Toward a relative q-entropy

Abstract

We address the question and related controversy of the formulation of the $q$-entropy, and its relative entropy counterpart, for models described by continuous (non-discrete) sets of variables. We notice that an $L_p$ normalized functional proposed by Lutwak-Yang-Zhang (LYZ), which is essentially a variation of a properly normalized relative Rényi entropy up to a logarithm, has extremal properties that make it an attractive candidate which can be used to construct such a relative $q$-entropy. We comment on the extremizing probability distributions of this LYZ functional, its relation to the escort distributions, a generalized Fisher information and the corresponding Cramér-Rao inequality. We point out potential physical implications of the LYZ entropic functional and of its extremal distributions.

Minor changes in this version. 23 pages. No figures. LaTeX2e. To be published in Physica A