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

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 equivalent to 1/(iscale) , where i(scale) 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 + 2i(scale). 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. (C) 2021 Elsevier Ltd. All rights reserved.