The 'wavelet' entropic index q of non-extensive statistical mechanics and superstatistics

Abstract Generalized entropies developed for non-extensive statistical mechanics are derived from the Boltzmann-Gibbs-Shannon entropy by a real number q that is a parameter based on q-calculus; where q is called ‘the entropic index’ and determines the degree of non-extensivity of a system in the interval between 1 and 3. In a very recent study, we introduced a new calculation method of the entropic index q of non-extensive statistical mechanics. In this study, we show the mathematical proof of this calculation method of the entropic index. Firstly, we propose that the number of degrees of freedom, n is proportional to the inverse of the wavelet scale index, n ≡ 1 i s c a l e , where i s c a l e is a wavelet based parameter called wavelet scale index that quantitatively measures the non-periodicity of a signal in the interval between 0 and 1. Then, by applying this proposition to the superstatistics approach, we derive the equation that expresses the relationship between the entropic index and the wavelet scale index, q = 1 + 2 i s c a l e . Therefore, we name this q -index as the ‘wavelet’ entropic index. Lastly, we calculate the Abe entropy, Landsberg-Vedral entropy and q-dualities of the Tsallis entropy of the Logistic Map and Hennon Map using the ‘wavelet’ entropic index, and based on our results, compare and discuss these generalized entropies.

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