Mean field limit for bosons and propagation of Wigner measures

TL;DR

This paper establishes that Wigner measures associated with bosonic quantum states evolve under nonlinear Hartree flow in the mean field limit, extending previous results to more general classes of states.

Contribution

It proves the transport of Wigner measures by nonlinear flows for a broad class of quantum states in the mean field limit of bosonic systems.

Findings

Wigner measures are transported by nonlinear Hartree flow.

The result applies to a more general class of quantum states.

The work extends previous infinite-dimensional analysis results.

Abstract

We consider the N-body Schr\"{o}dinger dynamics of bosons in the mean field limit with a bounded pair-interaction potential. According to the previous work \cite{AmNi}, the mean field limit is translated into a semiclassical problem with a small parameter $\epsilon\to 0$, after introducing an $\epsilon$-dependent bosonic quantization. The limit is expressed as a push-forward by a nonlinear flow (e.g. Hartree) of the associated Wigner measures. These object and their basic properties were introduced in \cite{AmNi} in the infinite dimensional setting. The additional result presented here states that the transport by the nonlinear flow holds for rather general class of quantum states in their mean field limit.