Kinetic equations through the lens of Fisher information: Landau, multi-species, and Fermi-Dirac variants

We consider the evolution of a dilute particle system in gases and plasmas, modelled by the homogeneous Landau equation. We focus on the monotonicity of the Fisher information, a functional that helps overcome the supercriticality arising from singular interactions. Since McKean brought Fisher information into kinetic equations, it has now been established by Guillen-Silvestre and Imbert-Silvestre-Villani that the Fisher information is monotone along Landau and Boltzmann flows, by refining the Bakry-Émery geometric criterion on the real projective space. Pushing this viewpoint further, we will discuss two extensions: multi-species systems, where natural symmetries break down, and the Fermi-Dirac quantum variant, where the Pauli exclusion principle modifies the entropy and the associated Fisher information.