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

Physica A 392 (2013) 3920–3944

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Tsallis non-extensive statistics, intermittent turbulence, SOC and chaos in the solar plasma. Part two: Solar flares dynamics L.P. Karakatsanis a,∗ , G.P. Pavlos a , M.N. Xenakis b a

Department of Electrical and Computer Engineering, Democritus University of Thrace, 67100 Xanthi, Greece

b

German Research School for Simulation Sciences, Aachen, Germany

highlights • • • •

The solar flare index embedded in the non-extensive statistical theory of Tsallis. We show the phase transition process in the solar dynamics. We show the intermittent solar turbulence and anomalous diffusion solar process. The agreement of Tsallis statistical theory with the experimental estimations.

article

info

Article history: Received 4 April 2012 Received in revised form 8 May 2013 Available online 22 May 2013 Keywords: Tsallis non-extensive statistics Non-Gaussian solar process Low dimensional chaos Self-organized criticality (SOC) Intermittent turbulence Solar flares dynamics

abstract In this study which is the continuation of the first part (Pavlos et al. 2012) [1], the nonlinear analysis of the solar flares index is embedded in the non-extensive statistical theory of Tsallis (1988) [3]. The q-triplet of Tsallis, as well as the correlation dimension and the Lyapunov exponent spectrum were estimated for the singular value decomposition (SVD) components of the solar flares timeseries. Also the multifractal scaling exponent spectrum f (a), the generalized Renyi dimension spectrum D(q) and the spectrum J (p) of the structure function exponents were estimated experimentally and theoretically by using the q-entropy principle included in Tsallis non-extensive statistical theory, following Arimitsu and Arimitsu (2000) [25]. Our analysis showed clearly the following: (a) a phase transition process in the solar flare dynamics from a high dimensional non-Gaussian self-organized critical (SOC) state to a low dimensional also non-Gaussian chaotic state, (b) strong intermittent solar corona turbulence and an anomalous (multifractal) diffusion solar corona process, which is strengthened as the solar corona dynamics makes a phase transition to low dimensional chaos, (c) faithful agreement of Tsallis non-equilibrium statistical theory with the experimental estimations of the functions: (i) non-Gaussian probability distribution function P (x), (ii) f (a) and D(q), and (iii) J (p) for the solar flares timeseries and its underlying non-equilibrium solar dynamics, and (d) the solar flare dynamical profile is revealed similar to the dynamical profile of the solar corona zone as far as the phase transition process from self-organized criticality (SOC) to chaos state. However the solar low corona (solar flare) dynamical characteristics can be clearly discriminated from the dynamical characteristics of the solar convection zone. © 2013 Elsevier B.V. All rights reserved.

1. Introduction In the first part of this study [1] we introduced a new theoretical framework for studying solar plasma dynamics by using experimental timeseries observations. The novelty of this framework includes the new concepts of Tsallis q-statistics theory

∗

Corresponding author. Tel.: +30 2541079110. E-mail addresses: [email protected], [email protected] (L.P. Karakatsanis).

0378-4371/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physa.2013.05.010

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coupled with intermittent turbulence theory [2] and chaos theory. These theoretical concepts were used for the extension of an existing chaotic algorithm for experimental timeseries analysis [1]. In part one, the sunspot dynamics was studied under a new aspect of Tsallis q-statistics of solar dynamics coupled with intermittent turbulence of the convective zone solar plasma, the self-organized criticality (SOC) and low dimensional perspectives. In this second part of our study we extend the analysis of part one to the solar flare plasma dynamics. As the Tsallis q-statistics extension of Boltzmann–Gibbs statistics is harmonized with the fractal generalization of dynamics [3–5], our extended algorithm of timeseries analysis revealed new possibilities for the verification of significant theoretical concepts. More specifically the experimental verification of chaotic multiscale and multifractal dynamics underlying the experimental timeseries observations is of high theoretical interest because the coupling of q-statistics with fractal dynamics and low dimensional chaos introduces new and significant knowledge about the solar flare dynamics. Moreover the solar flare dynamical system is of high interest since at solar flare regions strong turbulent magnetic energy dissipation and strong acceleration of charged particles as electrons, protons and heavy ions, can exist. These bursty phenomena are followed by hard X-ray luminosity, while the underlying mechanism of solar flare dynamics is some kind of magnetic reconnection process. However, the physical explanations of the classical magnetic reconnection process are inefficient to explain the structural dynamics of magnetic field dissipation [6,7]. In this study, we indicate the existence of a fractal dissipation–acceleration mechanism according to Pavlos [2]. In Section 2 we present the pre-supposed theoretical tools for the analysis of solar flare index timeseries. In Section 3 we present the results of the data analysis, while in Section 4 we summarize the data analysis results. Finally in Section 5 we discuss our results. 2. Theoretical concepts and methodology of data analysis In Section 2 of the first part of this study [1] we have presented the theoretical pre-suppositions for the data analysis. In these pre-suppositions we have included analytically the following: (a) The physical meaning of Tsallis non-extensive entropy theory (Part one, Sections 2.11–2.13) [1]. It is significant to remember that far from equilibrium the development of spatiotemporal plasma structures including long-range correlations could be possible. This can be indicated by the estimation of Tsallis q-triplet. Especially, far from equilibrium the q-triplet (qsen , qstat , qrel ) can differ significantly from the equilibrium Gaussian profile where qsen = qstat = qrel = 1. (b) Tsallis non-extensive entropy theory is related to the multifractal and multiscale character of the underlying phase space which far from equilibrium includes anomalous topology connected with multi-scaling and multifractality. This character is related to the corresponding multifractal and anomalous topology of the dissipation regions in the physical space–time. For this, we estimate the multifractal spectrum of dimensions f (a) and Dq according to the multifractal theory as well as the structure function (Sp ) and its scaling exponent spectrum [J (p)] which can differ from Kolmogorov’s first theory (K41) predictions. According to Frisch [8] for non-Gaussian multifractal and intermittent dissipation processes, the scaling exponent spectrum J (p) satisfies the relation: dJ (p) dp

= h∗ (p) ̸= 0

(1)

where h∗ (p) is related to the fractal dimension D(h) of the fractal dissipation region by the relation: D′ (h∗ (p)) = p.

(2)

The fractal dimension D(h) and the exponents of the structure function J (p) are related to a Legendre transformation: J (p) = inf[ph + 3 − D(h)].

(3)

(h)

This relation indicates the fact that, when the dissipation region is multifractal then D(h) ̸= 3 and dJ (p)/dp ̸= p. Also, according to the theory of Kolmogorov (K41), the dissipation region is mono-fractal. (c) The correlation dimension is given by the relation D = lim

r →0 m→∞

Dm , where Dm = limr →0

d ln Cm (r,m) d ln(r )

and Cm (r , m) =

i =1 j=1 Θ (r − ∥x(i) − x(j)∥), also ∥x(i) − x(j)∥ denotes the distance between the states x(i), x(j) in the m-dimensional reconstructed phase space. For the estimation of D we follow the theories of Takens [9], Grassberger and Procaccia [10] and Theiler [11]. (d) The Lyapunov exponent spectrum (λi ) is obtained by the evolution of small perturbations of the dynamical orbit in the reconstructed state space [12]. (e) The color noise pseudo chaos profile was discriminated from the low dimensional chaotic data exploiting the Theiler method of surrogate data [13]. 2 N (N −1)

N N

According to the above theoretical concepts and the analytic description of Section 2 of the first part [1], in the following we present the plan for the experimental data analysis of the solar flare index:

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(I) Use of the singular value decomposition (SVD) for the discrimination of deterministic and noisy (stochastic) components included in the observed signals, as well as for the discrimination of distinct dynamical components. (II) Estimation of the Flatness coefficient F . (III) Estimation of the qstat index of the q-Gaussian, through linear correlation fitting of lnq p(z ) versus z 2 , where lnq p(z ) is the q-logarithmic function of the probability function p(z ), as z = zn+1 − zn , (n = 1, . . . , N) corresponds to the first difference of the experimental solar flares timeseries data [14]. (IV) Determination of the qrel index of the q-statistics according to the relation: ddtΩ = − T 1 Ω qrel , where Ω corresponds qrel

(V)

(VI)

(VII) (VIII) (IX)

to the autocorrelation functions or the mutual information function of the experimental timeseries [3]. Determination of the qsen index according to the relation: q 1 = a 1 − a 1 , where amin and amax correspond to the sen max min zero points of the multifractal exponent spectrum. The index is related to the Kolmogorov-signal entropy production and Pesin theory [15,16]. Determination (a) of the structure functions S (p) = ⟨|δ u|p ⟩, where δ u is the spatial variation of the bulk plasma flow velocity and (b) of the scaling exponent spectrum J (p) according to the relation: S (p) ∼ lJ (p) , where l is the length scale of the dissipation process [2]. Determination of the correlation dimension (D) by using the saturation value of the slopes (Dm ) of the correlation integrals (Cm ). Determination of the Lyapunov exponent’s spectrum. Determination of the significance (σ ) of the discriminating statistics by using the surrogate method of Theiler [13]. For σ values >3 the null hypothesis can be rejected with confidence higher than 99%.

3. Results of data analysis In this section we present results concerning the analysis of data included in the solar flares index following the methodology presented in the previous section of this study. The daily Flare Index of solar activity was determined using the final grouped solar flares obtained by the NGDC (National Geophysical Data Center). It is calculated for each flare using the formula: Q = (i ∗ t ), where ‘‘i’’ is the importance coefficient of the flare and ‘‘t’’ is the duration of the flare in minutes. To obtain final daily values, the daily sums of the index for the total surface are divided by the total time of observation of that day. The data covers a time period from 1/1/1996 to 31/12/2007. 3.1. Timeseries and flatness coefficient F Fig. 1(a) represents the solar flares timeseries that was constructed concerning the period of 11 years. Fig. 1(b) presents the flatness coefficient (F ), estimated for the solar flare data during the same period. The values of F reveal continuous changes of the statistics of flares between Gaussian profile (F ∼ 3) to strong non-Gaussian profile (F ∼ 4–25). Fig. 1(c) 15 presents the first (V1 ) SVD component of the solar flare index and Fig. 1(e) presents the sum V2–15 = i=2 Vi of the next SVD components estimated for the solar flares timeseries. The estimation of the flatness coefficient F (V1 ) and F (V2–15 ) is shown in Fig. 1(d) and (f), correspondingly. As we notice in these figures, the statistics of the V1 component is clearly discriminated from the statistics of the V2–15 component, as the F (V1 ) flatness coefficient almost everywhere had low values (∼3–5), while the F (V2–15 ) reached much higher values (∼5–30). The low values (∼3–4) of the F (V1 ) coefficient indicate for the solar activity a near Gaussian dynamical process, underlying the V1 SVD component. Oppositely, the high values of the F (V2–15 ) coefficient strongly indicate a non-Gaussian solar corona dynamical process underlying the V2–15 SVD component. 3.2. The Tsallis q-statistics In this section we present results concerning the computation of the Tsallis q-triplet, including the three-index set

(qstat , qsen , qrel ) estimated for the original solar flare index timeseries, as well as for its V1 and V2–15 SVD components (presented in Figs. 2–4). 3.2.1. Determination of qstat index of the q-statistics In Fig. 2(a) we show (by open circles) the experimental probability distribution function (PDF) p(z ) vs. z, where z corresponds to the first difference timeseries Zn+1 − Zn , (n = 1, 2, . . . , N ). Fig. 2(b) presents the best linear correlation between lnq [p(z )] and z 2 . The best fitting was found for the value of qstat (orig) = 1.87 ± 0.06. This value was used to estimate the q-Gaussian distribution presented in Fig. 2(a) by the solid black line. Fig. 2(c), (d) and (e), (f) are similar to Fig. 2(a), (b) but for the V1 and V2–15 SVD components correspondingly. As far as the SVD components is concerned, the qstat values were found to be: qstat (V1 ) = 1.28 ± 0.04 and qstat (V2–15 ) = 2.02 ± 0.15. Observing these results the following relation is satisfied: 1 < qstat (V1 ) < qstat (orig) < qstat (V2–15 ), where qstat (orig) corresponds to the original solar flare index timeseries.

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Fig. 1. (a) Timeseries of solar flares index concerning the period of 11 years. (b) The coefficient F estimated for solar flares index timeseries (tms). (c) The SVD V1 component of solar flares index tms. (d) The coefficient F estimated for the SVD V1 component of solar flares index tms. (e) The SVD V2–15 component of solar flares index tms. (f) The coefficient F estimated for the SVD V2−15 component of solar flares index tms.

3.2.2. Determination of qrel index of the q-statistics (a) Relaxation of autocorrelation functions. Fig. 3 presents the best log plot fitting of the autocorrelation function C (τ ) estimated for the original solar flare index (Fig. 3(a)), its V1 SVD component (Fig. 3(c)) and its V2–15 SVD component (Fig. 3(e)). The three qrel values were found to satisfy the relation: 1 < qcrel (V2–15 ) < qcrel (orig) < qcrel (V1 ) as: qcrel (V2–15 ) = 4.546 ± 0.173,

qcrel (orig) = 5.329 ± 0.135,

qcrel (V1 ) = 42.666 ± 2.950.

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Fig. 2. (a) PDF P (zi ) vs. zi q Gaussian function that fits P (zi ) for the solar flares index timeseries. (b) Linear correlation between lnq p(zi ) and (zi )2 where q = 1.87 ± 0.06 for the solar flares index timeseries. (c) PDF P (zi ) vs. zi q Gaussian function that fits P (zi ) for the V1 SVD component. (d) Linear correlation between lnq p(zi ) and (zi )2 where q = 1.28 ± 0.04 for the V1 SVD component. (e) PDF P (zi ) vs. zi q Gaussian function that fits P (zi ) for the V2–15 SVD component. (f) Linear correlation between lnq p(zi ) and (zi )2 where q = 2.02 ± 0.15 for the V2–15 SVD component.

(b) Relaxation of mutual information. Fig. 3(b), (d), (f) are similar to Fig. 3(a), (c), (e) but correspond to the relaxation time of the mutual information I (τ ). In particular, the plot of I (τ ) in logarithmic scale (log–log) for the solar flare index timeseries, its V1 SVD component and its V2–15 SVD component is shown. The best log–log (linear) fitting was found for the values: qIrel (V2–15 ) = 3.207 ± 0.086, qIrel (orig) = 9.333 ± 0.395, qIrel (V1 ) = 9.849 ± 0.410. These three values satisfy the following relation: 1 < qIrel (V2–15 ) < qIrel (orig) < qIrel (V1 ). Comparing the qrel indices, as they were estimated from the autocorrelation function and the mutual information function, we found differences for all the cases, because the mutual information includes nonlinear characteristics of the underlying dynamics in contrast to the autocorrelation function which is a linear statistical index.

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Fig. 3. (a) Log–log plot of the self-correlation coefficient C (τ ) vs. time delay τ for the solar flares index timeseries. We obtain the best fit with qrel = 5.329 ± 0.135. (b) Log–log plot of the mutual information I (τ ) vs. time delay τ for the solar flares timeseries. We obtain the best fit with qrel = 9.333 ± 0.395. (c) Log–log plot of the self-correlation coefficient C (τ ) vs. time delay τ for the V1 SVD component. We obtain the best fit with qrel = 42.666 ± 2.950. (d) Log–log plot of the mutual information I (τ ) vs. time delay τ for the V1 SVD component. We obtain the best fit with qrel = 9.849 ± 0.410. (e) Log–log plot of the self-correlation coefficient C (τ ) vs. time delay τ for the V2–15 SVD component. We obtain the best fit with qrel = 4.546 ± 0.173. (f) Log–log plot of the mutual information I (τ ) vs. time delay τ . for the V2–15 SVD component. We obtain the best fit with qrel = 3.207 ± 0.086.

3.2.3. Determination of qsen index of the q-statistics Fig. 4 presents the estimation of the generalized dimension Dq and their corresponding multifractal (or singularity) spectrum f (α). The qsen index was estimated by using the relation 1/(1 − qsens ) = 1/amin − 1/amax for the original solar flares index (Fig. 4(a)–(b)), as well as for its V1 (Fig. 4(c)–(d)) and V2–15 (Fig. 4(e)–(f)) SVD components. The three index qsen values were found to satisfy the relation: qsen (V1 ) < qsen (V2–15 ) < qsen (orig) < 1, where: qsen (V1 ) = −0.540, qsen (V2–15 ) = 0.192, and qsen (orig) = 0.308. 3.2.4. Comparison of experimental and theoretical estimations about Dq, f (a) functions In this section, as well as the next Section 3.3, we follow Arimitsu and Arimitsu [17,18] and N. Arimitsu et al. [19] for the theoretical estimation of the generalized fractal dimension function Dq, the singularity spectrum f (a) as well as the structure function scaling spectrum J (p). In accordance with the theoretical description of intermittent turbulence processes dissipative nonlinear dynamics can produce self-organization and long range correlations in space and time. In this case we

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can imagine the mirroring relationship between the phase space multifractal attractor and the corresponding multifractal turbulence dissipation process of the dynamical system in the physical space. Multifractality and multi-scaling interaction, chaoticity and mixing or diffusion (normal or anomalous), all of them can be manifested in both the state (phase) space and the physical (natural) space as the mirroring of the same complex underlying dynamics.   r and the The multifractal character of turbulence can be characterized: (a) in terms of local velocity increments δ ul ⃗ p

structure function Sp (l) ≡ δ ul first introduced by Kolmogorov [20], and (b) in terms of local dissipation and local scale





a

a

⃗′ = λ 3 u⃗, t ′ = λ1− 3 t [21–23]. The turbulent flow invariance of HD or MHD equations upon scale transformations ⃗ r ′ = λ⃗ r, u is assumed to process a range of scaling exponent, hmin ≤ h ≤ hmax while for each h there is a subset of points of R3 of fractal dimension D(h) such that:   δ ul ⃗r ∼ lh as l → 0.

(4)

The multifractal assumption can be used to derive the structure function of order P by the relation: Sp (l) ∼



dµ(h) lph l3−D(h)

 

(5)

where dµ(h) gives the probability weight of the different scaling exponents, while the factor l3−D(h) is the probability of being at a distance l in the fractal subset of R3 with dimension D(h). By using the method of steepest descent [2] we can derive the power-law

  Sp (l) ≡ δ uPl ∼ lJ (P ) ,

l→0

(6)

J (p) = min [ph + 3 − D(h)] .

(7)

where: h

The above relation is the Legendre transformation between J (P ) and D(h) as D(h) can be derived by the relation: D(h) = min [ph + 3 − J (p)] .

(8)

p

The multifractal character of the turbulent state can be apparent in the spectrum of the structure function scaling exponents J (p) by the relation: dJ (p) dp

  dh∗ (p) = h∗ (p) + p − D′ (h∗ (p)) = h∗ (p)

(9)

dp

as the minimum value of the relation (8) corresponds to the maximum of the 3 − J (p) function for which: d(3 − J (p)) dp

=

d dp

(D − ph) = 0

(10)

and D′ (h∗ (p)) = p.

(11)

According to Frisch [24] the dissipation is said to be multifractal if there is a function F (a) which maps real scaling exponents a to scaling dimensions F (a) ≤ 3 such that:

  εl ⃗r ∼ la−1 as l → 0

(12)

for a subset of points ⃗ r of R with dimension F (a). In correspondence with the structure function theory presented above in the case of multifractal energy dissipation the q¯ moments εl follow the power laws: 3

  ¯ εlq ∼ lq¯

(13)

where the scaling exponents σ (¯q) are given by the relation:

σ (¯q) = min [q¯ (a − 1) + 3 − F (a)] .

(14)

a

In the case of a one dimensional dissipation process the multifractal character is described by the function f (a) instead of F (a) where f (a) = F (a) − 2 [24]. In the language of Re’nyi’s generalized dimensions and multifractal theory the dissipation σ (¯q) multifractal turbulence process corresponds to Re’nyi’s dimensions Dq¯ according to the relation Dq¯ = q¯0−1 + 3 [5]. Also, according to Frisch [24] the relation between the dissipation multifractal formalism and the multifractal turbulent velocity increments formalism is given by the following relations: h=

a 3

,

D(h) = F (a) = f (a) + 2,

J (p) =

p 3

+σ

p 3

.

(15)

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The theoretical description of turbulence in the physical space is based upon the concept of the invariance of the HD or MHD equations upon scaling transformations to the space–time variables (σ⃗ , t ) and velocity (⃗ u) and corresponding similar scaling relations for other physical variables [21–23]. In the next part of this section we follow Arimitsu and Arimitsu [25,17] connecting the Tsallis non-extensive statistics and intermittent turbulence process. Under the scale transformation:

εn ∼ ε0 (ln \ l0 )α−1

(16)

the original eddies of size l0 can be transformed to constituting eddies of different size ln = l0 δn , n = 0, 1, 2, 3, . . . after n steps of the cascade. If we assume that at each step of the cascade eddies break into δ pieces with 1δ diameter then the size ln = l0 δ −n . If δ un = δ u (ln ) represents the velocity difference across a distance r ∼ ln and εn represents the rate of energy transfer from eddies of size ln to eddies of size ln+1 then we have:

δ un = δ u

  3a ln

εn = ε

and

l0

 a−1 ln

(17)

l0

where a is the scaling exponent under the scale transformation (16).

  ∂ u(x)



 

The scaling exponent a describes the degree singularity in the velocity gradient  ∂ x = limln →0 δlun  as the first n equation in (17) reveals. The singularities a in the velocity gradient fill the physical space of dimension d, (a < d) with a fractal dimension F (a). Similar to the velocity singularity other frozen fields can reveal singularities in the d-dimensional natural space. After this, the multifractal (intermittency) character of the HD or the MHD dynamics consists in supposing that the scaling exponent α included in relations (16) takes on different values at different interwoven fractal subsets of the d-dimensional physical space in which the dissipation field is embedded. The exponent α and for values a < d is related ∂ A(x) with the degree of singularity in the field’s gradient ( ∂ x ) in the d-dimensional natural space [17]. Generally, the gradient singularities cause the anomalous diffusion in the physical or phase space of the dynamics. The total dissipation occurring in a d-dimensional space of size ln scales also with a global dimension Dq¯ for powers of different order q¯ as follows:



(¯q−1)Dq¯

εnq¯ ldn ∼ ln

= lτn (¯q) .

(18)

n

Supposing that the local fractal dimension of the set dn(a) which corresponds to the density of the scaling exponents in the region (α, α + dα) is a function fd (a) according to the relation: dn(a) ∼ ln−fd (a) da

(19)

where d indicates the dimension of the embedding space, then we can conclude the Legendre transformation between the mass exponent τ (¯q) and the multifractal spectrum fd (a): fd (a) = aq¯ − (¯q − 1)(Dq¯ − d + 1) + d − 1 d . [(¯q − 1)(Dq¯ − d + 1)] a=  dq¯



(20)

For linear intersections of the dissipation field, that is d = 1 the Legendre transformation is given as follows: f (a) = aq¯ − τ (¯q),

d

a=

dq¯

[(q − 1)Dq ] =

d dq¯

τ (¯q), q¯ =

df (a) da

.

(21)

The relations (16)–(17) describe the multifractal and multiscale turbulent process in the physical space. Moreover the similar relations:

Dq¯ =

log

1

lim q¯ − 1 λ→0

Nλ 

q¯

pi

i =1

log L

,

τ (¯q) ≡ (¯q − 1)Dq¯ = min[¯qa − f (a)]  q¯  ′ pi = da′ p(a′ )λ−f (α ) da′ a(¯q) =

d[τ (¯q)] dq¯

f (a) = minq [q¯ a − T (¯q)] ,

(22) (23) (24) (25) (26)

describe the multifractal and multiscale process on the attracting set of the phase space of the system dynamics. From this physical point of view, we suppose the physical identification of the magnitudes Dq¯ , a, f (a) and τ (¯q) estimated in the

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physical space and the corresponding phase space of the dynamics. By using experimental timeseries we can construct the function Dq¯ of the generalized Rényi d-dimensional space dimensions, while the relations (6) allow the calculation of the fractal exponent (a) and the corresponding multifractal spectrum fd (a). For homogeneous fractals of the turbulent dynamics the generalized dimension spectrum Dq¯ is constant and equal to the fractal dimension of the support [22]. Kolmogorov [20] supposed that Dq¯ does not depend on q¯ as the dimension of the fractal support is d = 3. In this case the multifractal spectrum consists of the single point (a = 1 and f (1) = 3). The singularities of degree (a) of the dissipated fields fill the physical space of dimension d with a fractal dimension F (a), while the probability P (a)da, to find a point of singularity (a) is specified by d−F (a) the probability density P (a)da ∼ ln . The filling space fractal dimension F (a) is related with the multifractal spectrum function fd (a) = F (a) − (d − 1), while according to the distribution function Πdis (εn ) of the energy transfer rate associated with the singularity a it corresponds to the singularity probability as Πdis (εn )dεn = P (a)da [25].  q¯ Moreover the partition function i Pi of the Rényi fractal dimensions estimated by the experimental timeseries includes information for the local and global dissipation process of the turbulent  q dynamics as well as for the local and global dynamics of the attractor set, as it is transformed to the partition function i Pi = Zq of the Tsallis q-statistical theory. In the following we follow Arimitsu and Arimitsu [25,17], presenting the theoretical estimation of significant quantitative relations which can also be estimated experimentally as the probability singularity distribution P (a) can be estimated by extremizing the Tsallis entropy functional Sq . According to Arimitsu and Arimitsu [25] the extremizing probability density function P (a) is given as a q-exponential function: P (a) = Zq

−1

(a − a0 )2 1 − (1 − q) 2X / ln 2



 1−1 q (27)

where the partition function Zq is given by the relation: Zq =

2X /[(1 − q) ln 2] B(1/2, 2/1 − q),



(28)

and B(a, b) is the Beta function. The partition function Zq as well as the quantities X and q can be estimated by using the following equations:

√

 √  2X = + (1 − q) − (1 − q) / b .  −(1−q) b = (1 − 2 )/[(1 − q) ln2 ] 



a20

2

(29)

We can conclude for the exponent’s spectrum f (a) by using the relation P (a) ≈ lnd−F (a) as follows:



f (a) = D0 + log2 1 − (1 − q)

(a − ao )2 2X / ln 2

 (1 − q)−1

(30)

where a0 corresponds to the q-expectation (mean) value of a through the relation:

⟨(a − a0 ) ⟩q = 2



daP (a) (a − a0 ) q

q

 

daP (a)q

(31)

while the q-expectation value a0 corresponds to the maximum of the function f (a) as df (a)/da|a0 = 0. For the Gaussian dynamics (q → 1) we have a mono-fractal spectrum f (a0 ) = D0 . The mass exponent τ (¯q) can also be estimated by using the inverse Legendre transformation: τ (¯q) = aq¯ − f (a) (relations 20–21) and the relation (30) as follows: 2X q¯ 2

τ (¯q) = q¯ a0 − 1 −

1+



Cq¯

−

1 1−q

[1 − log2 (1 +



Cq¯ )],

(32)

where Cq¯ = 1 + 2q¯ 2 (1 − q)X ln 2. The relation between a and q¯ can be found by solving the Legendre transformation equation q¯ = df (a)/da. Also if we use the Eq. (30) we can obtain the relation: aq¯ − a0 = (1 −

Cq¯ )/[¯q(1 − q) ln 2].



(33)

The q-index is related to the scaling transformations of the multifractal nature of turbulence according to the relation q = 1 − a. Arimitsu and Arimitsu [17] estimated the q-index by analyzing the fully developed turbulence state in terms of Tsallis statistics as follows: 1 1−q

=

1 a−

−

1 a+

(34)

where a± satisfy the equation f (a± ) = 0 of the multifractal exponents spectrum f (a). This relation can be used for the estimation of the qsen -index included in the Tsallis q-triplet (see the next section). The above analysis based on the extremization of Tsallis entropy can be also used for the theoretical estimation of the structure functions scaling exponent spectrum J (p), where p = 1, 2, 3, 4, . . . . The structure functions were first introduced

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Table 1 Experimentally estimated values (a0 , qsen ) for the timeseries of solar flares and SVD components. Signal

a0

qsen

Original solar flares V1 SVD component V2–15 SVD component

1.138 1.038 1.179

−0.540

0.308 0.192

by Kolmogorov [20] defined as statistical moments of the field increments:

− →

⃗) − u(⃗r )|p ⟩ = ⟨|δ un |p ⟩ Sp ( r ) = ⟨|u(⃗ r +d

(35)

Sp (⃗ r ) = ⟨|u(⃗ r + 1⃗ r ) − u(⃗ r )| ⟩.

(36)

p

After discretization of the 1⃗ r displacement the above relation can be identified as: Sp(ln) = ⟨|δ un |p ⟩.

(37)

− → The field values u( x ) can be related with the energy dissipation values εn by the general relation εn = (δ un )3 /ln in order to obtain the structure functions as follows: Sp (ln ) = ⟨(εn /ε0 )p ⟩ = ⟨δnp(a−1) ⟩ = δnJ (p)

(38)

where the averaging process ⟨. . .⟩ is defined by using the probability function P (a)da as ⟨. . .⟩ = scaling exponent J (p) of the structure functions is given by the relation:



J (p) = 1 + τ q¯ =

p 3



da(. . .)P (a). By this, the

.

(39)

By following Arimitsu [25,17] the relation (15) leads to the theoretical prediction of J (p) after extremization of Tsallis entropy as follows: J (p) =

a0 p 3

−

2Xp2 q(1 +

Cp/3 )



−

1 1−q

[1 − log2 (1 +



Cp/3 )].

(40)

The first term a0 p/3 corresponds to the original of the known Kolmogorov theory (K41) according to which the dissipation of field energy εn is identified with the mean value ε0 according to the Gaussian self-similar homogeneous turbulence dissipation concept, while a0 = 1 according to the previous analysis for homogeneous turbulence. According to this concept the multifractal spectrum consists of a single point. The next terms after the first in the relation (40) correspond to the multifractal structure of intermittence turbulence indicating that the turbulent state is not homogeneous across spatial scales. That is, there is a greater spatial concentration of turbulent activity at smaller than at larger scales. According to Abramenko [26] the intermittent multifractal (inhomogeneous) turbulence is indicated by the general scaling exponent of the structure functions according to the relation: J (p) =

p 3

+ T (u) (p) + T (F ) (p),

(41)

where the T (u) (p) term is related with the dissipation of kinetic energy and the T (F ) (p) term is related to other forms of the field’s energy dissipation as magnetic energy at MHD turbulence [26]. The scaling exponent spectrum J (p) can also be used for the estimation of the intermittency exponent µ according to the relation: S (p = 2) ≡ ⟨ε 2 /ε⟩ ∼ δnµ = δnJ (2) from which we conclude that µ = J (2). The intermittency turbulence correction to the law P (f ) ∼ f spectrum of Kolmogorov’s theory is given by using the intermittency exponent: P (f ) ∼ f −(5/3+µ) .

(42) −5/3

of the energy

(43)

The previous theoretical description can be used for the theoretical interpretation of the experimentally estimated structure function, as well as for relating physically the results of data analysis with Tsallis statistical theory, as it is described previously. According to relations (20, 21, 27–30) concerning the theoretical estimation of the singular spectrum f (a), as well as the estimation of the generalized dimension function Dq, we need to know the values of (α0 , q = qsen and X ). In Fig. 4(a)–(f) we present the comparison of theoretically and experimentally estimated functions f (a), Dq by using the Tsallis theory, according to Arimitsu and Arimitsu [17], as well as the p-model estimation. In this case we have used the experimental values of (α0 and q = qsen ), presented in Table 1 for the estimation of the theoretical f (a) and Dq functions after the estimation of X values by the relation (26).

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L.P. Karakatsanis et al. / Physica A 392 (2013) 3920–3944 Table 2 Theoretically estimated values (α0 , X , qsen ) by using different values of µ. The values in bold format correspond to the experimentally estimated values of α0 .

µ

α0

X

qsen

0.068 0.10 0.20 0.22 0.23 0.25 0.30 0.31 0.35 0.40 0.45 0.50

1.038 1.056 1.115 1.126 1.138 1.144 1.173 1.179 1.202 1.231 1.259 1.287

0.077 1.115 0.232 0.261 0.285 0.297 0.357 0.369 0.417 0.476 0.533 0.590

−0.610 −0.214 0.288 0.343 0.390 0.412 0.503 0.519 0.574 0.631 0.678 0.718

As we find out the theoretical estimation presented in Fig. 4(a)–(e) is faithful with high precision on the left part of the experimental function f (a), according to Arimitsu and Arimitsu [18,19]. In Fig. 4(a), (c), (e) the experimentally estimated spectral function f (α) is compared with a polynomial of sixth order (solid line) as well as by the theoretically estimated function f (α) (dashed line), by using the Tsallis q-entropy principle and following Arimitsu and Arimitsu [25]. As we find out the theoretical estimation is faithful with high precision on the left part of the experimental function f (a). However, the fit of theoretical and experimental data are less faithful for the right part of f (a), especially for the original solar flares timeseries (Fig. 4(a)) and its SVD components V1 (Fig. 4(c)). Finally, the coincidence of theoretically and experimentally data is excellent for the V2–15 SVD component (Fig. 4(e)). A similar comparison of the theoretical prediction and the experimental estimation of the generalized dimensions function D(q) is shown in Fig. 4(b), (d), (f). In these figures the solid brown line corresponds to the p-model prediction according to Ref. [27], while the solid red line corresponds to the D(q) function estimation according to Tsallis theory [3]. The correlation coefficient of the fitting was found higher than 0.9 for all cases. These results indicate the turbulence cascade of solar corona plasma, the partial mixing and asymmetric (intermittent) fragmentation process of the energy dissipation. Moreover, we notice the relation between the 1αmin,max values where 1αmin,max = amax − αmin , which was found to satisfy the following ordering relation: 1α(V1 ) = 0.77 < 1α(V2–10 ) = 1.37 < 1α(orig) = 2.15. According to N. Arimitsu et al. [19] and T. Arimitsu et al. [28], the left half of the experimentally estimated spectrum function f (α) corresponds to the tail part of the PDF for energy dissipation rates, and the right half of the experimental f (α) is related to the center part of the PDF. However, the theoretical formula for the spectrum function is applicable only to the part of the PDF constructed mainly from the intermittent singular element of turbulence. Since the tail-part of the PDF is responsible for the intermittent singular behavior of turbulence, the left half of theoretical spectrum f (α) is the part which should be compared with the experimentally estimated spectral function. On the other hand, since the center part of the PDF contains the element of thermal fluctuations originated from the dissipative term in the Navier–Stokes equation in addition to the contribution from the singular element, the right half of the theoretical spectrum f (α) has usually deviated from the experimentally estimated spectrum. Fig. 4(a), (c), (e) present quite well the characteristics of f (α) given above. The difference between the right-half theoretical spectrum f (α) and the right-half experimentally estimated spectrum tells us in what degree the element of fluctuation contributes to the center part of the PDF compared with the intermittent singular element. In this respect, all of Fig. 4(a), (c), (e) provide us equally important information about the system. The theoretical estimated values of the generalized dimension D(q) are presented in Fig. 4(b), (d), (f), with the appropriately adjusted values of α0 , qsen and X . Similarly according to Arimitsu et al. [28] for the function D(q) the deviation between the theoretical and experimental D(q) is due to the element of fluctuation contributing to the center part of the PDF. For larger positive q, the theoretical D(q) may deviate from the experimental one. This is due to the cut-off of the data point of the experimental PDF which is indispensable since the experimental PDF terminates itself at a certain tail point, i.e., causing a lack of the contribution from larger deviation. The values of (α0 , q = qsen , X ) estimated theoretically by using different values of the intermittency exponent µ are presented in Table 2. Fig. 4(g)–(l) present the comparison of the experimentally and theoretically estimated functions f (a), Dq corresponding to those values of µ, at which there is coincidence between the theoretical (predicted) and experimental (estimated) values of α0 . However in this case, the experimental values of qsen reveal noticeable deviation for theoretical values of qsen as we can conclude comparing Tables 1 and 2. The 1D−∞,+∞ and 1αmin,max values were estimated separately for the first fifteen SVD components. The 1D−∞+∞ (Vi ) values vs. Vi , (i = 1, . . . , 15) are shown in Fig. 5(a) and the 1αmin.max (Vi ) vs. Vi , (i = 1, . . . , 15) are shown in Fig. 5(b). The spectra of 1Dq and 1Da shown in these figures (Fig. 5(a), (b)) reveal a positive and increasing profile as we pass from the first to the last SVD component. This clearly indicates the solar intermittent turbulent process underlying all the SVD components of the solar flares index. However the intermittency character becomes stronger at the large SVD components.

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Fig. 4. (a) Multifractal spectrum of solar flares timeseries. The solid line is a sixth degree polynomial. We calculate the qsen = 0.308. (b) D(q) vs. q of solar flares timeseries. (c) Multifractal spectrum of V1 SVD component. We calculate the qsen = −0.540. The solid line is a sixth degree polynomial (d) D(q) vs. q of V1 SVD component. (e) Multifractal spectrum of V2–15 SVD component. We calculate the qsen = 0.192. The solid line is a sixth degree polynomial (f) D(q) vs. q of V2–15 SVD component. (g) Multifractal spectrum of solar flares timeseries. We calculate the qsen = 0.390 using µ = 0.23. (h) D(q) vs. q of solar flares timeseries. (i) Multifractal spectrum of V1 SVD component. We calculate the qsen = −0.610, using µ = 0.068. (j) D(q) vs. q of V1 SVD component. (k) Multifractal spectrum of V2–15 SVD component. We calculate the qsen = 0.519, using µ = 0.31. (l) D(q) vs. q of V2–15 SVD component. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

3.3. Determination of structure function spectrum 3.3.1. Intermittent solar turbulence Fig. 6(a)–(c)–(e) show the structure function S (p) plotted versus time lag(τ ) estimated for the original solar flare index signal (Fig. 6(a)) as well as for its SVD components V1 (Fig. 6(c)) and V2–15 (Fig. 6(e)). At low values of time lag(τ ), we can

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Fig. 4. (continued)

observe a scaling profile for all the cases of the original timeseries and its SVD components for every p value. Fig. 6(b)–(d)–(f) presents the best linear fitting of the scaling regions in the log time interval 1 log(τ ) = 0–1.4. Fig. 7(a) shows the exponent J (p) spectrum of structure function vs. the pth order, which was estimated separately for the first seven SVD components Vi , (i = 1, . . . , 15). In the same figure we present the exponent J (p) spectrum of the structure function S (p), according to the theory of Kolmogorov [20] of Gaussian turbulence, known as p/3 theory. This result shows clearly discrimination from the Gaussian turbulence K41 theory for all the SVD components. Moreover, we observe significant dispersion of the J (p) spectrum from the first to the fifteenth SVD component, especially for the higher orders (p > 10). Fig. 7(c) is similar to Fig. 7(a) estimated for the V1 SVD component and the summarized V2–15 SVD component. This shows significant discrimination between the V1 and V2–15 SVD components while both of them reveal pth order structure functions with their own index different from the p/3 values given by the K41 theory. In order to study in more detail the divergence from the Gaussian behavior of the turbulence underlying the solar flares index signal, we present in Fig. 7(e)–(f) the derivatives h(p) = dJ (p)/dp of the structure functions estimated for the spectra S (p) of the V1 and V2–10 SVD components.

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Fig. 5. (a) The differences ∆(Dq(Vi )) versus the Vi SVD components of solar flares timeseries for i = 1, 2, . . . , 15. (b) The differences ∆(Da(Vi )) versus the Vi SVD components of solar flares timeseries for i = 1, 2, . . . , 15.

For both cases the nonlinearity of the functions SV1 (p), SV2–10 (p) is apparent as their derivatives are strongly dependent upon the p values. 3.3.2. Comparison of solar turbulence with non-extensive q-statistics In this section we present interesting results concerning the comparison of the structure function experimental estimation with the theoretical predictions according to Arimitsu and Arimitsu [25] by using the Tsallis non-extensive statistical theory, as it was presented in Section 2 of part one [1]. Fig. 8(a) presents the structure function J (p) vs. the order parameter p, estimated for the solar flares index timeseries (black line), the K41 theory (dashed line), and the theoretically predicted values of the structure function (red line) by using q-statistics of Tsallis theory, as well as the differences 1J (p) vs. p, between the experimentally and theoretically produced values. Fig. 8(b)–(c) are similar to Fig. 8(a) but corresponding to the V1 and V2–15 SVD components of the original solar flares index timeseries. In all cases presented in Fig. 8(a)–(c) the theoretically estimated J (p) values (red lines) correspond to the HD turbulent dissipation of the solar plasma, revealing values lower than the K41 prediction in accordance with the HD intermittent turbulence. According to the previous theoretical description (Section 3.2.4), the experimentally produced structure function spectrum of the original signal and its SVD components is caused by including kinetic and magnetic dissipation simultaneously [7] of the MHD solar turbulence. In this case the values of the structure function spectrum are different to the values of the corresponding HD intermittent turbulence due to the dissipation of the solar corona magnetic field. As we see in Fig. 8(a)–(b) the best fitting to the differences 1J (p) shows the existence of the linear relation: 1J (p) ≈ α(p + b) at high values of p for all cases (the solar flares index and its V1 , V2–15 SVD components). In accordance to Arimitsu and Arimitsu [17], the experimentally estimated values J (p = 6) can be used for the estimation of the intermittency exponent µ by using the equations µ = 1 + τ (2), J (6) = 1 − τ (2), resulting to J (6) = 2 − µ. Table 3 present the theoretically estimated values of µ by using the J (6) values extracted by the structure function scaling exponent spectrum as well as the J ∗ (6) values estimated by the Tsallis theory in accordance with Arimitsu and Arimitsu [17]. As we can observe in Table 3 there is sensible differentiation of the µ values estimated for the experimental values of J (6) and the values µ∗ estimated for the theoretically predicted values J ∗ (6). The µ∗ values are lower than 0.5, while the µ values are higher than 1.0. 3.4. Determination of correlation dimension 3.4.1. Correlation dimension of the solar flares timeseries Fig. 9(a) shows the slopes of the correlation integrals vs. log r estimated for the solar flares index timeseries for embedding dimension m = 6–10. We notice that there is no tendency for low value saturation of the slopes, which increase continuously as the embedding dimension (m) increases. Fig. 9(b)–(c) present the comparison of the slopes with surrogate data. Specifically Fig. 9(b) presents the slopes of the original signal at embedding dimension m = 7 (red line), as well as the slopes estimated for a group of corresponding surrogate data. The significance of the statistics was shown in Fig. 9(d). As the significance of the discriminating statistics remains much lower than two sigmas, we cannot reject the null hypothesis of a high dimensional Gaussian and linear dynamics, underlying the solar flares index of the solar activity.

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Fig. 6. (a) The log–log plot of structure function S p of the solar flare index timeseries vs. time lag τ for various values of the order parameter p. (b) The first linear scaling of the log–log plot. (c) The log–log plot of structure function S p of the V1 SVD component vs. time lag τ for various values of the order parameter p. (d) The first linear scaling of the log–log plot. (e) The log–log plot of structure function S p of the V2–15 SVD component vs. time lag τ for various values of the order parameter p. (f) The first linear scaling of the log–log plot.

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Fig. 7. (a) The scaling exponent J (p) versus p of the independent V1 − V15 SVD components of the solar flares timeseries and compared with the Kolmogorov p/3 prediction (dashed line). (b) The zoom in the area of p = 12–16. The scaling exponent J (p) versus p of the V1 , V2–15 SVD components of the solar flares timeseries and compared with the Kolmogorov p/3 prediction (dashed line). (d) The zoom in the area of p = 8–16. (e) The h(p) = dJ (p)/dp function versus p for the V1 SVD component of solar flares timeseries. (f) The h(p) = dJ (p)/dp function versus p for the V2–15 SVD component of solar flares timeseries.

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Fig. 8. (a) The J (p) function versus p for solar flares timeseries and compared with the Kolmogorov (p/3) prediction (dashed line), the theoretical curve of Tsallis theory and its differences between experimental and theoretical. (b) The J (p) function versus p for V1 SVD component compared with the Kolmogorov (p/3) prediction (dashed line), the theoretical curve of Tsallis theory and its differences between experimental and theoretical results. (c) The J (p) function versus p for V2–10 SVD component and compared with the Kolmogorov (p/3) prediction (dashed line), the theoretical curve of Tsallis theory and its differences between experimental and theoretical results. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) Table 3 Experimentally and theoretically estimated values (µ, J (6)) for the timeseries of solar flares and SVD components. Signal

J (6 )

µ = (2 − J (6))

J (6 )∗

µ∗ = (2 − J ∗ (6))

Original solar flares V1 SVD component V2–15 SVD component

0.597 0.477 0.485

1.403 1.523 1.515

1.814 1.582 1.908

0.186 0.418 0.092

∗ Correspond to theoretically estimated values of Tsallis theory.

Fig. 10 shows the results of the estimation correlation dimensions of the V1 and V2–15 SVD components of the original solar flares signal. 3.4.2. Correlation dimension of the V1 SVD component Fig. 10(a) and (b) present the slopes of the correlation integrals estimated for the V1 SVD component and its corresponding surrogate data. The slope profiles of the V1 SVD component reveal no saturation. However, the corresponding slope profiles

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Fig. 9. (a) Slopes D of the correlation integrals estimated for the solar flares timeseries. (b) Slopes D of the correlation integrals estimated for the surrogate timeseries. (c) Slopes of the correlation integrals of the solar flares timeseries and its thirty (30) surrogates estimated for delay time τ = 500 and for embedding dimension m = 10, as a function of Ln(r ). (d) The significance of the statistics for the solar flares timeseries and its 30 surrogates. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

of the surrogate data are similar with the slope profile of the V1 SVD component. Fig. 10(c) shows the slopes of the V1 SVD component and a group of corresponding surrogate data at the embedding dimension m = 10, while in Fig. 10(d) we present the significance of the statistics. As the significance remains at low values (<2 sigma) for small values of log r, we cannot reject at this point the null hypothesis. 3.4.3. Correlation dimension of the V2–10 SVD component Fig. 10(e)–(h) is similar to Fig. 10(a)–(d) concerning the V2–10 SVD component. The profile of slopes of this signal reveals saturation at values lower than ∼5.5 (Fig. 10(e)), with strong discrimination from the corresponding surrogate data (Fig. 10(f), (g)). The significance of the discriminating statistics is much higher than two sigma (Fig. 10(h)). This allows the rejection of the null hypothesis by confidence >99%. 3.5. Determination of the spectra of the Lyapunov exponents In this section we present the estimated spectra of Lyapunov exponents for the original timeseries of the solar flare index and its SVD components. Fig. 11(a) shows the spectrum Li , i = 1–6 of the Lyapunov exponents estimated for the original timeseries of the solar flares index. As we can observe in Fig. 11(a) there is no positive Lyapunov exponent, while the largest one approaches the value of zero, from negative values. Fig. 11(b) is similar to Fig. 11(a) but for the surrogate data corresponding to the original timeseries. The similarity between the original signal of the Lyapunov exponent spectra and its surrogate data is obvious. The null hypothesis cannot be rejected as the significance of the discriminating statistics was estimated to be lower than two sigma. The V1 SVD component of the original timeseries in Fig. 11(c)–(d) is similar to the previous figures and semantically different from surrogate data because the significance of the statistics obtains values much higher than two sigma, while the largest Lyapunov exponent obtains zero value. Finally in Fig. 11(e)–(f), we present the Lyapunov exponent spectrum for the V2–15 SVD component (Fig. 10(e)) and its surrogate data (Fig. 10(f)). In the case of the V2–15 SVD component there is also observed the possibility for a strong discrimination from the surrogate data, as the significance of the statistics was found much higher than two sigma. Here the largest Lyapunov exponent was estimated to be clearly positive. Table 4 indicates that there is clearly discrimination between the dynamical characteristics of sunspot, flare and solar processes.

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Fig. 10. (a) Slopes D of the correlation integrals estimated for V1 SVD component. (b) Slopes D of the correlation integrals estimated for the surrogate timeseries. (c) Slopes of the correlation integrals of the V1 SVD component and its thirty (30) surrogates estimated for delay time τ = 500 and for embedding dimension m = 10, as a function of Ln(r ). (d) The significance of the statistics for the V1 SVD component timeseries and its 30 surrogates. (e) Slopes D of the correlation integrals estimated for V2–15 SVD component. (f) Slopes D of the correlation integrals estimated for the surrogate timeseries. (g) Slopes of the correlation integrals of the V2–15 SVD component and its thirty (30) surrogates, estimated for delay time τ = 500 and for embedding dimension m = 10, as a function of Ln(r ). (h) The significance of the statistics for the V1 SVD component timeseries and its 30 surrogates.

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Fig. 11. (a) The spectrum of the Lyapunov exponents Li as a function of embedding dimension for the solar flares timeseries. (b) The spectrum of the Lyapunov exponents Li as a function of embedding dimension for the surrogate data. (c) The spectrum of the Lyapunov exponents Li as a function of embedding dimension for the V1 SVD component (d) The spectrum of the Lyapunov exponents Li as a function of embedding dimension for the surrogate data. (e) The spectrum of the Lyapunov exponents Li as a function of embedding dimension for the V2–15 SVD component. (f) The spectrum of the Lyapunov exponents Li as a function of embedding dimension for the surrogate data.

4. Summary of data analysis results In this study we used the SVD analysis in order to discriminate the dynamical components, underlying the solar flare index timeseries. We applied an extended algorithm for the nonlinear analysis of the original solar flare index timeseries, its V1 (first) SVD component and the signal V2–10 composed from the sum of the higher SVD components. The analysis was expanded to include the estimation of: (a) flatness coefficients as a measure of Gaussian, and non-Gaussian dynamics, (b) the q-triplet of Tsallis non-extensive statistics, (c) the correlation dimension, (d) the Lyapunov exponent spectrum, and (e) the spectrum of the structure function scaling exponent. The results of data analysis presented in Section 3 are summarized as follows:

• Clear distinction was observed between the two dynamics everywhere: (a) the solar flare dynamics underlying the first (V1 ) and (b) the solar flares dynamics underlying the (V2–10 ) SVD component of the solar flare index timeseries. • The non-Gaussian and non-extensive statistics were found to be effective for the original solar flare index as well as its SVD components V1 , V2–10 of the solar flares process, indicating non-extensive solar dynamics.

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• The Tsallis q-triplet (qsen , qstat , qrel ) was found in every case to verify the expected scheme qsen ≤ 1 ≤ qstat ≤ qrel . •

• • •

• •

•

•

• • •

• •

Moreover the Tsallis q-triplet estimation showed clear distinction between the various dynamics underlying the SVD analyzed solar flare signal as follows: qk (V1 ) < qk (original) < qk (V2–10 ), for all the k ≡ (sen, stat, rel). The multifractal character was verified for the original signal and its SVD components. Also, the multifractality was found to be intensified as we pass from the V1 to the V2–10 SVD component in accordance with the relation: 0 < 1a(V1 ) < 1a(V2–10 ), 0 < 1Dq(V1 ) < 1Dq(V2–10 ), where 1a = amax − amin , and 1Dq = Dq=−∞ − Dq=+∞ . Efficient agreement between Dq and p-model was discovered indicating intermittent (multifractal) solar turbulence which is intensified for the solar dynamics underlying the V2–10 SVD component of the solar flares index. Generally the differences 1a(Vi ) and 1Dq (Vi ) increase passing from lower to higher SVD components Vi , i = 1, 2, . . . , 7. The qrel index was estimated for two distinct relaxation magnitudes, the autocorrelation function C (τ ) and the mutual information I (τ ), indicating a good agreement taking into account the linear–nonlinear character of the C (τ ) and I (τ ) respectively. The correlation dimension was estimated at low values DCD = 4–5 for the V2–10 SVD component. For the V1 SVD component of the original signal the correlation dimension was found to be higher than the value ∼8. Also the null hypothesis of non-chaotic dynamics and nonlinear distortion of white-noise was rejected only for the V2–10 SVD component. For the original signal and its V1 SVD component the rejection was insignificant (s < 2). These results indicate nonlinearity low dimensional determinism solar dynamics underlying the V2–10 SVD component and high dimensional self-organized criticality (SOC) solar dynamics for the V1 SVD component. The estimation of the Lyapunov exponent spectrum showed for the V2–10 SVD component one positive Lyapunov exponent (λ1 > 0) clearly discriminated from the signal of surrogate data. For the original solar flares signal and its V1 SVD component the discrimination with surrogate data was inefficient. These results indicate low-dimensional and chaotic deterministic dynamics underlying the V2–10 SVD (λ1 > 0) component and weak chaos solar dynamics underlying the V1 SVD component related to a SOC process at the edge of chaos (λ1 = 0). The structure function scaling exponents spectrum J (p) was estimated as values lower than the corresponding p/3 values of the K41 theory, in both cases, the V1 and the V2–10 SVD components, as well as for the original signal of the solar flare index. This result indicates the intermittent (multifractal) character of the solar corona turbulence dissipation. The slopes dJ (p)/dp of the scaling exponent function were found to be decreasing as the order p increases. This result confirms the intermittent and multifractal character of the low solar corona turbulence dissipation process. The SVD analysis of the solar flare dynamics in relation to the structure functions exponent spectrum J (p) estimated for the SVD components showed clearly the distinction of the dynamics underlying the first and the higher SVD components. The difference 1J (p) between the experimental and theoretical values of the scaling exponent spectrum of the structure functions was found to follow regionally a linear profile: 1J (p) = ap + b for the original solar flares index and its V1 and the V2–10 SVD components. Adding the 1J (p) = ap + b function to the theoretically estimated J (p) values by using Tsallis theory we obtain an excellent agreement of the theoretically predicted and the experimentally estimated exponent spectrum values J (p). We also found noticeable agreement of Tsallis theory and the experimental estimation of the functions f (a), D(q), and J (p).

5. Discussion and theoretical interpretation The results of previous data analysis showed clearly: (a) The non-Gaussian and non-extensive statistical character of the low solar corona dynamics underlying the solar flare index timeseries. (b) The intermittent and multifractal turbulent character of the solar low corona system, in accordance with non-extensive statistics of Tsallis theory. (c) The phase transition process between different solar flare dynamical profiles (that is from solar corona plasma selforganized criticality (SOC) process to solar corona plasma low dimensional chaotic dynamics). (d) Novel agreement of q-entropy principle and the experimental estimation of solar flare intermittent turbulence indices: f (a), D(q) and J (p). (e) Clear discrimination of the solar flare dynamics from solar flare and solar corona activity dynamics through the q-triplet of Tsallis and structure functions exponent spectrum, as we can conclude by Table 4 and Fig. 12. The experimental results of this study indicate clearly the non-Gaussian and non-extensive, as well as the multifractal and multiscale dynamics of the solar flare process. According to these results we can conclude for solar flares the existence of a new mechanism of anomalous kinetic and magnetic energy dissipation and anomalous charged particle acceleration at the solar flare regions. This mechanism can be characterized as a fractal dissipation–fractal acceleration mechanism as the regions of dissipation–acceleration corresponds to fractal fields–particles distributions. This mechanism is further described in the study of Pavlos et al. [2]. It is also significant to notice the similarity between the low solar corona dynamics and the dynamics of the solar convection zone, concerning the phase transition process from a high dimensional SOC state to a low dimensional chaos

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Table 4 Summarized parameter values of solar dynamics including the sunspot and the solar flare dynamics: From the top to the bottom we show: changes of the ranges 1α of the multifractal profile, the q-triplet (qsen , qstat , qrel ) of Tsallis, the values of the maximum Lyapunov exponent (Li ), the next Lyapunov exponent and the correlation dimension (D).

1α = αmax −αmin qsen q_stat q_rel (C (τ )) q_rel (I (τ )) L1 Li , (i > 2) D (cor. Dim.)

Solar flares TMS

Sunspot index TMS

Solar flares V1 compon.

Sunspot index V1 compon.

Solar flares V2–10 compon.

Sunspot index V2–15 compon.

2.15 0.308 1.87 ± 0.06 5.329 ± 0.135 9.333 ± 0.395 ≈0 <0 >8–10

1.752 0.368 1.53 ± 0.04 5.672 ± 0.127 2.522 ± 0.044 ≈0 <0 >8

0.77 −0.540 1.28 ± 0.04 42.666 ± 2.950 9.849 ± 0.410 ≈0 <0 >6–8

1.113 0.055 1.40 ± 0.08 29.571 ± 0.794 5.255 ± 0.308 ≈0 <0 >6

1.37 0.192 2.02 ± 0.15 4.546 ± 0.173 3.207 ± 0.086 >0 <0 (L2 > 0) ≈5.5

1.940 0.407 2.12 ± 0.20 4.115 ± 0.134 2.426 ± 0.054 >0 <0 (L2 > 0) ≈6

state. It is however possible to discriminate two dynamical systems by following the differentiation of the various dynamical characteristics as they are summarized in Table 4. Concerning the comparison of the experimental values of (f (a), Dq, J (p), µ) with the corresponding theoretically estimated values by using Tsallis non-extensive statistical theory, we conclude the necessity to extend the hydrodynamical turbulence theory to the MHD turbulence theory in accordance with Arimitsu and Arimitsu [25,17–19,28]. 5.1. Solar corona phase transition process The critical dynamics of distributed systems include the possibility of first or second order phase transition processes [29]. From this point of view Pavlos et al. [30,31] showed the existence of two distinct critical states of the magnetospheric plasma. The first critical state corresponds to a second order phase transition related to a high dimensional SOC process, while the second state corresponds to a first order phase transition process related to a low dimensional chaotic process. The results of this study reveal similar results for the case of the solar magnetic activity caused by the solar corona dynamics. The term ‘‘phase transition process’’ included in this study must be understood as a ‘‘meta-phase transition’’ process and it is used in order to characterize the change of the metaequilibrium critical state of the solar corona zone dynamics from the high dimensional solar SOC state to the low dimensional solar chaos state. This ‘‘meta-phase transition’’ process was concluded in this study by using SVD analysis of experimental timeseries. The first component (V1 ) of the SVD analysis of the solar flare and solar corona activity index, including the main power of the solar corona activity timeseries as it is associated with the higher singular value (σ1 ), can be related to the solar dynamics at quiet periods. At this critical state the solar corona lives at the edge of the chaos as it was indicated by the zero value of the largest Lyapunov exponent, while the dimensionality of the dynamical degrees of freedom was found to be higher than m = 6 without any low saturation profile of the slopes of the correlation integrals. The sum V2–10 of the next SVD components corresponds to solar active periods, where the critical state corresponds clearly to low dimensional chaos with positive Lyapunov exponent and low dimensional nonlinear deterministic dynamics. The SVD components {Vi } corresponding to the non-zero (higher than the noise level) singular values {σ } describe significant dynamical components underlying the experimental timeseries. According to Arimitsu and Arimitsu [32] the probability distribution function of experimental signals of a turbulent process can be divided into two parts at every scale (n) (n) level ln = l0 δn , Π (n) (Xn ) = ΠN (Xn ) + ΠSn (Xn ). The normal part ΠN (Xn ) stems from thermal Gaussian dissipation and the (n)

singular part ΠS (Xn ) stems from non-Gaussian multifractal distribution of the singularities (α) corresponding at different interwoven fractal subsets of the turbulent state. The normal part can be related with the noise level singular values and SVD components, while the singular part corresponds to the dynamical SVD components corresponding to non-zero singular values. As we conclude by the analysis of experimental signals in the next sections of this study, the SVD method can reveal distinct singular parts corresponding to distinct multifractal structures underlying the solar flares signal, caused by a phase transition process of the solar system, as we explain in the last section of this study. As the solar system passes from quiet to active periods the critical state changes its profile. Also according to Arimitsu [32] the singular part of the PDF can be discriminated into two distinct parts Πs (V1 ), Πs (V2–10 ). The Πs (V1 ) PDF describes the solar SOC state, while the Πs (V2–10 ) describes the solar chaos state. Similar to the magnetospheric plasma the solar SOC state corresponds to a second order phase transition, while the solar chaos state corresponds to a first order phase transition. 5.2. Non-extensive statistics and phase transition of the solar dynamics The estimated values of the Tsallis q-triplet corresponding to the solar activity showed clearly non-extensive statistical character for both critical states SOC and chaos of the solar corona activity. However, the distinct profile of the q-triplet values was shown clearly as the solar corona system changes its critical states. At the quiet period of the SOC state the values of the q-triplet were found to be: qsen = −0.54, qstat = 1.28, qrel = 9.84, while the q-triplet values at the solar

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active period were found to be qsen = 0.19, qstat = 2.02, qrel = 4.54. The quiet state is clearly of non-Gaussian profile (qstat = 1.28 > 1) but it clearly increases to higher values (qstat = 2.02) as the solar system passes to active periods. The corresponding and simultaneous increase of qsen from the value −0.54 to the value of 0.19 and the decrease of qrel from the value 9.84 to the lower value qrel = 4.54 are in agreement with the change from SOC to the low dimensional chaos as the largest Lyapunov exponent changes from zero to positive values. This implies a decrease of the relaxation time while the attractive set of the phase space obtains strange and strong multifractal topological character causing the increase of the qsen and the largest Lyapunov exponent values. Also as the low dimensional chaotic state corresponds to a first order phase transition process, the change from quiet to active profile of the solar dynamics is accompanied with the development of long range correlations. This is in accordance with the observed strengthening of the non-Gaussian profile of dynamics and increasing of the qstat value. 5.3. Renormalization group (RNG) theory and solar phase transition The multifractal and multiscale intermittent turbulent character of the solar dynamics in the solar corona was verified in this study after the estimation of the structure function scaling exponents spectrum J (p) and the spectrum f (a) of the pointwise dimensions or singularities (α). This justifies the application of RNG theory for the description of the scale invariance and the development of long-range correlation of the solar intermittent turbulence state. Generally the solar plasma can be described by generalized Langevin stochastic equations of the general type:

∂ϕi = fi (ϕ, x, t ) + ni (x, t ) ∂t

i = 1, 2, . . .

(44)

where fi corresponds to the deterministic process as concerns the plasma dynamical variables φ(⃗ x, t ) and ni to the stochastic components (fluctuations). Generally, fi are nonrandom forces corresponding to the functional derivative of the free energy functional of the system. According to Chang [33,34] and Chang et al. [35] the behavior of a nonlinear stochastic system far from equilibrium can be described by the density functional P, defined by P (ϕ (x, t )) =



   ˙ D (x) exp −i · L (ϕ, ϕ, x) dx dt

(45)

˙ ϕ, x) is the stochastic Lagrangian of the system, which describes the full dynamics of the stochastic system. where L(ϕ, Moreover, the far from equilibrium renormalization group theory applied to the stochastic Lagrangian L generates the singular points (fixed points) in the affine space of the stochastic distributed system. At fixed points the system reveals the character of criticality, as near criticality the correlations among the fluctuations of the random dynamic field are extremely long-ranged and there exist many correlation scales. Also, close to dynamic criticality certain linear combinations of the parameters, characterizing the stochastic Lagrangian of the system, correlate with each other in the form of power laws and the stochastic system can be described by a small number of relevant parameters characterizing the truncated system of equations with low or high dimensionality. According to these theoretical results, the stochastic solar plasma system can exhibit low dimensional chaotic or high dimensional SOC-like behavior, including fractal or multifractal structures with power law profiles. The power laws are connected to the near-criticality phase transition process which creates spatial and temporal correlations as well as strong or weak reduction (self-organization) of the infinite dimensionality corresponding to a spatially distributed system. First and second phase transition processes can be related to discrete fixed points in the affine dynamical (Lagrangian) space of the stochastic dynamics. The SOC-like behavior of plasma dynamics corresponds to the second phase transition process as a high dimensional process at the edge of chaos. The process of strong and low dimensional chaos can be related to a first order phase transition process. The probabilistic solution (Eq. (44)) of the generalized magnetospheric Langevin equations may include Gaussian or non-Gaussian processes as well as normal or anomalous diffusion processes depending upon the critical state of the system. From this point of view, a SOC or low dimensional chaos interpretation or distinct q-statistical states with different values of the Tsallis q-triplet depends upon the type of the critical fixed (singular) point in the functional solution space of the system. When the stochastic system is externally driven or perturbed, it can be moved from a particular state of criticality to another characterized by a different fixed point and different dimensionality or scaling laws. Thus, the old SOC theory could be a special kind of critical dynamics of an externally driven stochastic system. After all SOC and low dimensional chaos can coexist in the same dynamical system as a process specified by different kinds of fixed (critical) points in its solution space. Due to this fact, the solar corona dynamics may include a high dimensional SOC process or low dimensional chaos or other more general dynamical process corresponding to various q-statistical states. 5.4. q-extension and fractal extension of dynamics The well known Boltzmann’s formula S = k log W where S is the entropy of the system and W is the number of the microscopic states corresponding to a macroscopic state indicates the priority of statistics over dynamics. However, Einstein

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Fig. 12. The J (p) function versus p for sunspot timeseries with V1 , V2–10 SVD components, solar flares timeseries with V1 , V2–15 SVD components and compared with the Kolmogorov (p/3) prediction (dashed line). S

preferred to put dynamics in priority over statistics [36] using the inverse relation, i.e. W = e k where entropy is not a t Ω statistical but a dynamical magnitude. Already, Boltzmann himself was using the relation WR = limT →∞ TR = ΩR where tR is the total amount of time that the system spends during the time T in its phase space trajectory in the region R while ΩR is the phase space volume of region R and Ω is the total volume of phase space [31]. The above concepts of Boltzmann and Einstein were innovative as concerns modern q-extension of statistics which is internally related to the fractal extension of dynamics. According to Zaslavsky [4,5] the fractal extension of dynamics includes simultaneously the q-extension of statistics as well as the fractal extension RNG theory in the Fraction at Fokker–Planck–Kolmogorov equation (FFPK):

∂ β P (x, t ) ∂α ∂ a+1 [B(x)P (x, t )] . [A(x)P (x, t )] + = a β ∂ t ∂ (−x) ∂ (−x)a+1

(46)

As concerns the space plasma dynamics the plasma particle and fields magnitude correspond to the (x) variable in Eq. (46), while P (x, t ) describes the probability distribution of the particle–fields variables. The variables A(x), B(x) corresponds to the first and second moments of probability transfer and describe the wandering process in the fractal space β

α

(phase space) and time. The fractional space and time derivatives ∂∂t β , ∂∂xα are caused by the multifractal (strange) topology of phase space which can be described by the anomalous phase space Renormalization Transform [5]. We must notice here that the multifractal character of phase space is the mirroring of the phase space strange topology in the spatial multifractal distribution of the dynamical variables. The q-statistics of Tsallis corresponds to the metaequilibrium solutions of the FFPK equation [5,24]. Also, the metaequilibrium states of the FFPK equation correspond to the fixed points of Chang non-equilibrium RNG theory for space plasmas [33,34]. The anomalous topology of phase space dynamics includes inherently the statistics as a consequence of its multiscale and multifractal character. From this point of view the non-extensive character of thermodynamics constitutes a kind of unification between statistics and dynamics. From a wider point of view the FFPK equation is a partial manifestation of a general fractal extension of dynamics. According to Tarasov [5] Zaslavsky’s equation can be derived from a fractional generalization of the Liouville and BBGKI equations. According also to Tarasov [5] the fractal extension of dynamics including the dynamics of particles or fields is based on the fact that the fractal structure of matter (particles, fluids, fields) can be replaced by a fractional continuous model. In this generalization the fractional integrals can be considered as approximations of integrals on fractals. Also, the fractional derivatives are related to the development of long range correlations and localized fractal structures. In this direction the solar dynamo theory must be based on the extended fractal plasma theory including anomalous magnetic transport and diffusion, magnetic percolation and magnetic Levy random walk [37]. Also from a more extreme point of view, the fractal environment for the anomalous turbulence dissipation of magnetic field and plasma flows is the fractality of the space–time itself according to Shlesinger [38], Nottale [39], and Chen [40]. Finally we must say that the solar phase transition process corresponds to the topological phase transition process of the attracting set in the phase space of the solar dynamics. Acknowledgments The authors of this paper wish to express their gratitude for the useful comments of the referees which contributed to the improvement of this manuscript. Furthermore, special thanks are due to Professor N. Arimitsu for his fruitful suggestions on the theoretical explanation of the experimental results.

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