On semi-classical limit of spatially homogeneous quantum Boltzmann equation: weak convergence

TL;DR

This paper provides a mathematical justification for the semi-classical limit of the spatially homogeneous quantum Boltzmann equation, showing convergence to the Fokker-Planck-Landau equation as the Planck constant approaches zero.

Contribution

It introduces a new framework to analyze weak convergence using the weak projection gradient and explores the symmetric structure of collision operators.

Findings

Established weak convergence of quantum Boltzmann to Fokker-Planck-Landau equation

Developed a novel analytical framework involving weak projection gradients

Identified symmetry properties in collision operator cubic terms

Abstract

It is expected in physics that the homogeneous quantum Boltzmann equation with Fermi-Dirac or Bose-Einstein statistics and with Maxwell-Boltzmann operator (neglecting effect of the statistics) for the weak coupled gases will converge to the homogeneous Fokker-Planck-Landau equation as the Planck constant $\hbar$ tends to zero. In this paper and the upcoming work \cite{HLP2}, we will provide a mathematical justification on this semi-classical limit. Key ingredients into the proofs are the new framework to catch the {\it weak projection gradient}, which is motivated by Villani \cite{V1} to identify the $H$-solution for Fokker-Planck-Landau equation, and the symmetric structure inside the cubic terms of the collision operators.