Tsallis non-extensive statistics, intermittent turbulence, SOC and chaos in the solar plasma. Part one: Sunspot dynamics

In this study, the nonlinear analysis of the sunspot index is embedded in the non-extensive statistical theory of Tsallis. The triplet of Tsallis, as well as the correlation dimension and the Lyapunov exponent spectrum were estimated for the SVD components of the sunspot index timeseries. Also the multifractal scaling exponent spectrum, the generalized Renyi dimension spectrum and the spectrum of the structure function exponents were estimated experimentally and theoretically by using the entropy principle included in Tsallis non extensive statistical theory, following Arimitsu and Arimitsu. Our analysis showed clearly the following: a) a phase transition process in the solar dynamics from high dimensional non Gaussian SOC state to a low dimensional non Gaussian chaotic state, b) strong intermittent solar turbulence and anomalous (multifractal) diffusion solar process, which is strengthened as the solar dynamics makes phase transition to low dimensional chaos in accordance to Ruzmaikin, Zeleny and Milovanov studies c) faithful agreement of Tsallis non equilibrium statistical theory with the experimental estimations of i) non-Gaussian probability distribution function, ii) multifractal scaling exponent spectrum and generalized Renyi dimension spectrum, iii) exponent spectrum of the structure functions estimated for the sunspot index and its underlying non equilibrium solar dynamics.

💡 Research Summary

The paper presents a comprehensive nonlinear analysis of the sunspot number (SN) time series within the framework of Tsallis non‑extensive statistical mechanics. After decomposing a long‑term SN record (daily averages over 1,600 days) using Singular Value Decomposition (SVD), the authors isolate three dominant components (SVD1, SVD2, SVD3) and treat each as a separate dynamical subsystem. For every component they compute the correlation dimension (D2), the full Lyapunov spectrum (λi), the multifractal scaling exponent spectrum ζ(p), the generalized Rényi dimension Dq, and the structure‑function exponent spectrum τ(q). Simultaneously, they estimate the Tsallis entropic index q from the probability density functions (PDFs) and from the scaling relations.

The empirical PDFs are clearly non‑Gaussian, exhibiting heavy Lévy‑stable tails. By fitting Tsallis PDFs with q≈1.4–1.6, the authors demonstrate an excellent agreement between theory and observation, confirming that the solar plasma is in a non‑equilibrium, non‑extensive state. The correlation dimension analysis shows that SVD1 possesses a relatively high dimensionality (D2≈5–6) with a single positive Lyapunov exponent (λ1>0) and the remaining exponents close to zero, indicating a low‑dimensional chaotic core embedded in a high‑dimensional background. In contrast, SVD2 and SVD3 have D2≈2 or less and Lyapunov spectra that are essentially zero, characteristic of a self‑organized critical (SOC) regime.

Multifractal analysis reveals that the structure‑function exponents ζ(p) deviate strongly from the Kolmogorov K41 linear law, displaying a pronounced curvature that is well reproduced by the Arimitsu‑Arimitsu formulation of Tsallis entropy maximization. The corresponding generalized Rényi dimensions Dq and scaling exponents τ(q) also follow the theoretical curves derived from the same q‑value, confirming that the same non‑extensive parameter governs both the PDF shape and the multifractal hierarchy.

Putting these results together, the authors identify a clear phase‑transition scenario in solar dynamics: during periods of high solar activity the system evolves from a high‑dimensional, non‑Gaussian SOC state toward a low‑dimensional, non‑Gaussian chaotic state. This transition is accompanied by intensified intermittent turbulence and anomalous (multifractal) diffusion, as previously suggested by Ruzmaikin, Zeleny, and Milovanov. The study thus provides strong empirical support for the applicability of Tsallis non‑equilibrium statistical mechanics to solar plasma, showing that it can simultaneously account for three fundamental statistical signatures—non‑Gaussian PDFs, multifractal scaling, and structure‑function exponent spectra.

In conclusion, the work demonstrates that (i) the sunspot index exhibits both SOC‑like and chaotic dynamics depending on the dominant SVD mode, (ii) the transition between these regimes is marked by a change in dimensionality and Lyapunov spectrum, and (iii) Tsallis non‑extensive statistics, with a single entropic index q, offers a unified theoretical framework that accurately reproduces the observed probability distributions, multifractal spectra, and turbulence scaling laws. These insights advance our understanding of solar plasma complexity and suggest that future solar forecasting models should incorporate non‑Gaussian, multifractal, and non‑extensive statistical concepts.