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

• Published in: Physica A Vol. 392; no. 15; pp. 3140 - 3154
• Main Author: Bercher, J.-F.
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.08.2013; Elsevier
• Subjects: Additives; Complement; Computer Science; Condensed Matter; Derivatives; Generalized [formula omitted]-Gaussian distributions; Generalized Fisher information; Generalized Rényi and Tsallis entropies; Heat equations; Information theoretic inequalities; Information Theory; Mathematical Physics; Mathematics; Minimization; Nonlinearity; Optimization; Physics; Statistical Mechanics; Generalized Rényi and Tsallis entropies; Generalized q-Gaussian distributions; Generalized Fisher information; Information theoretic inequalities
• ISSN: 0378-4371; 1873-2119; 0378-4371
• DOI: 10.1016/j.physa.2013.03.062

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 generalizedq-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≥1. 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. •We present two generalized Fisher information in nonextensive statistics.•Generalized q-Gaussians are characterized via generalized Fisher information.•The interplay between generalized Fisher information, q-Gaussians and q-entropies is described.•Properties of generalized Fisher information and of their minimization are studied.