On the Weak Coupling Limit of Quantum Many-body Dynamics and the Quantum Boltzmann Equation

TL;DR

This paper investigates the weak coupling limit of quantum many-particle systems and shows that under certain regularity assumptions, the limit satisfies the quantum Maxwell-Boltzmann hierarchy rather than the Uehling-Uhlenbeck equation, highlighting the need for different regularity conditions.

Contribution

It rigorously demonstrates that the weak coupling limit leads to the Maxwell-Boltzmann hierarchy instead of the Uehling-Uhlenbeck hierarchy under specific regularity assumptions.

Findings

Weak coupling limit converges to Maxwell-Boltzmann hierarchy.

Regularity assumptions determine the limiting hierarchy.

Uehling-Uhlenbeck equation requires lower regularity bounds.

Abstract

The rigorous derivation of the Uehling-Uhlenbeck equation from more fundamental quantum many-particle systems is a challenging open problem in mathematics. In this paper, we exam the weak coupling limit of quantum N-particle dynamics. We assume the integral of the microscopic interaction is zero and we assume W^{4,1} per-particle regularity on the coressponding BBGKY sequence so that we can rigorously commute limits and integrals. We prove that, if the BBGKY sequence does converge in some weak sense, then this weak-coupling limit must satisfy the infinite quantum Maxwell-Boltzmann hierarchy instead of the expected infinite Uehling-Uhlenbeck hierarchy, regardless of the statistics the particles obey. Our result indicates that, in order to derive the Uehling-Uhlenbeck equation, one must work with per-particle regularity bound below W^{4,1}.