Fisher Information in Kinetic Theory

2025/01/01 by Cédric Villani, Villani, Cédric · 5 voices · 7 citations
Mathematics · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #Classical mechanics #Computer science #Econometrics #Fisher information #Kinetic energy #Mathematical economics #Mathematics #Physics #Statistics #math-ph #math.AP

paper · pdf · doi:10.48550/arxiv.2501.00925

published in arXiv (Cornell University) (Cornell University)

openalex publication_date 2025/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Cited by

• On the existence of solutions to the multi-species Landau equation
• Stability of the spatially homogeneous Landau equation in relative entropy and applications to score-based numerical methods
• Deficit functions and the log Sobolev inequality
• Fisher information for solutions of the Boltzmann equation
• The fuzzy Landau equation: global well-posedness and Fisher information
• The Landau equation and Fisher information
• Nonlocal Fisher information: lifting, local limit, and the Blachman-Stam inequality

Discussions

• Looks to be a really nice read; clear that the man enjoys writing! arxiv.org/abs/2501.00925 'Fisher Information in Kinetic Theory' - Cédric Villani [bsky, 54 points, 2 comments]
• Oooh arxiv.org/abs/2501.00925 [bsky, 7 points, 0 comments]
• For those interested, it happened on January 1st, there: arxiv.org/pdf/2501.00925 [bsky, 1 points, 0 comments]
• Fisher Information in Kinetic Theory [hn, 1 points, 0 comments]
• "on this occasion I consider it fair to thank the french politicians … who ensured, willingly or unwillingly, my defeat at the 2022 French Parliamentary elections, even though by a statistically nonsi [bsky, 0 points, 0 comments]

Related

• On the monotonicity of the Fisher information for the Boltzmann equation
• Fisher information for solutions of the Boltzmann equation
• Uniform estimates on the Fisher information for solutions to Boltzmann and Landau equations
• The Landau equation and Fisher information
• Dissipation estimates of the Fisher information for the Landau equation