From the quantum Boltzmann operator to the quantum Landau operator

TL;DR

This paper derives the quantum Landau operator as a weak-coupling limit of the quantum Boltzmann operator, incorporating quantum statistics and providing explicit convergence rates.

Contribution

It introduces a novel two-parameter scaling to capture quantum effects and derives the quantum Landau operator from the quantum Boltzmann operator for Fermi-Dirac and Bose-Einstein statistics.

Findings

Derived the quantum Landau operator from the quantum Boltzmann operator.

Introduced a new scaling preserving quantum effects in the limit.

Provided explicit convergence rates depending on potential regularity.

Abstract

In this manuscript we derive the quantum Landau operator as the weak-coupling limit of the quantum Boltzmann operator (also known as the Uehling-Uhlenbeck operator). We consider both Fermi-Dirac and Bose-Einstein statistics. Our approach is inspired by the work by Benedetto and Pulvirenti, where the classical Landau operator was derived from the quantum Boltzmann operator. To capture the ternary term in the quantum Landau operator, we introduce a new two-parameter scaling that preserves the quantum effects in the limit. Furthermore, we provide an explicit rate of convergence that depends on the regularity of the interaction potential.