Multi-species kinetic models: GENERIC formulation and Fisher information

TL;DR

The paper addresses how to consistently formulate multi-species Boltzmann and Landau equations, including Bose-Einstein, Maxwell-Boltzmann, and Fermi-Dirac statistics, within the GENERIC framework, ensuring thermodynamic consistency of reversible and irreversible dynamics. It develops explicit GENERIC building blocks for both the inhomogeneous Boltzmann equation and its grazing-limit Landau counterpart, providing gradient-structured representations and entropy-energy relations. A main contribution is proving that the Fisher information for the spatially homogeneous multi-species Boltzmann equation is non-increasing in time under suitable collision kernels and symmetry assumptions, extending single-species decay results to the multispecies setting. The work leverages lifting/doubling techniques and angular-Fisher information inequalities to establish dissipation, with Appendix detailing the grazing limit to the multi-species Landau equation and linking variational structures to the dynamics.

Abstract

In this paper, we study the GENERIC structures of multi-species spatially inhomogeneous Boltzmann and Landau equations with Bose-Einstein, Maxwell-Boltzmann, and Fermi-Dirac statistics. In addition, under suitable assumptions on the collision kernels, we show that the Fisher information for the multi-species spatially homogeneous Boltzmann equation is non-increasing in time.

Theorems & Definitions (5)

• Remark 2.1: Linearised equations
• Theorem 3.1
• proof
• Lemma A.1
• proof : Proof of Lemma \ref{['lem:grazing']}