From Fisher information decay for the Kac model to the Landau-Coulomb hierarchy

TL;DR

This paper demonstrates that Fisher information decreases over time in the Kac model for Coulomb interactions, enabling the derivation of weak solutions for the Landau hierarchy.

Contribution

It establishes Fisher information decay for the Kac model with Coulomb potential, linking it to the Landau hierarchy and providing a new analytical tool.

Findings

Fisher information is monotonically decreasing in the model

Monotonicity leads to compactness and weak solutions

Connects particle system behavior to kinetic hierarchy

Abstract

We consider the Kac model for the space-homogeneous Landau equation with the Coulomb potential. We show that the Fisher information of the Liouville equation for the unmodified $N$-particle system is monotonically decreasing in time. The monotonicity ensures the compactness to derive a weak solution of the Landau hierarchy.