A $q$-Caputo Fractional Generalization of Tsallis Entropy

Abstract

In this work we build a $q$-Caputo fractional generalization of the Tsallis entropy functional $S_q$. The construction starts from the already known Jackson derivative representation of $S_q$ and replaces the Jackson derivative with a Caputo type $q$-fractional operator ${}^{C}D_q^α$. The parameter $q$ keeps its standard role as the nonextensive index, while the fractional parameter $α$ gives a nonlocal, memory like contribution in the entropy generating procedure. The resulting functional, denoted by $S_q^α$, is written as a convergent series of logarithms in the probabilities and is defined for $0<α<1$, $q>0$, and $p_i>0$. We show explicitly that the standard Tsallis entropy is recovered in the local limit $α\to 1$. We then start rewriting $S_q^α$ in trace form and analyze its main structural properties. Non-negativity is tested numerically in a binary and ternary probability distributions. These tests show positive values in the respective domains, but they are interpreted just as numerical evidence rather than as a global/formal proof. Concavity is studied through the second derivative of the trace form contribution yielding to a parameter dependent condition. A global statement requires control over the full probability domain. We also discuss Lesche stability and expansibility. Both properties depend on the behavior of the fully resumed functional near its boundary \(p\to 0^+\), because the entropy contains powers of \(\ln p_i\). Finally, we derive the composition rule for statistically independent subsystems. The logarithmic structure produces mixed moments, so the standard Tsallis pseudo-additive law is not directly inherited for \(0<α<1\). As an illustration, we evaluate the proposed entropy in a two-state fractional relaxation process with memory.