Around the Quantum Lenard-Balescu equation

TL;DR

This paper justifies the quantum Lenard-Balescu equation's validity over a logarithmic timescale and demonstrates its convergence to the classical Landau equation in the semi-classical limit for dimensions two and higher.

Contribution

It provides a rigorous justification of the quantum Lenard-Balescu equation's validity time and establishes its convergence to the classical Landau equation in the semi-classical limit.

Findings

Justifies the quantum Lenard-Balescu equation up to logarithmic timescales.

Proves convergence of quantum to classical solutions in the semi-classical limit.

Connects quantum kinetic equations to classical counterparts via grazing collision limit.

Abstract

In the mean-field regime, a gas of quantum particles with Boltzmann statistics can be described by the Hartree-Fock equation. This dynamics becomes trivial if the initial distribution of particle is invariant by translation. However, the first correction is given on time of order $O(N)$ by the quantum Lenard--Balescu equation. In the first part of the present article, we justify this equation until time of order $O((\log N)^{1-\delta})$ (for any $\delta\in(0,1)$). A similar phenomenon exists in the classical setting (with a similar validity time obtained by Duerinckx \cite{Duerinckx}). In a second time, we prove the convergence for dimension $d\geq 2$ of the solutions of the quantum Lenard--Balescu equation to the solutions of its classical counterpart in the semi-classical limit. This problem can be interpreted as a grazing collision limit: the quantum Lenard--Balescu equation looks like a cut-off Boltzmann equation, when the classical one looks like the Landau equation.