Derivation of the Euler equations from many-body quantum mechanics

TL;DR

This paper rigorously derives the compressible Euler equations from many-body quantum mechanics for fermions, using a quantum entropy method without intermediate classical approximations, under certain technical conditions.

Contribution

It provides a novel, direct derivation of classical fluid dynamics equations from quantum mechanics, avoiding semi-classical or kinetic intermediate models.

Findings

Convergence of quantum dynamics to Euler equations in the hydrodynamic limit

Use of a quantum entropy method and virial theorem in the derivation

Explicit expression for pressure from quantum statistical mechanics

Abstract

The Heisenberg dynamics of the energy, momentum, and particle densities for fermions with short-range pair interactions is shown to converge to the compressible Euler equations in the hydrodynamic limit. The pressure function is given by the standard formula from quantum statistical mechanics with the two-body potential under consideration. Our derivation is based on a quantum version of the entropy method and a suitable quantum virial theorem. No intermediate description, such as a Boltzmann equation or semi-classical approximation, is used in our proof. We require some technical conditions on the dynamics, which can be considered as interesting open problems in their own right.