q-Calculus Framework For Entropy In Multifractal Distributions

TL;DR

This paper develops a $q$-calculus framework linking Tsallis entropy with multifractal distributions, introducing $q$-analogues of Shannon and Kullback entropies, and exploring their properties and underlying lattice structures.

Contribution

It establishes a mathematical connection between Tsallis entropy and $q$-calculus, defining new $q$-entropic measures and analyzing their properties.

Findings

Derived a $q$-analogue of Shannon's entropy.

Demonstrated $q$-additivity of the new entropy.

Defined a $q$-analogue of Kullback relative entropy.

Abstract

The connection between Tsallis entropy for a multifractal distribution and Jackson's $q$-derivative is established. Based on this derivation and definition of a homogeneous function, a $q$-analogue of Shannon's entropy is discussed. $q$-additivity of this entropy is shown. We also define $q$-analogue of Kullback relative entropy. The implications of lattice structure beneath $q$-calculus are highlighted in the context of $q$-entropy.