Fisher Information in Kinetic Theory

TL;DR

This paper reviews Fisher information in kinetic theory, generalizes a recent monotonicity theorem, and proves decay of Fisher information for the Boltzmann equation, solving a longstanding regularity problem for soft potentials.

Contribution

It introduces a generalized perspective on Fisher information in kinetic theory and proves its decay for the Boltzmann equation with singular collision kernels.

Findings

Fisher information decays along the spatially homogeneous Boltzmann equation.

The decay result applies to all relevant interactions, including very soft potentials.

This work resolves a longstanding regularity problem for singular collision kernels.

Abstract

These notes review the theory of Fisher information, especially its use in kinetic theory of gases and plasmas. The recent monotonicity theorem by Guillen--Silvestre for the Landau--Coulomb equation is put in perspective and generalised. Following my joint work with Imbert and Silvestre, it is proven that Fisher information is decaying along the spatially homogeneous Boltzmann equation, for all relevant interactions, and from this the once longstanding problem of regularity estimates for very singular collision kernels (very soft potentials) is solved.