The Compressible Euler and Acoustic Limits from quantum Boltzmann Equation with Fermi-Dirac Statistics

TL;DR

This paper rigorously derives the compressible Euler and acoustic limits from the quantum Boltzmann equation with Fermi-Dirac statistics using the Hilbert approach, highlighting differences from classical forms and introducing new nonlinear estimates.

Contribution

It provides a rigorous proof of the quantum to classical limit transition for the quantum Boltzmann equation with Fermi-Dirac statistics, employing novel nonlinear estimates and analysis of implicit transformations.

Findings

Rigorous derivation of Euler and acoustic limits from quantum Boltzmann equation

Identification of differences between classical and quantum derived Euler systems

Development of new nonlinear estimates for the analysis

Abstract

This paper justifies the compressible Euler and acoustic limits from quantum Boltzmann equation with Fermi-Dirac statistics (briefly, BFD) rigorously. This limit was formally derived in Zakrevskiy's thesis \cite{Zakrevskiy} by moment method. We employ the Hilbert approach. The forms of the classical compressible Euler system and the one derived from BFD are different. Our proof is based on the analysis of the nonlinear implicit transformation of these two forms, and a few novel nonlinear estimates.