Tsallis statistics and magnetospheric self-organization

Abstract

In this study we use Tsallis non-extensive statistics for a new understanding the magnetospheric dynamics and the magnetospheric self-organization during quiet and intensive superstorm periods. The $q_{sens}$, $q_{stat}$, and $q_{rel}$ indices set known as the Tsallis $q$-triplet was estimated during both quiet and strongly active periods, as well as the correlation dimensions and Lyapunov exponents spectrum for magnetospheric bulk plasma flows data. The results obtained by our analysis clearly indicate the magnetospheric phase transition process from a high-dimensional quiet SOC state to a low-dimensional global chaotic state when superstorm events are developed. During such a phase transition process the non-extensive statistical character of the magnetospheric plasma is strengthened as the values of the $q$-triplet indices changes obtaining higher values than their values during the quiet periods.

Introduction

The thermodynamical equilibrium state can be described by Boltzmann–Gibbs (BG) statistics. Far away from thermodynamical equilibrium the Tsallis non-extensive statistics or its generalizations of non-BG statistics can exist [1]. Recently, Pavlos et al. [2] showed the presence of Tsallis non-extensive $q$-statistics at the earth magnetospheric plasma during magnetospheric superstorms. As we know, the magnetospheric plasma system is strongly coupled externally to the solar wind plasma flow revealing also dissipative internal non-equilibrium and non-linear dynamics [3]. Until now the main theme of the magnetospheric dynamics remains the development of magnetospheric superstorms during which strong plasma flows can be developed along the magnetotail. The plasma flow includes also frozen magnetic field structures flowing tailward or earthward [4], [5], [6], [7]. The dynamics of magnetospheric superstorms was explained by Pavlos [8] as the manifestation of magnetospheric chaos and the existence of a magnetospheric strange attractor dynamics. According to our original hypothesis: “$\cdots$ The magnetospheric system is open, as it exchanges mass and energy with the ionosphere and the solar wind, while it remains far from thermodynamic equilibrium. For this reason the magnetosphere belongs to the class of dissipative chaotic systems. An important consequence of chaos theory for these systems is the possibility of existence of strange attractors in phase space. Our failure until now to develop a sufficient local-character description of magnetospheric dynamics, though not extinguishing the hope for future achievements, supports our suggestion that chaos theory may constitute a powerful tool for a global holistic comprehension of magnetospheric dynamics. A central question to be answered through chaos theory is how far the transition of the magnetospheric system from quiet state to the growth phase and subsequently to the explosive phase of superstorms corresponds to a transition from a simple attractor to a chaotic or strange attractor $\cdots$” [8]. However the reductionist and bottom to top point of view was dominating at the space physics community during the late eighties. Our original try to publish theoretical concepts, as well as experimental evidence concerning the far from equilibrium magnetospheric self-organization, concluded by Haken, Nikolis and Prigogine complexity theory, was encountered with refusal. Later strong evidence for the magnetospheric chaos was given by Baker et al. [9], Vassiliadis et al. [10], and Pavlos et al. [11], [12]. Also the analysis by Angelopoulos [5], [6], Angelopoulos [7], Lui [13], [14] and Nagai et al. [15] in situ observations of magnetospheric plasma flows revealed non-Gaussian distributions and intermittent plasma processes. In this direction, Consolini et al. [16](a) concluded the intermittent turbulent magnetospheric state during magnetospheric superstorms. As the classical MHD approach to the magnetospheric dynamics is not always adequate to describe the highly irregular and multiscale magnetospheric dynamics [16](b), the SOC theory was suggested by Chang [17], [18], Chang et al. [19] and Klimas et al. [20] in order explain the intermittent turbulence, the multi-fractal spectra, the sporadic localized reconnections and other non-linear multiscale or non-Gaussian manifestations of the magnetospheric dynamics [17], [18], [19], [20], [21]. An excellent review of modern non-equilibrium statistical theory applied to magnetospheric dynamics is given by Consolini et al. [16](c). The SOC process is a multiscale high-dimensional stochastic and non-chaotic process efficient to produce temporal correlations with long time memory persistence [22], [23], [24]. Also the SOC process is a processes at the edge of chaos [25], [26]. In contrast to the SOC concept as the ruling theoretical paradigm of the magnetospheric dynamics the coexistence of SOC and chaos was supported theoretically and experimentally by Pavlos et al. in a series of studies [11], [12], [27], [28].

In this paper we estimate Tsallis non-extensive statistics for a new understanding of the magnetospheric dynamics. The $q_{sens},q_{stat}$, and $q_{rel}$ indices set known as the Tsallis $q$-triplet was estimated during quiet and strong activity periods as well as correlation dimensions and Lyapunov exponent spectrum for magnetospheric bulk plasma flow data. The results obtained by our analysis indicate clearly the magnetospheric phase transition process from a high-dimensional SOC quiet state to a low-dimensional chaotic state during superstorm events. During the phase transition process the $q$-triplet changes obtaining higher values than the $q$-triplet values during the quiet periods. The strengthening of the intermittent magnetospheric turbulence during superstorms can be concluded also.

The organization of this paper is as follows. In Section 2.2 we present results concerning the $q$-statistics of Tsallis after the determination of the magnetospheric Tsallis $q$-triplet at the quiet and superstorm active magnetosphere. In Section 2.3 we present the estimation of the correlation dimension. The estimation of the Lyapunov exponent spectrum for the quiet and the superstorm active magnetosphere and the results of a null hypothesis testing are presented in Section 24 and 2.5. Finally in section 3 we include a summary of the data analysis and in Section 4 their theoretical interpretation.