Some Considerations on the derivation of the nonlinear Quantum Boltzmann Equation

TL;DR

This paper rigorously derives a nonlinear quantum Boltzmann equation for a system of weakly interacting particles, demonstrating convergence of a perturbative series to the equation's solution and recovering Fermi's Golden Rule transition rates.

Contribution

It provides a rigorous derivation of the quantum Boltzmann equation from first principles in any dimension greater than 2, including convergence results and physical rate recovery.

Findings

Convergence of a perturbative series to the Boltzmann equation solution.

Recovery of Fermi's Golden Rule transition rates.

Applicability in space dimensions greater than 2.

Abstract

In this paper we analyze a system of N identical quantum particles in a weak-coupling regime. The time evolution of the Wigner transform of the one-particle reduced density matrix is represented by means of a perturbative series. The expansion is obtained upon iterating the Duhamel formula. For short times, we rigorously prove that a subseries of the latter, converges to the solution of the Boltzmann equation which is physically relevant in the context. In particular, we recover the transition rate as it is predicted by Fermi's Golden Rule. However, we are not able to prove that the quantity neglected while retaining a subseries of the complete original perturbative expansion, indeed vanishes in the limit: we only give plausibility arguments in this direction. The present study holds in any space dimension greater than 2.