Some properties of generalized Fisher information in the context of nonextensive thermostatistics

TL;DR

This paper introduces extended forms of Fisher information within nonextensive thermostatistics, revealing their relationship with $q$-Gaussians and $q$-entropies, and establishing new variational and thermodynamic-like properties.

Contribution

It develops generalized Fisher information measures compatible with nonextensive statistics, linking them to $q$-Gaussians, $q$-entropies, and nonlinear heat equations, with new variational characterizations.

Findings

Generalized Fisher information minimized by $q$-Gaussians under constraints.

Derived a generalized de Bruijn identity involving $q$-entropies.

Showed convexity and bounds of the generalized Fisher information.

Abstract

We present two extended forms of Fisher information that fit well in the context of nonextensive thermostatistics. We show that there exists an interplay between these generalized Fisher information, the generalized $q$-Gaussian distributions and the $q$-entropies. The minimum of the generalized Fisher information among distributions with a fixed moment, or with a fixed $q$-entropy is attained, in both cases, by a generalized $q$-Gaussian distribution. This complements the fact that the $q$-Gaussians maximize the $q$-entropies subject to a moment constraint, and yields new variational characterizations of the generalized $q$-Gaussians. We show that the generalized Fisher information naturally pop up in the expression of the time derivative of the $q$-entropies, for distributions satisfying a certain nonlinear heat equation. This result includes as a particular case the classical de Bruijn identity. Then we study further properties of the generalized Fisher information and of their minimization. We show that, though non additive, the generalized Fisher information of a combined system is upper bounded. In the case of mixing, we show that the generalized Fisher information is convex for $q\geq1.$ Finally, we show that the minimization of the generalized Fisher information subject to moment constraints satisfies a Legendre structure analog to the Legendre structure of thermodynamics.