiscale representations
3
2 RATE/RMS models
1 RATE equations 0 W(q,p) −1 Multiscale −2 150 representations 150 100 100 50 50 Conclusions 0 0 coordinate (q) momentum (p)
Figure 35: Entangled-like pattern. Fusion
Fedorova, Zeitlin
Outline 8
6 Multiphysics
4
W(q,p) 2 Hopes
0 Introduction −2 30 30 20 25 20 BBGKY hierarchy 10 15 10 5 0 0 coordinate (q) momentum (p) Multiscale analysis
Variational approach Figure 36: Localized pattern: waveleton. Modeling of patterns
Vlasov-Poisson system
Multiscale representations
1
0.8 RATE/RMS models 0.6 RATE equations 0.4
0.2 Multiscale
distribution function F(q,p) 0 60 representations 60 40 40 20 Conclusions 20 0 0 coordinate (q) momentum (p)
Figure 37: Localized pattern: waveleton. Introduction Fusion Fedorova, Zeitlin
Outline
Multiphysics
Hopes
Introduction
BBGKY hierarchy
Multiscale analysis
Variational approach We consider the application of a new numerical/analytical technique Modeling of patterns based on local nonlinear harmonic analysis approach for the description Vlasov-Poisson system of complex non-equilibrium behaviour of statistical ensembles, Multiscale considered in the framework of the general BBGKY hierarchy of kinetics representations equations and their different truncations/reductions. RATE/RMS models RATE equations
Multiscale representations
Conclusions Fusion
Fedorova, Zeitlin ◮ Kinetic theory is an important part of general statistical physics related to phenomena which cannot be understood on the Outline thermodynamic or fluid models level. Multiphysics ◮ We restrict ourselves to the rational/polynomial type of Hopes nonlinearities (with respect to the set of all dynamical variables) Introduction that allows to use our results, which are based on the so called BBGKY hierarchy multiresolution framework and the variational formulation of initial Multiscale analysis nonlinear (pseudodifferential) problems. Variational approach Modeling of patterns ◮ Wavelet analysis is a set of mathematical methods which give a Vlasov-Poisson system
possibility to work with well-localized bases in functional spaces Multiscale and provide the maximum sparse forms for the general type of representations operators (differential, integral, pseudodifferential) in such bases. RATE/RMS models ◮ It provides the best possible rates of convergence and minimal RATE equations Multiscale complexity of algorithms inside and, as a result, saves CPU time representations and HDD space. Conclusions ◮ Our main goals are an attempt of classification and construction of a possible zoo of nontrivial (meta) stable states: high-localized (nonlinear) eigenmodes, complex (chaotic-like or entangled) patterns, localized (stable) patterns (waveletons). Fusion We start from the corresponding qualitative definitions: Fedorova, Zeitlin ◮ By a localized state (localized mode) we mean the corresponding Outline (particular) solution (or generating mode) which is localized in Multiphysics maximally small region of the phase space. Hopes Introduction ◮ By a chaotic pattern we mean some solution (or asymptotics of BBGKY hierarchy solution) which has random-like distributed energy spectrum in a Multiscale analysis full domain of definition. In quantum case we need to consider Variational approach additional entangled-like patterns, roughly speaking, which cannot Modeling of patterns be separated into pieces of sub-systems. Vlasov-Poisson system ◮ By a localized pattern (waveleton) we mean (asymptotically) Multiscale (meta) stable solution localized in relatively small region of the representations whole phase space (or a domain of definition). In this case an RATE/RMS models energy is distributed during some time (sufficiently large) between RATE equations Multiscale only a few localized modes (from point 1). We believe, it is a good representations
image for plasma in a fusion state (energy confinement). Conclusions ◮ In all cases above, by the system under consideration in the classical case we mean the full BBGKY hierarchy or some cut-off of it. Our construction of cut-off of the infinite system of equations is based on some criterion of convergence of the full solution. Fusion ◮ This criterion is based on a natural norm in the proper functional Fedorova, Zeitlin space, which takes into account (non-perturbatively) the underlying multiscale structure of complex Outline statistical dynamics. According to our approach the choice of the Multiphysics underlying functional space is important to understand the Hopes corresponding complex dynamics. Introduction BBGKY hierarchy ◮ It is obvious that we need to fix accurately the space, where we Multiscale analysis construct the solutions, evaluate convergence etc. and where the Variational approach very complicated infinite set of operators, appeared in the BBGKY Modeling of patterns formulation, acts. Vlasov-Poisson system ◮ We underline that many concrete features of the complex dynamics Multiscale are related not only to the concrete form/class of representations operators/equations but depend also on the proper choice of RATE/RMS models function spaces, where operators act. It should be noted that the RATE equations Multiscale class of smoothness (related at least to the appearance of representations
chaotic/fractal-like type of behaviour) of the proper functional Conclusions space under consideration plays a key role in the following. ◮ Our main goals are an attempt of classification, construction and control of a possible zoo of nontrivial states/patterns. ◮ Localized (meta)stable pattern (waveleton) is a good image for fusion state in plasma (energy confinement). Fusion
Fedorova, Zeitlin
Outline Our constructions can be applied to the following general Hamiltonians: Multiphysics
N Hopes p2 i Introduction HN = + Ui (q) + Uij (qi , qj ), 2m i=1 1≤ i≤j≤N BBGKY hierarchy Multiscale analysis where the potentials Ui (q) = Ui (q1,..., qN ) and Uij (qi , qj ) Variational approach are restricted to rational functions on the coordinates. Let Ls and Lij be Modeling of patterns the Liouvillean operators and Vlasov-Poisson system
Multiscale FN (x1,..., xN ; t) representations RATE/RMS models be the hierarchy of N-particle distribution function, satisfying the RATE equations standard BBGKY hierarchy (V is the volume): Multiscale representations s ∂Fs 1 Conclusions + Ls Fs = d s+1 Li,s+1Fs+1 ∂t υ i=1 Fusion
Fedorova, Zeitlin
Our key point is the proper nonperturbative generalization of the Outline previous perturbative multiscale approaches. The infinite hierarchy of Multiphysics distribution functions is: Hopes Introduction
F = {F0, F1(x1; t),..., FN (x1,..., xN ; t),... }, BBGKY hierarchy where Multiscale analysis Variational approach p Modeling of patterns Fp(x1,..., xp; t) ∈ H , Vlasov-Poisson system 0 p 2 6p H = R, H = L (R ), Multiscale representations ∞ 0 1 p F ∈ H = H ⊕ H ⊕ ⊕ H ⊕ ... RATE/RMS models with the natural Fock space like norm (guaranteeing the positivity of RATE equations Multiscale the full measure): representations
Conclusions i 2 2 (F , F ) = F + Fi (x1,..., xi ; t) ℓ. 0 i ℓ=1 Fusion ◮ Multiresolution decomposition naturally and efficiently introduces Fedorova, Zeitlin
the infinite sequence of the underlying hidden scales, which is a Outline 2 sequence of increasing closed subspaces Vj ∈ L (R): Multiphysics
Hopes
...V−2 ⊂ V−1 ⊂ V0 ⊂ V1 ⊂ V2 ⊂ ... Introduction ◮ Our variational approach reduces the initial problem to the problem BBGKY hierarchy of solution of functional equations at the first stage and some Multiscale analysis algebraical problems at the second. Variational approach Modeling of patterns ◮ As a result, the solutions of the (truncated) hierarchies have the Vlasov-Poisson system
following multiscale decomposition via high-localized eigenmodes Multiscale l s m representations (ωl ∼ 2 , km ∼ 2 ) RATE/RMS models
RATE equations i j F (t, x1, x2,... ) = aij U ⊗ V (t, x1,... ), Multiscale representations (i, j)∈Z 2 Conclusions j j,slow j V (t) = VN (t) + Vl (ωl t), l≥N i i,slow i s U (xs ) = UM (xs ) + Um(kmxs ), m ≥M Fusion which corresponds to the full multiresolution expansion in all underlying Fedorova, Zeitlin time/space scales. ◮ In this way one obtains contributions to the full solution from each Outline scale of resolution or each time/space scale or from each nonlinear Multiphysics eigenmode. Hopes Introduction ◮ It should be noted that such representations give the best possible BBGKY hierarchy
localization properties in the corresponding (phase) space/time Multiscale analysis
coordinates. Variational approach ◮ Numerical calculations are based on compactly supported wavelets Modeling of patterns and related wavelet families and on evaluation of the accuracy on Vlasov-Poisson system the level N of the corresponding cut-off of the full system regarding Multiscale norm above. representations RATE/RMS models ◮ Numerical modeling shows the creation of different internal RATE equations
structures from localized modes, which are related to (meta)stable Multiscale or unstable type of behaviour and the corresponding patterns representations (waveletons) formation. Reduced algebraical structure provides the Conclusions pure algebraical control of stability/unstability scenario. ◮ CONTROLLABLE (META) STABLE WAVELETON CONFIGURATION REPRESENTS A REASONABLE APRROXIMATION FOR THE REALIZABLE CONFINEMENT STATE. Fusion
Fedorova, Zeitlin
5 4 Outline 3
2 Multiphysics 1 Hopes 0
−1 Introduction 150 120 100 100 BBGKY hierarchy 80 50 60 40 20 Multiscale analysis 0 0 Variational approach Figure 38: Waveleton Pattern Modeling of patterns Vlasov-Poisson system
Multiscale representations
RATE/RMS models 5
4 RATE equations
3 Multiscale 2 representations 1
0 Conclusions
−1 150 120 100 100 80 50 60 40 20 0 0
Figure 39: Waveleton Pattern BBGKY hierarchy Fusion Fedorova, Zeitlin
Outline We start from set-up for kinetic BBGKY hierarchy and present ex- Multiphysics plicit analytical construction for solutions of hierarchy of equations, Hopes which is based on tensor algebra extensions of multiresolution repre- Introduction sentation and variational formulation. We give explicit representa- BBGKY hierarchy tion for hierarchy of n-particle reduced distribution functions in the Multiscale analysis base of high-localized generalized coherent (regarding underlying Variational approach affine group) states given by polynomial tensor algebra of wavelets, Modeling of patterns which takes into account contributions from all underlying hidden Vlasov-Poisson system multiscales from the coarsest scale of resolution to the finest one Multiscale representations to provide full information about stochastic dynamical process. RATE/RMS models
RATE equations
So, our approach resembles Bogolubov and related approaches but we Multiscale don’t use any perturbation technique (like virial expansion) or representations linearization procedures. Conclusions Numerical modeling shows the creation of different internal (coher- ent) structures from localized modes, which are related to stable (equilibrium) or un stable type of behaviour and corresponding pat- tern (waveletons) formation. Fusion Let M be the phase space of ensemble of N particles (dimM =6N) with coordinates Fedorova, Zeitlin
Outline
Multiphysics
Hopes
xi =(qi , pi ), i =1, ..., N, Introduction 1 2 3 3 qi =(qi , qi , qi ) ∈ R , BBGKY hierarchy 1 2 3 3 Multiscale analysis pi =(pi , pi , pi ) ∈ R , Variational approach q q q R3N =( 1,..., N ) ∈ . Modeling of patterns
Vlasov-Poisson system
Multiscale representations
RATE/RMS models
RATE equations Individual and collective measures are: Multiscale representations
Conclusions
N
i = dxi = dqi pi , = i i=1 Distribution function Fusion Fedorova, Zeitlin
DN (x1,..., xN ; t) Outline
Multiphysics
Hopes
Introduction
BBGKY hierarchy satisfies Liouville equation of motion for ensemble with Hamiltonian HN : Multiscale analysis
Variational approach
Modeling of patterns
Vlasov-Poisson system ∂DN = {HN , DN } Multiscale ∂t representations
RATE/RMS models
RATE equations
Multiscale and normalization constraint representations
Conclusions
DN (x1,..., xN ; t)d =1 where Poisson brackets are: Fusion Fedorova, Zeitlin
Outline
N Multiphysics H D H D ∂ N ∂ N ∂ N ∂ N Hopes {HN , DN } = − ∂qi ∂pi ∂pi ∂qi i=1 Introduction BBGKY hierarchy
Multiscale analysis Our constructions can be applied to the following general Hamiltonians: Variational approach Modeling of patterns
Vlasov-Poisson system
Multiscale representations N 2 pi RATE/RMS models HN = + Ui (q) + Uij (qi , qj ) 2m RATE equations i=1 1≤ i≤j≤N Multiscale representations
Conclusions where potentials Ui (q) = Ui (q1,..., qN ) and Uij (qi , qj ) are not more than rational functions on coordinates. Fusion
Fedorova, Zeitlin Let Ls and Lij be the Liouvillean operators (vector fields)
Outline
Multiphysics
Hopes s pj ∂ ∂uj ∂ Introduction Ls = − − Lij m ∂qj ∂q ∂pj BBGKY hierarchy j=1 1≤i leqj≤s Multiscale analysis
Variational approach
∂Uij ∂ ∂Uij ∂ Modeling of patterns Lij = + ∂qi ∂pi ∂qj ∂pj Vlasov-Poisson system
Multiscale representations
RATE/RMS models
For s=N we have the following representation for Liouvillean vector field RATE equations
Multiscale representations
Conclusions
LN = {HN , } Fusion and the corresponding ensemble equation of motion: Fedorova, Zeitlin
Outline
Multiphysics
Hopes ∂DN + LN DN =0 Introduction ∂t BBGKY hierarchy
Multiscale analysis
Variational approach
LN is self-adjoint operator regarding standard pairing on the set of Modeling of patterns phase space funct ions. Let Vlasov-Poisson system Multiscale representations
RATE/RMS models
RATE equations FN (x1,..., xN ; t) = DN (x1,..., xN ; t) Multiscale SN representations Conclusions
be the N-particle distribution function (SN is permutation group of N element s). Then we have the hierarchy of reduced distribution functions (V s is the corresponding normalized volume factor) Fusion
Fedorova, Zeitlin
Outline
Multiphysics
Fs (x1,..., xs ; t) = Hopes
s Introduction V DN (x1,..., xN ; t) i BBGKY hierarchy s+1 ≤i≤N Multiscale analysis
Variational approach
Modeling of patterns
After standard manipulations we arrived to BBGKY hierarchy: Vlasov-Poisson system
Multiscale representations s ∂Fs 1 RATE/RMS models + Ls Fs = d s+1 Li,s+1Fs+1 ∂t υ RATE equations i=1 Multiscale representations
Conclusions It should be noted that we may apply our approach even to more general formulation. Fusion For s=1,2 we have: Fedorova, Zeitlin
Outline
Multiphysics
Hopes ∂F1(x1; t) p1 ∂ 1 + F1(x1; t) = dx2L12F2(x1, x2; t) Introduction ∂t m ∂q1 υ BBGKY hierarchy
Multiscale analysis
∂F2(x1, x2; t) p1 ∂ p2 ∂ Variational approach + + − L12 ∂t m ∂q1 m ∂q2 Modeling of patterns 1 Vlasov-Poisson system F2(x1, x2; t) = dx3(L13 + L23)F3(x1, x2; t) Multiscale υ representations
RATE/RMS models As in the general as in particular situations (cut-off, e.g.) we are RATE equations interested in the cases when Multiscale representations k Conclusions
Fk (x1,..., xk ; t) = F1(xi ; t) + Gk (x1,..., xk ; t), i=1 where Gk are correlators, really have additional reductions as in the simplest case of one-particle Vlasov/Boltzmann-like systems. Fusion
Fedorova, Zeitlin
Outline
Multiphysics
Hopes
Introduction Then by using such physical motivated reductions or/and during the BBGKY hierarchy corresponding cut-off procedure we obtain instead of linear and Multiscale analysis pseudodifferential (in general case) equations their finite-dimensional Variational approach but nonlinear approximations with the polynomial type of nonlinearities Modeling of patterns (more exactly, multilinearities). Our key point in the following Vlasov-Poisson system Multiscale consideration is the proper generalization of naive perturbative representations multiscale Bogolubov’s structure. RATE/RMS models
RATE equations
Multiscale representations
Conclusions Multiscale analysis Fusion Fedorova, Zeitlin
Outline
The infinite hierarchy of distribution functions satisfying BBGKY Multiphysics
system in the thermodynamical limit is: Hopes
Introduction
BBGKY hierarchy
Multiscale analysis
F = {F0, F1(x1; t), F2(x1, x2; t),..., Variational approach
FN (x1,..., xN ; t),... } Modeling of patterns Vlasov-Poisson system
Multiscale representations
RATE/RMS models
where RATE equations
Multiscale representations
Conclusions p 0 p 2 6p Fp(x1,..., xp; t) ∈ H , H = R, H = L (R )
(or any different proper functional space), Fusion F ∈ H∞ = H0 ⊕ H1 ⊕ ⊕ Hp ⊕ ... Fedorova, Zeitlin
Outline
Multiphysics
Hopes with the natural Fock-space like norm (of course, we keep in mind the Introduction positivity of the full measure): BBGKY hierarchy
Multiscale analysis
Variational approach
Modeling of patterns i 2 2 Vlasov-Poisson system (F , F ) = F0 + Fi (x1,..., xi ; t) ℓ Multiscale i ℓ=1 representations
RATE/RMS models
k RATE equations
Fk (x1,..., xk ; t) = F1(xi ; t) Multiscale representations i=1 Conclusions
First of all we consider F = F (t) as function on time variable only, F ∈ L2(R), via multiresolution decomposition which naturally and efficiently introduces the infinite sequence of underlying hidden scales into the game. Fusion Because affine group of translations and dilations is inside the ap- proach, this method resembles the action of a microscope. We Fedorova, Zeitlin have contribution to final result from each scale of resolution from Outline the whole infinite scale of spaces. Multiphysics Hopes
Introduction Let the closed subspace Vj (j ∈ Z) correspond to level j of resolution, or to scale j. We consider a multiresolution analysis of L2(R) (of course, BBGKY hierarchy we may consider any different functional space) which is a sequence of Multiscale analysis Variational approach increasing closed subspaces Vj : Modeling of patterns
Vlasov-Poisson system
Multiscale representations ...V−2 ⊂ V−1 ⊂ V0 ⊂ V1 ⊂ V2 ⊂ ... RATE/RMS models
RATE equations
Multiscale representations satisfying the following properties: let Wj be the orthonormal Conclusions complement of Vj with respect to Vj+1:
Vj+1 = Vj Wj then we have the following decomposition: Fusion Fedorova, Zeitlin
Outline
Multiphysics
{F (t)} = Wj Hopes
−∞ BBGKY hierarchy Multiscale analysis Variational approach or in case when V0 is the coarsest scale of resolution: Modeling of patterns Vlasov-Poisson system Multiscale ∞ representations RATE/RMS models {F (t)} = V0 Wj , j=0 RATE equations Multiscale representations Conclusions Subgroup of translations generates basis for fixed scale number: j/2 j spank∈Z {2 Ψ(2 t − k)} = Wj . Fusion The whole basis is generated by action of full affine group: Fedorova, Zeitlin Outline Multiphysics j/2 j Hopes span {2 Ψ(2 t − k)} = span {Ψj k } = {F (t)} k∈Z,j∈Z k,j∈Z , Introduction BBGKY hierarchy Multiscale analysis Variational approach Let sequence Modeling of patterns t t 2 {Vj }, Vj ⊂ L (R) Vlasov-Poisson system correspond to multiresolution analysis on time axis, Multiscale representations xi RATE/RMS models {Vj } RATE equations correspond to multiresolution analysis for coordinate xi , then Multiscale representations Conclusions n+1 x1 xn t Vj = Vj ⊗ ⊗ Vj ⊗ Vj corresponds to multiresolution analysis for n-particle distribution fuction Fusion Fn(x1,..., xn; t). E.g., for n = 2: Fedorova, Zeitlin Outline Multiphysics Hopes V 2 f f x x a 2 x k x k a 2 Z 2 0 = { : ( 1, 2) = k1,k2 φ ( 1 − 1, 2 − 2), k1,k2 ∈ ℓ ( )}, Introduction k ,k 1 2 BBGKY hierarchy Multiscale analysis Variational approach where Modeling of patterns 2 1 2 1 2 Vlasov-Poisson system φ (x1, x2) = φ (x1)φ (x2) = φ ⊗ φ (x1, x2), Multiscale i xi representations and φ (xi ) ≡ φ(xi ) form a multiresolution basis corresponding to {Vj }. If RATE/RMS models 1 RATE equations {φ (x1 − ℓ)}, ℓ ∈ Z Multiscale form an orthonormal set, then representations Conclusions 2 φ (x1 − k1, x2 − k2) 2 form an orthonormal basis for V0 . Action of affine group provides us by multiresolution representation of L2(R2). After introducing detail spaces 2 Wj , we have, e.g. 2 2 2 V1 = V0 ⊕ W0 . 2 Fusion Then 3-component basis for W0 is generated by translations of three functions Fedorova, Zeitlin Outline Multiphysics Hopes 2 1 2 Ψ1 = φ (x1) ⊗ Ψ (x2), Introduction 2 1 2 Ψ2 = Ψ (x1) ⊗ φ (x2), BBGKY hierarchy 2 1 2 Multiscale analysis Ψ3 = Ψ (x1) ⊗ Ψ (x2) Variational approach Modeling of patterns Vlasov-Poisson system Multiscale Also, we may use the rectangle lattice of scales and one-dimentional representations wavelet decomposition : RATE/RMS models RATE equations Multiscale representations Conclusions f (x1, x2) = < f , Ψi,ℓ⊗Ψj,k > Ψj,ℓ⊗Ψj,k (x1, x2) i ,ℓ;j,k where bases functions Ψi,ℓ ⊗ Ψj,k depend on two scales 2−i and 2−j . Fusion Fedorova, Zeitlin Outline Multiphysics Hopes Introduction BBGKY hierarchy We obtain our multiscale/multiresolution representations below) via the Multiscale analysis variational wavelet approach for the following formal representation of Variational approach the BBGKY system (or its finite-dimensional nonlinear approximation Modeling of patterns for the n-particle distribution functions) with the corresponding obvious Vlasov-Poisson system Multiscale constraints on the distribution functions. representations RATE/RMS models RATE equations Multiscale representations Conclusions Variational approach Fusion Fedorova, Zeitlin Outline Multiphysics Let L be an arbitrary (non)linear differential/integral operator with Hopes matrix dimension d (finite or infinite), which acts on some set of Introduction n functions from L2(Ω⊗ ): BBGKY hierarchy 1 d Multiscale analysis Ψ ≡ Ψ(t, x1, x2,... ) = Ψ (t, x1, x2,... ),... ,Ψ (t, x1, x2,... ) , Variational approach 6 xi ∈ Ω ⊂ R , n is the number of particles: Modeling of patterns Vlasov-Poisson system L L Q t x t x Ψ ≡ ( , , i )Ψ( , i )=0, Multiscale representations Q ≡ Qd ,d ,d ,...(t, x1, x2,...,∂/∂t,∂/∂x1,∂/∂x2,..., k ) = 0 1 2 RATE/RMS models RATE equations d ,d ,d ,... 0 1 2 i i i ∂ 0 ∂ 1 ∂ 2 Multiscale qi0i1i2...(t, x1, x2,... ) ... k representations ∂t ∂x1 ∂x2 i0,i1 ,i2, =1 Conclusions Let us consider now the N mode approximation for the solution as the following ansatz: Fusion Fedorova, Zeitlin N Outline N Ψ (t, x1, x2,... ) = ai0i1i2...Ai0 ⊗ Bi1 ⊗ Ci2 ... (t, x1, x2,... ) Multiphysics i0,i1 ,i2, =1 Hopes Introduction We shall determine the expansion coefficients from the following BBGKY hierarchy conditions (different related variational approaches are considered also): Multiscale analysis Variational approach N N ℓk k k ≡ (LΨ )Ak (t)Bk (x1)Ck (x2)dtdx1dx2 =0 0, 1, 2,... 0 1 2 Modeling of patterns Vlasov-Poisson system n n Thus, we have exactly dN algebraical equations for dN unknowns Multiscale representations ai0,i1,.... This variational approach reduces the initial problem to the problem of solution of functional equations at the first stage and some RATE/RMS models algebraical problems at the second. We consider the multiresolution RATE equations Multiscale expansion as the second main part of our construction. The solution is representations parametrized by the solutions of two sets of reduced algebraical Conclusions problems, one is linear or nonlinear (depending on the structure of the operator L) and the rest are linear problems related to the computation of the coefficients of the algebraic equations. These coefficients can be found by some methods by using the compactly supported wavelet basis functions. Fusion Fedorova, Zeitlin As a result the solution has the following multiscale/multiresolution decomposition via nonlinear high-localized eigenmodes Outline Multiphysics Hopes Introduction i j BBGKY hierarchy F (t, x1, x2,... ) = aij U ⊗ V (t, x1, x2,... ) Multiscale analysis (i, j)∈Z 2 Variational approach j j,slow j l V (t) = VN (t) + Vl (ωl t), ωl ∼ 2 Modeling of patterns l≥N Vlasov-Poisson system i i,slow i s s m Multiscale U (xs ) = UM (xs ) + Um(kmxs ), km ∼ 2 , representations m ≥M RATE/RMS models RATE equations Multiscale representations which corresponds to the full multiresolution expansion in all underlying Conclusions time/space scales. These formulae give the expansion into a slow and fast oscillating parts. So, we may move from the coarse scales of resolution to the finest ones for obtaining more detailed information about the dynamical process. Fusion Fedorova, Zeitlin In this way we give contribution to our full solution from each Outline scale of resolution or each time/space scale or from each non- Multiphysics linear eigenmode. It should be noted that such representations Hopes give the best possible localization properties in the corresponding Introduction (phase)space/time coordinates. BBGKY hierarchy Multiscale analysis In contrast with different approaches our formulae do not use Variational approach perturbation technique or linearization procedures. Numerical Modeling of patterns calculations are based on compactly supported wavelets and related Vlasov-Poisson system Multiscale wavelet families and on evaluation of the accuracy regarding norm: representations RATE/RMS models RATE equations Multiscale F N+1 − F N ≤ ε representations Conclusions Modeling of patterns Fusion Fedorova, Zeitlin Outline To summarize, the key points are: Multiphysics 1. The ansatz-oriented choice of the (multidimensional) bases related to Hopes some polynomial tensor algebra. Introduction 2. The choice of proper variational principle. A few projection/ BBGKY hierarchy Galerkin-like principles for constructing (weak) solutions are considered. Multiscale analysis The advantages of formulations related to biorthogonal (wavelet) Variational approach decomposition should be noted. Modeling of patterns 3. The choice of bases functions in the scale spaces Wj from wavelet Vlasov-Poisson system zoo. They correspond to high-localized (nonlinear) Multiscale oscillations/excitations, nontrivial local (stable) representations distributions/fluctuations, etc. Besides fast convergence properties it RATE/RMS models should be noted minimal complexity of all underlying calculations, RATE equations especially in case of choice of wavelet packets which minimize Shannon Multiscale representations entropy. Conclusions 4. Operator representations providing maximum sparse representations for arbitrary (pseudo) differential/ integral operators df /dx, dnf /dx n, T (x, y)f (y)dy), etc. 5. (Multi)linearization. Besides the variation approach we can consider also a different method to deal with (polynomial) nonlinearities: para-products-like decompositions. To classify the qualitative behaviour we apply standard methods from Fusion general control theory or really use the control. We will start from a Fedorova, Zeitlin priori unknown coefficients, the exact values of which will subsequently Outline be recovered. Roughly speaking, we will fix only class of nonlinearity Multiphysics (polynomial in our case) which covers a broad variety of examples of Hopes possible truncation of the systems. As a simple model we choose Introduction band-triangular non-sparse matrices (aij ) in particular case d = 2. BBGKY hierarchy These matrices provide tensor structure of bases in (extended) phase Multiscale analysis space and are generated by the roots of the reduced variational Variational approach (Galerkin-like) systems. As a second step we need to restore the Modeling of patterns coefficients from these matrices by which we may classify the types of Vlasov-Poisson system behaviour. We start with the localized mode, which is a base Multiscale mode/eigenfunction, Fig. 14, corresponding to def. 1, which was representations constructed as a tensor product of the two Daubechies functions. Fig. RATE/RMS models 15, corresponding to def. 2, presents the result of summation of series RATE equations Multiscale up to value of the dilation/scale parameter equal to six on the bases of representations symmlets with the corresponding matrix elements equal to one. The Conclusions size of matrix is 512x512 and as a result we provide modeling for one-particle distribution function corresponding to standard Vlasov-like 2 cut-off with F2 = F1 . So, different possible distributions of the root values of the generical algebraical system provide qualitatively different types of behaviour. The above choice provides us by a distribution with chaotic-like equidistribution. Fusion Fedorova, Zeitlin Outline Multiphysics Hopes But, if we consider a band-like structure of matrix (aij ) with the band Introduction along the main diagonal with finite size (≪ 512) and values, e.g. five, BBGKY hierarchy while the other values are equal to one, we obtain localization in a fixed Multiscale analysis finite area of the full phase space, i.e. almost all energy of the system is Variational approach concentrated in this small volume. This corresponds to definition 3 and Modeling of patterns is shown in Fig. 16, constructed by means of Daubechies-based wavelet Vlasov-Poisson system packets. Depending on the type of solution, such localization may be Multiscale representations present during the whole time evolution (asymptotically-stable) or up to RATE/RMS models the needed value from time scale (e.g. enough for plasma RATE equations fusion/confinement). Multiscale representations Conclusions Fusion Fedorova, Zeitlin Outline 0.8 0.6 Multiphysics 0.4 Hopes 0.2 distribution function F(q,p) Introduction 0 30 30 20 20 BBGKY hierarchy 10 10 0 0 coordinate (q) momentum (p) Multiscale analysis Variational approach Figure 40: Localized mode contribution to distribution function. Modeling of patterns Vlasov-Poisson system Multiscale representations 1 0.8 RATE/RMS models 0.6 RATE equations 0.4 0.2 Multiscale distribution function F(q,p) 0 60 representations 60 40 40 20 20 Conclusions 0 0 coordinate (q) momentum (p) Figure 41: Chaotic-like pattern. Fusion Fedorova, Zeitlin Outline Multiphysics Hopes 1 Introduction 0.8 BBGKY hierarchy Multiscale analysis 0.6 Variational approach 0.4 Modeling of patterns 0.2 Vlasov-Poisson system distribution function F(q,p) 0 Multiscale 60 representations 60 40 40 RATE/RMS models 20 20 RATE equations 0 0 coordinate (q) momentum (p) Multiscale representations Figure 42: Localized waveleton pattern. Conclusions Vlasov-Poisson system Fusion Fedorova, Zeitlin In this paper we consider the applications of numerical-analytical Outline approach based on multiscale variational wavelet technique to the Multiphysics systems with collective type behaviour described by some forms of Hopes Vlasov-Poisson/Maxwell equations. Such approach may be useful in all Introduction models in which it is possible and reasonable to reduce all complicated BBGKY hierarchy problems related with statistical distributions to the problems described Multiscale analysis by the systems of nonlinear ordinary/partial differential/integral Variational approach equations with or without some (functional) constraints. In periodic Modeling of patterns accelerators and transport systems at the high beam currents and Vlasov-Poisson system charge densities the effects of the intense self-fields, which are produced Multiscale by the beam space charge and currents, determinine (possible) representations equilibrium states, stability and transport properties according to RATE/RMS models underlying nonlinear dynamics. The dynamics of such space-charge RATE equations Multiscale dominated high brightness beam systems can provide the understanding representations of the instability phenomena such as emittance growth, mismatch, halo Conclusions formation related to the complicated behaviour of underlying hidden nonlinear modes outside of perturbative tori-like KAM regions. Our analysis is based on the variational-wavelet approach, which allows us to consider polynomial and rational type of nonlinearities. Fusion In some sense our approach is direct generaliztion of traditional nonlinear δF approach in which weighted Klimontovich representation Fedorova, Zeitlin Outline Multiphysics Hopes Nj Introduction f a w x x p p δ j = j ji δ( − ji )δ( − ji ) BBGKY hierarchy i=1 Multiscale analysis Variational approach Modeling of patterns or self-similar decompostion like Vlasov-Poisson system Multiscale representations RATE/RMS models Nj RATE equations δnj = bj wji s(x − xji ), Multiscale representations i=1 Conclusions where s(x − xji ) is a shape function of distributing particles on the grids in configuration space, are replaced by powerful technique from local nonlinear harmonic analysis, based on underlying symmetries of functional space such as affine or more general. Fusion Fedorova, Zeitlin The solution has the multiscale/multiresolution decomposition via nonlinear high-localized eigenmodes, which corresponds to the full Outline multiresolution expansion in all underlying time/phase space scales. Multiphysics Hopes Introduction Starting from Vlasov-Poisson equations, we consider the approach based BBGKY hierarchy on multiscale variational-wavelet formulation. We give the explicit Multiscale analysis representation for all dynamical variables in the base of compactly Variational approach supported wavelets or nonlinear eigenmodes. Our solutions are Modeling of patterns parametrized by solutions of a number of reduced algebraical problems Vlasov-Poisson system one from which is nonlinear with the same degree of nonlinearity as Multiscale initial problem and the others are the linear problems which correspond representations to the particular method of calculations inside concrete wavelet scheme. RATE/RMS models Because our approach started from variational formulation we can RATE equations control evolution of instability on the pure algebraical level of reduced Multiscale representations algebraical system of equations. This helps to control Conclusions stability/unstability scenario of evolution in parameter space on pure algebraical level. In all these models numerical modeling demonstrates the appearance of coherent high-localized structures and as a result the stable patterns formation or unstable chaotic behaviour. Fusion Fedorova, Zeitlin Analysis based on the non-linear Vlasov equations leads to more clear understanding of collective effects and nonlinear beam dynamics of high Outline intensity beam propagation in periodic-focusing and uniform-focusing Multiphysics transport systems. We consider the following form of equations Hopes Introduction BBGKY hierarchy Multiscale analysis ∂ ∂ ∂ ∂ψ ∂ + px + py − kx (s)x + − Variational approach s x y x p ∂ ∂ ∂ ∂ ∂ x Modeling of patterns ∂ψ ∂ Vlasov-Poisson system ky (s)y + fb(x, y, px , py , s)=0, ∂y ∂py Multiscale representations 2 2 ∂ ∂ 2πKb dp dp f RATE/RMS models 2 + 2 ψ = − x y b, ∂x ∂y Nb RATE equations Multiscale dxdydpx dpy fb = Nb representations Conclusions The corresponding Hamiltonian for transverse single-particle motion is given by Fusion Fedorova, Zeitlin Outline 1 2 2 1 2 Multiphysics H(x, y, px , py , s) = (px + py ) + [kx (s)x 2 2 Hopes 2 Introduction +ky (s)y ] + H1(x, y, px , py , s) + ψ(x, y, s), BBGKY hierarchy Multiscale analysis Variational approach Modeling of patterns where H1 is nonlinear (polynomial/rational) part of the full Hamiltonian and corresponding characteristic equations are: Vlasov-Poisson system Multiscale representations RATE/RMS models RATE equations 2 d x ∂ Multiscale + kx (s)x + ψ(x, y, s) = 0 ds2 ∂x representations d2y ∂ Conclusions + ky (s)y + ψ(x, y, s) = 0 ds2 ∂y Multiscale representations Fusion Fedorova, Zeitlin Outline We obtain our multiscale/multiresolution representations for solutions Multiphysics of these equations via variational-wavelet approach. We decompose the Hopes solutions as Introduction BBGKY hierarchy Multiscale analysis ∞ Variational approach i fb(s, x, y, px , py ) = ⊕δ f (s, x, y, px , py ) Modeling of patterns i=ic Vlasov-Poisson system ∞ Multiscale ψ(s, x, y) = ⊕δj ψ(s, x, y) representations RATE/RMS models j=jc RATE equations ∞ k Multiscale x(s) = ⊕δ x(s), representations k =kc Conclusions ∞ y(s) = ⊕δℓy(s) ℓ =ℓc Fusion where set Fedorova, Zeitlin (ic , jc , kc ,ℓc ) Outline corresponds to the coarsest level of resolution c in the full Multiphysics multiresolution decomposition Hopes Introduction BBGKY hierarchy Multiscale analysis Vc ⊂ Vc+1 ⊂ Vc+2 ⊂ ... Variational approach Modeling of patterns Vlasov-Poisson system W V Multiscale Introducing detail space j as the orthonormal complement of j with representations respect to RATE/RMS models RATE equations Multiscale representations Vj+1 : Vj+1 = Vj Wj , Conclusions we have for f ,ψ, x, y ⊂ L2(R) Fusion ∞ Fedorova, Zeitlin 2 L (R) = Vc Wj , Outline j=c Multiphysics Hopes Introduction In some sense it is some generalization of the old δF approach. Let L BBGKY hierarchy be an arbitrary (non) linear differential/integral operator with matrix Multiscale analysis dimension d, which acts on some set of functions Variational approach Modeling of patterns Vlasov-Poisson system 1 d Ψ ≡ Ψ(s, x) = Ψ (s, x),..., Ψ (s, x) , Multiscale representations s, x ∈ Ω ⊂ Rn+1 RATE/RMS models RATE equations Multiscale representations Conclusions from L2(Ω): LΨ ≡ L(R(s, x), s, x)Ψ(s, x)=0, Fusion Fedorova, Zeitlin (x are the generalized space coordinates or phase space coordinates, s is Outline “time” coordinate). After some anzatzes the main reduced problem Multiphysics may be formulated as the system of ordinary differential equations Hopes Introduction BBGKY hierarchy Multiscale analysis dfi Variational approach Qi (f ) = Pi (f , s), f =(f1, ..., fn), ds Modeling of patterns i =1,..., n, max deg Pi = p, max deg Qi = q i i Vlasov-Poisson system Multiscale representations RATE/RMS models or a set of such systems corresponding to each independent coordinate RATE equations Multiscale in phase space. They have the fixed initial (or boundary) conditions representations fi (0), where Pi , Qi are not more than polynomial functions of dynamical Conclusions variables fj and have arbitrary dependence on time. As result we have the following reduced algebraical system of equations on the set of k unknown coefficients λi of localized eigenmode expansion: Fusion Fedorova, Zeitlin Outline Multiphysics L(Qij ,λ,αI ) = M(Pij ,λ,βJ ), Hopes Introduction BBGKY hierarchy Multiscale analysis where operators L and M are algebraization of RHS and LHS of initial Variational approach problem and λ are unknowns of reduced system of algebraical equations Modeling of patterns (RSAE). After solution of RSAE we determine the coefficients of Vlasov-Poisson system wavelet expansion and therefore obtain the solution of our initial Multiscale problem. It should be noted that if we consider only truncated representations expansion with N terms then we have the system of N × n algebraical RATE/RMS models equations with degree RATE equations Multiscale ℓ = max{p, q} representations and the degree of this algebraical system coincides with degree of initial Conclusions differential system. So, we have the solution of the initial nonlinear (rational) problem in the form Fusion Fedorova, Zeitlin Outline N k Multiphysics fi (s) = fi (0) + λ fk (s), i Hopes k=1 Introduction BBGKY hierarchy Multiscale analysis k where coefficients λi are the roots of the corresponding reduced Variational approach algebraical (polynomial) problem RSAE. Consequently, we have a Modeling of patterns parametrization of solution of initial problem by the solution of reduced Vlasov-Poisson system algebraical problem. The obtained solutions are given in this form, Multiscale representations where fk (t) are basis functions obtained via multiresolution expansions RATE/RMS models and represented by some compactly supported wavelets. As a result the RATE equations solution of equations has the following multiscale/multiresolution Multiscale decomposition via nonlinear high-localized eigenmodes, which representations corresponds to the full multiresolution expansion in all underlying scales Conclusions starting from coarsest one For x =(x, y, px , py ) Fusion Fedorova, Zeitlin i j Ψ(s, x) = aij U ⊗ V (s, x), Outline (i, j)∈Z 2 Multiphysics j j slow j l , Hopes V (s) = VN (s) + Vl (ωl s), ωl ∼ 2 l≥N Introduction i i,slow i m BBGKY hierarchy U (x) = U (x) + Um(kmx), km ∼ 2 , M Multiscale analysis m ≥M Variational approach Modeling of patterns Vlasov-Poisson system Multiscale representations 0.08 RATE/RMS models 0.06 0.04 RATE equations 0.02 Multiscale 0 representations −0.02 −0.04 Conclusions −0.06 60 60 40 50 40 20 30 20 10 0 0 Figure 43: Eigenmode of level 1. Fusion Fedorova, Zeitlin 8 6 Outline 4 Multiphysics 2 Hopes 0 Introduction −2 30 30 BBGKY hierarchy 20 25 20 10 15 10 5 Multiscale analysis 0 0 Variational approach Figure 44: Stable waveleton pattern. Modeling of patterns Vlasov-Poisson system Multiscale representations RATE/RMS models 2 1.5 RATE equations 1 0.5 Multiscale 0 representations −0.5 Conclusions −1 −1.5 60 60 40 50 40 20 30 20 10 0 0 Figure 45: Chaotic-like behaviour. Fusion Fedorova, Zeitlin Outline Multiphysics slow This formula gives us expansion into the slow part ΨN,M and fast Hopes oscillating parts for arbitrary N, M. So, we may move from coarse scales Introduction of resolution to the finest one for obtaining more detailed information BBGKY hierarchy about our dynamical process. The first terms in the RHS correspond on Multiscale analysis the global level of function space decomposition to resolution space and Variational approach the second ones to detail space. It should be noted that such Modeling of patterns representations give the best possible localization properties in the Vlasov-Poisson system corresponding (phase)space/time coordinates. In contrast with different Multiscale approaches this formulae do not use perturbation technique or representations linearization procedures. So, by using wavelet bases with their good RATE/RMS models (phase) space/time localization properties we can describe RATE equations Multiscale high-localized (coherent) structures in spatially-extended stochastic representations systems with collective behaviour. Conclusions Fusion Fedorova, Zeitlin Outline Multiphysics Modelling demonstrates the appearance of stable patterns forma- Hopes tion from high-localized coherent structures or chaotic behaviour. Introduction On Fig. 17 we present contribution to the full expansion from coars- BBGKY hierarchy est level (waveleton) of decomposition. Fig. 18, 19 show the repre- Multiscale analysis sentations for full solutions, constructed from the first 6 eigenmodes Variational approach (6 levels in our formula), and demonstrate stable localized pattern Modeling of patterns formation and chaotic-like behaviour outside of KAM region. Vlasov-Poisson system Multiscale representations RATE/RMS models We can control the type of behaviour on the level of reduced algebraical RATE equations Multiscale system. representations Conclusions RATE/RMS models Fusion Fedorova, Zeitlin In this paper we consider the applications of a new numerical- Outline -analytical technique based on the methods of local nonlinear har- Multiphysics monic analysis or wavelet analysis to nonlinear rms/rate equations Hopes for averaged quantities related to some particular case of nonlinear Introduction Vlasov-Maxwell equations. BBGKY hierarchy Multiscale analysis Variational approach Modeling of patterns Our starting point is a model and approach proposed by R. C. Davidson Vlasov-Poisson system e.a.. We consider electrostatic approximation for a thin beam. This Multiscale approximation is a particular important case of the general reduction representations from statistical collective description based on Vlasov-Maxwell RATE/RMS models equations to a finite number of ordinary differential equations for the RATE equations Multiscale second moments related quantities (beam radius and emittance). In our representations case these reduced rms/rate equations also contain some disribution Conclusions averaged quantities besides the second moments, e.g. self-field energy of the beam particles. Such model is very efficient for analysis of many problems related to periodic focusing accelerators, e.g. heavy ion fusion and tritium production. So, we are interested in the understanding of collective properties, nonlinear dynamics and transport processes of intense non-neutral beams propagating through a periodic focusing field. Fusion Our approach is based on the variational-wavelet approach that allows Fedorova, Zeitlin to consider rational type of nonlinearities in rms/rate dynamical Outline equations containing statistically averaged quantities also. Multiphysics Hopes The solution has the multiscale/multiresolution decomposition via Introduction nonlinear high-localized eigenmodes (waveletons), which corre- BBGKY hierarchy sponds to the full multiresolution expansion in all underlying in- Multiscale analysis ternal hidden scales. We may move from coarse scales of resolu- Variational approach tion to the finest one to obtain more detailed information about Modeling of patterns our dynamical process. In this way we give contribution to our full Vlasov-Poisson system solution from each scale of resolution or each time/space scale or Multiscale representations from each nonlinear eigenmode. RATE/RMS models RATE equations Starting from some electrostatic approximation of Vlasov-Maxwell Multiscale system and rms/rate dynamical models we consider the approach based representations on variational-wavelet formulation. We give explicit representation for Conclusions all dynamical variables in the bases of compactly supported wavelets or nonlinear eigenmodes. Our solutions are parametrized by the solutions of a number of reduced standard algebraical problems. We present also numerical modelling based on our analytical approach. RATE equations Fusion Fedorova, Zeitlin In thin-beam approximation with negligibly small spread in axial Outline momentum for beam particles we have in Larmor frame the following Multiphysics electrostatic approximation for Vlasov-Maxwell equations: Hopes Introduction BBGKY hierarchy Multiscale analysis ∂F ′ ∂F ′ ∂F ∂ψ ∂F Variational approach + x + y − k(s)x + ′ ∂s ∂x ∂y ∂x ∂x Modeling of patterns ∂ψ ∂F Vlasov-Poisson system − k(s)y + =0 y y ′ Multiscale ∂ ∂ representations RATE/RMS models 2 2 RATE equations ∂ ∂ 2πK ′ ′ dx dy F Multiscale 2 + 2 ψ = − ∂x ∂y N representations Conclusions where ψ(x, y, s) is normalized electrostatic potential and F (x, y, x ′, y ′, s) is distribution function in transverse phase space Fusion Fedorova, Zeitlin ′ ′ (x, y, x , y , s) Outline with normalization Multiphysics Hopes Introduction BBGKY hierarchy N = dxdyn, n(x, y, s) = dx ′dy ′F Multiscale analysis Variational approach Modeling of patterns Vlasov-Poisson system Multiscale where K is self-field perveance which measures self-field intensity. representations Introducing self-field energy RATE/RMS models RATE equations Multiscale representations 1 Conclusions E(s) = dxdy|∂2ψ/∂x 2 + ∂2ψ/∂y 2| 4πK we have obvious equations for root-mean-square beam radius R(s) Fusion R(s) =< x 2 + y 2 >1/2 Fedorova, Zeitlin Outline Multiphysics Hopes and unnormalized beam emittance Introduction BBGKY hierarchy Multiscale analysis Variational approach 2 ′2 ′2 2 2 ′ ′ ε (s)=4(< x +y >< x +y > − < xx −yy >), Modeling of patterns Vlasov-Poisson system Multiscale representations which appear after averaging second-moments quantities regarding RATE/RMS models distribution function F : RATE equations Multiscale representations Conclusions d2R(s) K(1+∆) ε2(s) k s R s R s 2 + ( ) ( ) − 2 ( ) = 3 ds 2R (s) 4R (s) dε2(s) dR K(1+∆) dE(s) +8R2(s) − =0, ds ds 2R(s) ds Fusion where the term K(1+∆)/2 Fedorova, Zeitlin may be fixed in some interesting cases, but generally we have it only as Outline average Multiphysics Hopes Introduction BBGKY hierarchy K(1+∆)/2 = − < x∂ψ/∂x + y∂ψ/∂y > Multiscale analysis Variational approach Modeling of patterns Vlasov-Poisson system Multiscale regarding distribution F . Anyway, the rate equations represent representations reasoanable reductions for the second-moments related quantities from RATE/RMS models the full nonlinear Vlasov-Poisson system. For trivial distributions RATE equations Davidson e.a. found additional reductions. For KV distribution Multiscale (step-function density) the second rate equation is trivial, representations Conclusions ε(s) = const and we have only one nontrivial rate equation for rms beam radius. The fixed-shape density profile ansatz for axisymmetric distributions also leads to similar situation: emittance conservation and the same envelope equation with two shifted constants only. Multiscale representations Fusion Fedorova, Zeitlin Accordingly to our approach which allows us to find exact solutions as Outline for Vlasov-like systems as for rms-like systems we need not to fix Multiphysics particular case of distribution function Hopes Introduction F (x, y, x ′, y ′, s). BBGKY hierarchy Our consideration is based on the following multiscale N-mode anzatz: Multiscale analysis Variational approach Modeling of patterns Vlasov-Poisson system N 5 Multiscale N ′ ′ ′ ′ representations F (x, y, x , y , s) = ai1,...,i5 Aik (x, y, x , y , s) RATE/RMS models i1,..., i5=1 k=1 RATE equations N 3 Multiscale N representations ψ (x, y, s) = bj1,j2,j3 Bjk (x, y, s) Conclusions j1, j2,j3=1 k=1 These formulae provide multiresolution representation for variational solutions of our system. Each high-localized mode/harmonics Aj (s) corresponds to level j of resolution from the whole underlying infinite scale of spaces: Fusion ... V−2 ⊂ V−1 ⊂ V0 ⊂ V1 ⊂ V2 ⊂ ..., Fedorova, Zeitlin Outline Multiphysics where the closed subspace Vj (j ∈ Z) corresponds to level j of resolution, Hopes or to scale j. The construction of such tensor algebra based multiscales Introduction bases is considered by us. We’ll consider rate equations as the following BBGKY hierarchy operator equation. Multiscale analysis Variational approach Let L, P, Q be an arbitrary nonlinear (rational in dynamical vari- Modeling of patterns ables) first-order matrix differential operators with matrix dimension Vlasov-Poisson system Multiscale d (d=4 in our case) representations RATE/RMS models corresponding to the system of equations, which act on some set of RATE equations functions Multiscale representations Conclusions Ψ ≡ Ψ(s) = Ψ1(s),..., Ψd (s) , s ∈ Ω ⊂ R from L2(Ω) : Q(R, s)Ψ(s) = P(R, s)Ψ(s) Fusion Fedorova, Zeitlin or Outline Multiphysics Hopes LΨ ≡ L(R, s)Ψ(s)=0 Introduction BBGKY hierarchy Multiscale analysis Variational approach where Modeling of patterns R ≡ R(s,∂/∂s, Ψ). Vlasov-Poisson system Let us consider now the N mode approximation for solution as the Multiscale representations following expansion in some high-localized wavelet-like basis: RATE/RMS models RATE equations Multiscale N representations N N Ψ (s) = ar φr (s) Conclusions r=1 We’ll determine the coefficients of expansion from the following Fusion variational condition: Fedorova, Zeitlin Outline Multiphysics N N Lk ≡ (LΨ )φk (s)ds =0 Hopes Introduction BBGKY hierarchy Multiscale analysis We have exactly dN algebraical equations for dN unknowns ar . So, Variational approach variational approach reduced the initial problem to the problem of Modeling of patterns solution of functional equations at the first stage and some algebraical Vlasov-Poisson system problems at the second stage. As a result we have the following reduced Multiscale algebraical system of equations (RSAE) on the set of unknown representations N coefficients ai of the expansion: RATE/RMS models RATE equations Multiscale representations N N Conclusions H(Qij , ai ,αI ) = M(Pij , ai ,βJ ), where operators H and M are algebraization of RHS and LHS of initial problem. Qij (Pij ) are the coefficients of LHS (RHS) of the initial system of differential equations and as consequence are coefficients of RSAE. Fusion Fedorova, Zeitlin I =(i1, ..., iq+2), J =(j1, ..., jp+1) Outline are multiindexes, by which are labelled αI and βI , the other coefficients Multiphysics of RSAE: Hopes Introduction BBGKY hierarchy Multiscale analysis βJ = {βj1...jp+1 } = φjk , Variational approach 1≤j k ≤p+1 Modeling of patterns Vlasov-Poisson system where p is the degree of polynomial operator P Multiscale representations RATE/RMS models RATE equations αI = {αi ...αi } = φi ...φ˙i ...φi , Multiscale 1 q+2 1 s q+2 representations i1,..., iq+2 Conclusions where q is the degree of polynomial operator Q, ˙ iℓ = (1, ..., q + 2), φis = dφis /ds. Fusion Fedorova, Zeitlin We may extend our approach to the case when we have additional Outline constraints on the set of our dynamical variables Multiphysics Hopes Ψ = {R, ε} Introduction and additional averaged terms also. BBGKY hierarchy In this case by using the method of Lagrangian multipliers we again Multiscale analysis may apply the same approach but for the extended set of variables. As Variational approach a result we receive the expanded system of algebraical equations Modeling of patterns analogous to our system. Then, after reduction we again can extract Vlasov-Poisson system Multiscale from its solution the coefficients of the expansion. It should be noted representations that if we consider only truncated expansion with N terms then we have RATE/RMS models the system of N × d algebraical equations with the degree RATE equations Multiscale ℓ = max{p, q} representations Conclusions and the degree of this algebraical system coincides with the degree of the initial system. So, after all we have the solution of the initial nonlinear (rational) problem in the form Fusion Fedorova, Zeitlin N Outline N N R (s) = R(0) + ak φk (s) Multiphysics k =1 Hopes N Introduction N N ε (s) = ε(0) + bk φk (s) BBGKY hierarchy k=1 Multiscale analysis Variational approach Modeling of patterns where coefficients Vlasov-Poisson system Multiscale N N representations ak , bk RATE/RMS models are the roots of the corresponding reduced algebraical (polynomial) RATE equations problem RSAE. Consequently, we have a parametrization of the solution Multiscale of the initial problem by solution of reduced algebraical problem. The representations problem of computations of coefficients Conclusions αI , βJ of reduced algebraical system may be explicitly solved in wavelet approach. The obtained solutions are given in the form, where φk (s) are proper Fusion wavelet bases functions (e.g., periodic or boundary). Fedorova, Zeitlin Outline It should be noted that such representations give the best possi- Multiphysics ble localization properties in the corresponding (phase)space/time Hopes coordinates. Introduction BBGKY hierarchy Multiscale analysis Variational approach In contrast with different approaches these formulae do not use Modeling of patterns perturbation technique or linearization procedures and represent Vlasov-Poisson system dynamics via generalized nonlinear localized eigenmodes expansion. Multiscale representations −3 RATE/RMS models x 10 10 RATE equations 5 Multiscale representations 0 Conclusions −5 30 30 20 25 20 10 15 10 5 0 0 Figure 46: Waveleton-like distribution. Fusion −1 −2 Fedorova, Zeitlin −3 −4 Outline −5 Multiphysics −6 Hopes −7 Introduction −8 BBGKY hierarchy −9 Multiscale analysis −10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Variational approach Figure 47: Multiscale decomposition. Modeling of patterns Vlasov-Poisson system Multiscale Our N mode construction gives the following general multiscale representations representation: RATE/RMS models RATE equations Multiscale representations slow i i Conclusions R(s) = RN (s) + R (ωi s), ωi ∼ 2 i≥N slow j j ε(s) = εN (s) + ε (ωj s), ωj ∼ 2 j≥N Fusion Fedorova, Zeitlin Outline Multiphysics Hopes i j where R (s), ε (s) are represented by some family of (nonlinear) Introduction eigenmodes and gives the full multiresolution/multiscale representation BBGKY hierarchy in the high-localized wavelet bases. Multiscale analysis The corresponding decomposition is presented on Fig. 21 and Variational approach two-dimensional localized mode (waveleton) contribution to distribution Modeling of patterns function is presented on Fig. 20. Vlasov-Poisson system Multiscale representations As a result we can construct different (stable) patterns from high- RATE/RMS models localized (coherent) structures in spatially-extended stochastic sys- RATE equations tems with complex collective behaviour. Multiscale representations Conclusions Instead of conclusions (a few remarks): Fusion Fedorova, Zeitlin Outline ◮ Confinement of magnetic lines Multiphysics vs. confinement of modes in (non) Hopes equilibrium ensemble (of particles with their own (arbitrary) Introduction individual dynamics). BBGKY hierarchy ◮ Lorentz/Flux Dynamics (dynamical variables/observables) Multiscale analysis vs. Variational approach Ensemble Dynamics (dynamics of partition functions). Modeling of patterns ◮ Non–U(1) Fourier Dispersion relations/ Pseudodispersion Vlasov-Poisson system Multiscale Relations: representations LOCALIZED SPECTRUM DOMINATES. RATE/RMS models ◮ As Discrete as Continuous WT/LNHA. RATE equations ◮ Multiscale Symmetries and Topology of Configuration (Flux) Space! representations ◮ (Non) linear MHD, (Alfen) Waves etc...Ansatzes. Conclusions ◮ Knots/Braids Theory: Chern-Simons Topological Field Theory (part of QCD, Gauge Fields (YM)), Quantum Groups, CFT. World of Symmetry (dim 2, 3, 4). ◮ Phenomenology: Grad-Shafranov (EWBH) vs. GL/NO (rmns.: Vlasov op. problems vs. ψDO). Conclusions Fusion Fedorova, Zeitlin Outline Multiphysics Let us summarize our main results: Hopes ◮ Physical Conjectures: Introduction BBGKY hierarchy ◮ P1 Multiscale analysis State of fusion (confinement of energy) in plasma physics may and Variational approach need be considered from the point of view of non-equilibrium Modeling of patterns statistical physics. According to this BBGKY framework looks Vlasov-Poisson system naturally as first iteration. Main dynamical variables are partitions. Multiscale representations ◮ P2 RATE/RMS models High localized nonlinear eigenmodes are real physical modes RATE equations important for fusion modeling. Intermode multiscale interactions Multiscale create various patterns from these fundamental building blocks, representations and determine the behaviour of plasma. Conclusions High localized (meta) stable patterns (waveletons), considered as long-living fluctuations, are proper images for plasma in fusion state. Fusion Fedorova, Zeitlin ◮ Mathematical framework: Outline ◮ M1 Problems under consideration, like BBGKY hierarchies or their Multiphysics reductions are considered as ΨDO problems in the framework of Hopes proper family of methods unified by effective multiresolution Introduction approach. BBGKY hierarchy ◮ M2 Multiscale analysis Formulae based on generalized dispersion relations (GDR) provide Variational approach exact multiscale representation for all dynamical variables Modeling of patterns (partitions, first of all) in the basis of high-localized nonlinear Vlasov-Poisson system Multiscale (eigen)modes. Numerical realizations in this framework are representations maximally effective from the point of view of complexity of all RATE/RMS models algorithms inside. GDR provide the way for the state control on RATE equations the algbraic level. Multiscale ◮ Realizability representations According to this approach, it is possible on formal level, in Conclusions principle, to control ensemble behaviour and to realize the localization of energy (confinement state) inside the waveleton configurations created from a few fundamentals modes only. Fusion Fedorova, Zeitlin ◮ Open Questions Outline ◮ Q1 Multiphysics Definitely, all above is only very naive ensemble approach. Current Hopes level of non-equilibrium statistical physics provides us only by Introduction BBGKY generic framework. All related internal problems are BBGKY hierarchy well-known but we still have nothing better. At the same time Multiscale analysis possible Vlasov-like reductions or phenomenological models also Variational approach look as very far from reasonable from the point of view of the Modeling of patterns fusion problem set-up. Vlasov-Poisson system ◮ Multiscale Q2 representations Considering for allusion successful areas of physics like RATE/RMS models superconductivity, for example, we may conclude that only RATE equations microscopic BCS formulation provides the full explanation although Multiscale Ginzburg-Landau (GL) phenomenological approach and even representations Froelich’s and London’s ones contributed to the general picture. Conclusions Whether Vlasov equations are the analogue of GL ones and whether it is possible to construct microscopic model for plasma, these two important questions remain unanswered at present time. Fusion Fedorova, Zeitlin ◮ Q3 Outline It may be natural also that approaches proposed in this paper and Multiphysics related ones are wrong because the proper and adequate framework Hopes for solution of fusion problem is related to confinement of magnetic Introduction lines or loops (new physical dynamical variables instead of BBGKY hierarchy partitions) or fluxes instead of confinement of localized point Multiscale analysis modes (attribute of any local field theory) considered as new and Variational approach really proper physical variables. Such approach demands the Modeling of patterns topological background related to proper mathematical Vlasov-Poisson system constructions. As allusion it is possible to consider the description Multiscale of (fractional) quantum Hall effect by means of representations Chern-Simons/anyon models which allow to describe the dynamics RATE/RMS models on (of) knots and braids analytically. Anyway, it is still possible to RATE equations apply successfull methods from (M1) and (M2) here too. Remarks Multiscale representations in Section 2.2. demonstrate interesting relations. Other open Conclusions possibility is related to taking into account internal quantum properties. From this point of view our approach is very useful because we unify quantum description and its classical counterpart in the general ΨDO framework. Fusion Fedorova, Zeitlin Outline Multiphysics Hopes Introduction BBGKY hierarchy Multiscale analysis Variational approach Antonina N. Fedorova and Michael G. Zeitlin, Modeling of patterns math.ipme.ru/zeitlin.html Vlasov-Poisson system mp.ipme.ru/zeitlin.html Multiscale representations RATE/RMS models RATE equations Multiscale representations Conclusions