On the Fermi-Dirac-type Fisher information

TL;DR

This paper introduces a generalized Fisher information for Fermi-Dirac-like kinetic models, demonstrating its decreasing behavior under certain initial conditions and exploring its evolution in related equations.

Contribution

It develops a new Fermi-Dirac-type Fisher information and analyzes its properties within kinetic equations, extending the understanding of entropy dissipation in quantum statistical models.

Findings

Fisher information decreases over time with suitable initial bounds

Monotonicity can fail without initial bounds

Evolution analyzed for heat and Landau-Fermi-Dirac equations

Abstract

We consider kinetic models for Fermi-Dirac-like particles obeying the exclusion principle. A generalized notion of Fisher information, tailored to kinetic equations of Fermi-Dirac-Fokker-Planck type, is introduced via the associated entropy dissipation identity. We show that, subject to a suitable upper bound on the initial data, this quantity decreases along solutions of the Fermi-Dirac-Fokker-Planck equation, while monotonicity can fail in the absence of such a bound. We also discuss the time evolution of this Fermi-Dirac-type Fisher information for the heat equation and the linear-type Landau-Fermi-Dirac equation with Maxwell molecules.