Some properties of generalized Fisher information in the context of nonextensive thermostatistics - Archive ouverte HAL Accéder directement au contenu
Article Dans Une Revue Physica A: Statistical Mechanics and its Applications Année : 2013

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

Résumé

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.
Fichier principal
Vignette du fichier
VariationalGenFisher_arxiv.pdf (232.2 Ko) Télécharger le fichier
Origine : Fichiers produits par l'(les) auteur(s)
Loading...

Dates et versions

hal-00766699 , version 1 (18-12-2012)
hal-00766699 , version 2 (22-05-2013)
hal-00766699 , version 3 (04-06-2013)

Identifiants

Citer

Jean-François Bercher. Some properties of generalized Fisher information in the context of nonextensive thermostatistics. Physica A: Statistical Mechanics and its Applications, 2013, 392 (15), pp.3140-3154. ⟨10.1016/j.physa.2013.03.062⟩. ⟨hal-00766699v3⟩
581 Consultations
914 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More