On the asymptotic behavior at the kinetic time of a weakly interacting Fermi gas

TL;DR

This paper justifies the quantum Boltzmann equation for a weakly interacting Fermi gas at kinetic times by analyzing the two-point correlation function near equilibrium.

Contribution

It proves that the leading order behavior of the correlation function is governed by the Boltzmann-Nordheim collision operator, confirming a prior prediction.

Findings

Correlation function's leading order determined by collision frequency

Justification of quantum Boltzmann equation from many-body quantum mechanics

Analysis valid for initial states close to equilibrium

Abstract

This paper is devoted to the dynamics of a weakly interacting Fermi gas at the kinetic time regime $t\sim \lambda^{-2}$ where $\lambda \ll 1$ is the strength of the interaction potential. We prove that if the initial state is close to equilibrium, then the two-point time correlation function of the many-body quantum dynamics can be computed effectively. In fact, we show that its leading order behavior is determined completely by the collisional frequency of the Boltzmann-Nordheim collision operator at equilibrium. This settles a prediction by Lukkarinen-Spohn, and thus gives a justification of the quantum Boltzmann equation from many-body quantum mechanics.