A $q$-Caputo Fractional Generalization of Tsallis Entropy: Series Representation and Non-Negativity Domains

Abstract

We introduce a fractional generalization of Tsallis entropy by acting with a $q$-Caputo operator on the generating family $\sum_i p_i^{\,x}$ evaluated at $x=1$. Concretely, we define $S_{q}^α$ through the $q$-Caputo differintegral of order $0<α<1$ and derive a closed series representation in terms of the $q$-Gamma function. The construction is anchored at the evaluation point, which ensures well-behaved limits: as $α\!\to\!1$ we recover the standard Tsallis entropy $S_q$. Finally we perform a numerical calculation to show the regions where the obtained $q$-fractional entropy $S^α_q$ can be non-negative (or negative) through the fractional parameter $α$ and the non extensive index $q$.

Figures (1)

• Figure 1: Non-negativity domain of the fractional Tsallis entropy $S_{q}^{\alpha}$ for two equiprobable microstates ($W=2$, $p_i=\tfrac{1}{2}$). Shaded (blue) region corresponds to $S_{q}^{\alpha}>0$, grey region to $S_{q}^{\alpha}<0$, and the black curve marks the implicit boundary $S_{q}^{\alpha}=0$. The boundary is obtained by evaluating the exact series definition in Eq. \ref{['Our entropylimited']} with the $q$-Gamma coefficients, sweeping the parameters space $(\alpha,q)$ with $0<\alpha<1$ and $0\le q\le 2$.