The applications of Complexity Theory and Tsallis Non-extensive Statistics at Space Plasma Dynamics

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1 The applications of Complexity Theory and Tsallis Non-extensive Statistics at Space Plasma Dynamics Thessaloniki, Greece Thusday George P. Pavlos Democritus University of Thrace Department of Electrical and Computer Engineering Xanthi, Greece

2 DRIVEN SYSTEMS (LOADING - UNLOADING) Solar Dynamics Description of Sun s Interior Input (Solar Radiation) Driven System (Photosphere) SunSpots Prominence Flares Description of Sun s Photosphere

3 Solar Wind

4 Complexity Theory Entropy Principle = W = max n k S q q min 1 = = = q q q q q Z n TS U F β 1 1 = = q p k p n p k S q i i q i q i q q q q q opt Z e Z q P Ε = Ε = β β 1/1 ] ) (1 [1 Ε = β Z q e q F q (Free energy) = minimum => Nonequilibrium stationary states (NESS) q=1 => Boltzman Gibbs Theory Tsallis Non-extensive Statistical Mechanics Nonequilibrium self-organization

5 THEORETICAL CONCEPTS Near Thermodynamic Equilibrium Linear Dynamics Euclidean Geometry-Topology Smooth Functions Smooth differential Equations Normal derivatives-integrals Far from Thermodynamic Equilibrium Non-Linear Dynamics Fractal Geometry -singular functions Fractional differential-integral equations MHD Vlason-Blotzman theory Normal diffusion-brownian motion Gaussian statistics-dynamics Normal Langevin-FP equations Extensive statistics BG entropy Infinite dimensional noise White-colored noise Normal Central Limit Theorem (CLT) Fractional MHD theory Anomalous diffusion motion Strange Kinetics Non-Gaussian statistics-dynamics Fractional Langevin-FP equations Non-Extensive statistics Extremization of Tsallis entropy q-extended CLT Intermittent turbulence Normal Liouville theory Locality in space and time Separation of time-spatial scales Microscopic-macroscopic locality Equilibrium RG Fractional Liouville theory (multi)fractal Topology Power laws mulitscale processes Memory long range correlations Nonlocality in space and time Non-Equilibrium RG

6 Renormalization Group Theory (RGT) (Chang, 1992) ϕ t i i (,, ) (, ) = f t + n t Non-Equilibrium Non-linear Space Plasma Physics φx i x i = 1, 2,... Generalized Langevin Stochastic Equation P ( φ( x, t) ) = { } ( ) exp (,, ) D x i L φφx dx dt Random Field Distribution Function Plasma System Stochastic Lagrangian Scale Invariance - RGT - Fixed Points (SOC, Chaos, Intermittency etc.) L/ l = RL L'/ dl = R L' dp / dl = Σ( R ) P ' ' m L mn n L L'( l) =Σ V( lu ) =ΣV exp( λ lu ) ή k k k0 k k Non-equilibrium Non-extensive Random Field Theory (Partition Function Theory)

x Dynamical Bifurcation dx / dt = f ( x, λ) Non Linear Dynamics (A) Gaussian Equilibrium Critical States Equilibrium Phase Transition Power Laws Α Β (B) Self-Organization Structure Dissipative

7 x Dynamical Bifurcation dx / dt = f ( x, λ) Non Linear Dynamics (A) Gaussian Equilibrium Critical States Equilibrium Phase Transition Power Laws Α Β (B) Self-Organization Structure Dissipative Structures Long Range Correlations C F C Β Α λ(i) λ Critical Points Bifurcation Points (C) Spatiotemporal Chaos Strange Dynamics (Attractors) Levy Processes Scale Invariance Intermittent Turbulence Tsallis Entropy Fractal Topology Anomalous Diffussion

8 Poincare Section of Phase Space Intermittency in Phase Space Stochastic Sea Fractal Set (Island, Cantori) Trapping Stickiness Levy flights Lyapunov Exponents 0 Strange Topology Strange Dynamics Scale Invariance (RGT) Levy Distribution Anomalous Diffusion

9 Basic Space Plasma Theory BBGKY Hierarchy Boltzmann - Vlassov, MHD Theory Scale Invariance - Singularities α 1 τ = λτ t = λ = 1 /3 ht t λ α /3 = = b = λ b α /3 h u λ u λ u Fractional Extension Fractional Real - Space Derivative Fractional Time Derivative x β β i β r 2β r f 1, = Γ 1 ( t r ) r β β r 2β 2β e ˆ i dx ( β ) x ( x x ) i, k i x i δ β, k β i Riesz Weyl operator x i i β i i i 2β x x β i Levy flights Levy walks Power Law distribution 1 ( ) = 1 2 ϕ + β β k f ( t, r ) t a f ( t, r ) = Γ 1 m dr m ( m a) t ( t r) Riemann Liouville operator t 0 1+ a m Fractal time random walks (FTRW) Fractal active time (Cantor set) f ( t, r ) 1 a r persistent (super-diffusive) process 0 a 1 anti-persistent (sub-diffusive) process Waiting time Power Law distribution φ 1 r ( r ) ~ 1+ a

10 FRACTIONAL KINETIC EQUATION (FRACTIONAL FOKKER PLANCK KOLMOGOROV EQUATION) Zaslavsky G.M., Chaos, fractional kinetics, and anomalous transport, Physics Reports 371, , 2002.

$Mean square displacement Normal Anomalous Diffusion < x 2 µ 2H ( t) > Dt = Dt, 0 H 1 β ds 2 µ =, 1 µ 2 α = df = 2 + θ < (Non Gaussian dynamics) µ = 2H, d s =spectral fractal dimension θ=connectivity$

11 Mean square displacement Normal Anomalous Diffusion < x 2 µ 2H ( t) > Dt = Dt, 0 H 1 β ds 2 µ =, 1 µ 2 α = df = 2 + θ < (Non Gaussian dynamics) µ = 2H, d s =spectral fractal dimension θ=connectivity index d f = Hausdorff fractal dimension of space H=Hurst exponent d w =1/H=fractal dimension of trajectories µ = 1, H = 1/2, dw = 2 Normal diffusion (Gaussian dynamics) μ={competition between FTRW Levy process} θ > 0, μ < 1 subdiffusion µ 1, θ < 0, μ > 1 superdiffusion

12 Fractional Solar Plasma (1999,2004) Solar Photosphere Magnetic Flux - Fractal Distribution Solar Wind Fracton Theory Intermittent Turbulence

13 Turbulence Intermittent Turbulence Uriel Frisch Turbulence (The Legacy of A.N. Kolmogorov) Eddies Transformation mother eddy daughters Space filling Cascade over Energy dissipation (K41)

14 Uriel Frisch Turbulence (The Legacy of A.N. Kolmogorov) mother eddy daughters β-model cascade (eddies without space filling) (β-model) Brownian self similarity (probability density) p-model

15 Multifractal intermittency velocity structure Uriel Frisch Turbulence (The Legacy of A.N. Kolmogorov) Bifractal β Model singularity manifold fractal set-fractal dimension bifractal model singular exponent multiscaling exponents spectrum

16 Multifractal β model Uriel Frisch Turbulence (The Legacy of A.N. Kolmogorov) Legendre transformation Legendre transform

$Multifractal Intermittent Energy Dissipation Velocity Energy Dissipation Multifractal Structures$

17 Multifractal Intermittent Energy Dissipation Velocity Energy Dissipation Multifractal Structures Space time scale envariance Intermittent vortex filaments. Three dimensional turbulence simulation

$Multifractal Theory Theiler J., Vol. 7, No. 6/June 1990/J. Opt. Soc. Am.$

18 Multifractal Theory Theiler J., Vol. 7, No. 6/June 1990/J. Opt. Soc. Am. A, 1055 generalized dimensions most-dence points least-dence points Information entropy Information dimension Rényi entropy

19 Multifractal Theory Theiler J., Vol. 7, No. 6/June 1990/J. Opt. Soc. Am. A, 1055

20 Tsallis Extension of Statistics Nonextensive Statistical Mechanics Microscopic Level Macroscopic Level Quantum Complexity Quantum Phase Transition (Q.P.T.) Equilibrium Phase Transition (E.P.T.) Non Equilibrium Phase Transition (N.E.P.T.)

21 Tsallis Theory Nonextensive Statistical Mechanics Generalized Fokker-Planck Equations Levy Distribution q-gaussian Distribution

22 q-extension of thermodynamics Z q q q( Ei Vq) e β q conf = q β = β / p β = 1/ KT conf i q q E pe/ p = U q conf i i i q conf q-extension of central limit theorem q-independent random variables 1 F U TS = ln qz β 1 Sq U q = ln qzq, = β T U q q q q δ U F Cq T = = T T T T 2 q q q 2 q

23 q-triplet Tsallis One-Dimensional Systems (Timeseries) q rel q ent =q sen q stat S q t

24 The q-triplet of Tsallis q - CLT (q k-1, q k, q k+1 )= qstat, qrel, qsen) Gaussian-BG equilibrium (qstat=qsen=qrel=1) Nonequilibrium (qstat, qsen, qrel) (qstat, qsen, qrel) Equilibrium PDF Metaequilibrium PDF (qstat) Equilibrium BG entropy production Metaequilibrium q-entropy production (qsen) Equilibrium relaxation process Metaequilibrium nonextensive relaxation process (qrel)

25 Intermittent Turbulence Tsallis Theory fd( a) = aq ( q 1)( Dq d + 1) + d 1 d a = [( q 1)( Dq d + 1)] dq f ( a) = aq τ ( q), d d a = [( q 1) Dq ] ( q) dq = dq τ q = Fractal dissipation of Energy n ε df ( a) da ~ ε α 1 ( l ) 0 l n / n q d ( q 1) Dn τ ( q) nln ~ ln ln 0 ε = dn d ( ) ( α ) ~ l f n α d α Tsallis Entropy Extremization - Singular Spectrum ( a a0) 1 q q Pa ( ) = Z [1 (1 q) ] 2X ln 2 2 ( a a ) f( a) = D0 + log 2[1 (1 q) o ] / (1 q) 2X ln 2 2 2Xq τ ( q) = qa 1 [1 log (1 Cq )] 1+ C 1 q + aq a0 = (1 Cq ) / [ q(1 q) ln 2] = 1 q a a q + Intermittence Turbulence Spectrum n Sp( l ) =< δu p n > p J( p) = 1 + τ ( q = ) 3 2 ap 0 2Xp 1 J( p) = [1 log 2 (1 + Cp/3 )] 3 q(1+ C 1 q p/3 p J p = + T p + T p 3 ( u) ( F) ( ) ( ) ( ) 1

FLOW CHART OF NON LINEAR ANALYSIS UNKNOWN Local Data = Timeseries Law of Internal Dynamics dx = FX ( ( t), u ( t), λ, wt ( )) dt Reconstructed Phase Space SVD analysis Dynamic Degrees of Freedom

26 FLOW CHART OF NON LINEAR ANALYSIS UNKNOWN Local Data = Timeseries Law of Internal Dynamics dx = FX ( ( t), u ( t), λ, wt ( )) dt Reconstructed Phase Space SVD analysis Dynamic Degrees of Freedom Dynamic Instability Strange Dynamics Fractal Topology - Percolation Chaos SOC - Turbulence Phenomenological Characteristics Time Frequency Domain Non-extensive Statistics (q-triplet Tsallis) Linearity Non-Linearity - Noise Non-Linear Prediction Models Spatiotemporal Modeling

$Embedding theory, metrics, fractals, multiple manifolds, probability information theory, dynamical systems and maps Notes: D.$

27 CHAOTIC ALGORITHM Takens Theorem (1981) State Space Reconstruction ORIGINAL STATE SPACE EMBEDDING RECONSTRUCTED STATE SPACE The Chaotic Algorithm is based in proficient mathematical concepts: Embedding theory, metrics, fractals, multiple manifolds, probability information theory, dynamical systems and maps Notes: D. Kugiumtzis, AUTH TIME SERIES m small, self-cutting m big, noise Autocorrelatrion Mutual Information

28 I. Analysis in the time-frequency domain COMPLEXITY AND TIME SERIES ANALYSIS Autocorrelation and Power Spectrum (noise-chaoticity) Mutual Information (Nonlinear Correlations) Flatness Coefficient (F) (Gaussian non-gaussian proccess) Hurst Exponent (temporal fractality singularities, persistence anti persistence, normal anomalous diffusion structure function (normal intermittent turbulence, singularity spectrum (h, D(h)) q-entropy production (S q ) II. Fractal geometry and fractional dynamics Singularity spectrum (a, f(a)) Generalized Spectrum ( qdq, ( )) P model (multiplicative process) Probability distribution function (Tsallis distributions) q-extension of CLT and Tsallis q-triplet parameters (q sen, q rel, q stat ) Tsallis modeling of non-equilibrium stationary states (NESS) III. Low dimensionality and Nonlinearity Correlation Dimension of reconstructed phase space and existence of strange attractor. Strange Dynamics and sensitivity of initial conditions (Lyapunov exponents spectrum) Discrimination of stochasticity and nonlinear randomnicity - chaoticity Discrimination of dynamical components

29 Synopsis Chaotic Analysis Dimensional Analysis Statistical Analysis Turbulence Analysis

30 Sunspot

31 Sunspot Index (Original SVD Component)

32 Sunspot Index (Correlation Dimension & Lyapunov Exponent) Correlation Dimension Lyapunov Spectrum Exponents

33 Sunspot Index (Tsallis Statistics) Sunspot TMS - qstat=1.53 ±0.04 V1 SVD Component - qstat=1.40 ±0.08 V2-10 SVD Component - qstat=2.12 ±0.20 Sunspot TMS TMS - qsen=0.368 V1 SVD Component - qsen= V2-10 SVD Component qsen=0.407 Δα= Δα= Δα= 1.940

34 Sunspot Index (Turbulence Analysis)

35 Sunspot Index (Parameters, Generalized Hurst Exponent & Entropy Production) Sunspot TMS V1 Comp. V2-10 Comp. Sunspot TMS V1 Comp. V2-10 Comp. Sunspot TMS V1 Comp. V2-10 Comp. Sunspot TMS V1 Comp. V2-10 Comp. Sunspot TMS V1 Comp. V2-10 Comp. Sunspot TMS V1 Comp. V2-10 Comp.

36 Solar Flares

37 Solar Flares Index (Original SVD Components)

38 Solar Flares Index (Correlation Dimension & Lyapunov Exponent) Correlation Dimension Lyapunov Spectrum Exponents

39 Solar Flares Index (Tsallis Statistics) Solar Flare TMS - qstat=1.87 ±0.05 V1 SVD Component - qstat=1.28 ±0.04 V2-15 SVD Component - qstat=2.02 ±0.15 Solar Flare TMS - qsen=0.308 V1 SVD Component - qsen= V2-15 SVD Component qsen=0.192 Δα= 2.15 q relaxation 9.333±0.395 Δα= 9.849± ±0.086 Δα= 1.37

40 Solar Flares Index (Turbulence Analysis)

41 Solar Flares Index (Parameters, Generalized Hurst Exponent & Entropy Production) Flares V1 Comp. V2-15 Comp. Flares V1 Comp. V2-15 Comp. Flares V1 Comp. V2-15 Comp. Flares V1 Comp. V2-15 Comp. V1 V2 V3 V4 V5 Flares V1 Comp. V2-15 Comp.

42 Solar Flares Event 1 X-rays intensity Power Spectrum

43 Solar Flares Event 1 (Chaotic Analysis) Correlation Dimension Lyapunov Spectrum Exponents CALM (m=10) SHOCK (m=7) DECAY(m=10) L1 maximum > L L L L Voltera Weiner Model

44 Solar Flares Event 2 (Statistical Analysis) X-rays intensity

45 Solar Flares Event 2 (Tsallis Statistics) Calm- qstat=1.04±0.02 Shock- qstat=1.43±0.05 Decay- qstat=1.08±0.02 Calm - qsen= Shock - qsen=0.376 Decay - qsen= Δ(α)= Δ(α)= Δ(α)= Calm Δ(Dq)=0.141 Shock Δ(Dq) =0.738 Decay Δ(Dq) =0.148

46 Solar Flares Event 2 (Parameters) Calm Shock Decay Calm Shock Decay Calm Shock Decay Calm Shock Decay Calm Shock Decay Calm Shock Decay

47 Solar Flares Event 2 (Turbulence Analysis)

48 Solar Energetic Particle Interplanetary Space

49 Interplanetary Observations (CME 7 March 2012) (Re -2,5) (Re -59,5)

50 Solar Energetic Particles

51 Solar Energetic Particle (Generalized Hurst Exponent & Entropy Production)

52 Solar Energetic Particle (Tsallis Statistics) Period 1 - qstat=1.08 ±0.03 Period 2- qstat=1.18 ±0.03 Period 3 - qstat=1.57 ±0.08 Period 4 - qstat=1.11 ±0.05

53 Solar Energetic Particle (Multifractal Spectrum) Period 1 - qsen= Period 2 - qsen= Period 3 - qsen=0.427 Period 4 - qsen= Δα=0.251 Δα=0.653 Δα=1.483 Δα=0.388 Period 1 - ΔDq=0.155 Period 2 - ΔDq =0.556 Period 3 - ΔDq =1.371 Period 4 - ΔDq =0.287 p-model=0.546 p-model=0.612 p-model=0.753 p-model=0.566

54 Solar Energetic Particle (Parameters) Phase Transition through Tsallis q-triplet Phase Transition through Multifractal Structure and p-model values

55 ACE/MAG Project - Solar Wind (Magnetic Field)

56 Solar Wind Magnetic Field (q-stationary & q-relaxation) Calm - qstat= 1.38 ±0.04 Shock - qstat=1.74 ±0.05 ICME - qstat = 1.17 ±0.02 Calm - qrel = Shock - qrel=5.74 ICME - qrel = 11.12

57 Solar Wind Magnetic Field (Multifractal Spectrum) Calm - qsen= Shock - qsen=0.117 ICME - qsen= Δα=0.679 Δα=1.057 Δα=0.227 f(α) f(α) f(α) α α α Calm - ΔDq=0.579 Shock - ΔDq =0.967 ICME - ΔDq =0.131 p-model=0.616 p-model=0.677 p-model=0.539

58 Sola Wind Magnetic Field (Parameters) Phase Transition through Tsallis q-triplet Phase Transition through Multifractal Structure and p-model values

59 Interplanetary Solar Magnetic Field (Re -2,5) (Re -59,5) ACE CLUSTER THEMIS-E THEMIS-C (B total ) (B total ) (B total ) (B total ) q stationary (CALM) 1.37 ± ± ± ± 0.06 q stationary (SHOCK) 1.74 ± ± ± ± 0.06

60 Solar Wind (High Time Resolution Ion Fluxes) Photo of BMSW instrument (Fast Monitor of the Solar Wind) and Typical SPECTR-R orbits evolution relative the Earth s magnetosphere in October 2011.

61 Solar Wind Ion Fluxes

62 Solar Wind Ion Fluxes (Chaotic Analysis) Time Series Power Spectrum Correlation Dimension Lyapunov Spectrum Exponents

63 Solar Wind Ion Fluxes (Tsallis Statistics) Calm - qstat= 1.37 ±0.05 Shock - qstat = 1.61±0.03 Calm - qsen= Δα=0.753 Shock - qsen=0.268 Δα=1.205 Calm - ΔDq=0.431 Shock - ΔDq =0.871

64 Solar Wind Ion Fluxes (Turbulence Analysis)

65 Magnetosphere July 2004 GEOTAIL Orbit

66 Magnetosphere July 2004 Bz Component Ex Component

67 Summarize parameter values of non-extensive statistics in Solar Plasmas The q-triplet (q_sen, q_stat, q_rel) of Tsallis. System q_sen q_stat q_rel (C(τ)) Solar Wind (Bz cloud) 0.484± ± ±2.31 Solar (Sunspot Index) 0.368± ± ±0.13 Solar (Flares Index) 0.308± ± ±0.22 Solar (Protons) ± ± ±0.23 Solar (Electrons) 0.860± ± ±0.22 Cosmic Stars (Brigthness) ± ± ±0.58 Cosmic Ray (C) ± ± ±18.43

68 Conclusions Everywhere in Solar plasma we have confirmed the following: Tsallis Sq entropy and non extensivity Nonequillibrium stationary states (NESS) Nonequillibrium topological phase transition processes Multifractal intermittent turbulence Anomalous diffusion Multiplicative processes Low Dimensional Chaos and/or SOC. These experimental results are concluded theoretically by: Strange Dynamics Fractal extension of MHD theory Fractional extension of magnetized plasmas Langevin and Fokker Planck equations Nonequillibrium Percolation Theory

69 THANK YOU for your attention

Tsallis non - Extensive statistics, intermittent turbulence, SOC and chaos in the solar plasma. Part two: Solar Flares dynamics

Tsallis non - Extensive statistics, intermittent turbulence, SOC and chaos in the solar plasma. Part two: Solar Flares dynamics L.P. Karakatsanis [1], G.P. Pavlos [1] M.N. Xenakis [2] [1] Department of