Derivation of Euler's equations of perfect fluids from von Neumann's equation with magnetic field

TL;DR

This paper rigorously derives the 2D incompressible Euler equations from the von Neumann equation with a magnetic field, establishing weak convergence through a novel energy functional approach.

Contribution

It provides the first rigorous derivation of classical fluid dynamics equations from quantum von Neumann equations with magnetic fields, using modulated energy methods.

Findings

Weak convergence of quantum dynamics to classical Euler equations

Use of modulated energy functional for convergence analysis

Application of recent functional inequalities in the proof

Abstract

We give a rigorous derivation of the incompressible 2D Euler equation from the von Neumann equation with magnetic field. The convergence is with respect to the modulated energy functional, and implies weak convergence in the sense of measures. This is the quantum counterpart of theorem 1.2 in [Key: 10]. Our proof is based on a Gronwall estimate for the modulated energy functional, which in turn heavily relies on a recent functional inequality due to [Key: 20].