From Quantum Many-Body Systems to Ideal Fluids

TL;DR

This paper rigorously derives the incompressible Euler equation from a many-body quantum bosonic system with Coulomb interactions in a supercritical scaling regime, connecting quantum many-body dynamics to classical fluid equations.

Contribution

It provides a novel, rigorous derivation of the incompressible Euler equation from quantum many-body systems under supercritical scaling, extending previous mean-field results.

Findings

Density converges to uniform density 1

Current converges to Euler solution u

Optimal scaling in 2D and heuristic in 3D

Abstract

We give a rigorous, quantitative derivation of the incompressible Euler equation from the many-body problem for $N$ bosons on $\mathbb{T}^d$ with binary Coulomb interactions in the semiclassical regime. The coupling constant of the repulsive interaction potential is $~1/(\varepsilon^2 N)$, where $\varepsilon \ll 1$ and $N\gg 1$, so that by choosing $\varepsilon=N^{-\lambda}$, for appropriate $\lambda>0$, the scaling is supercritical with respect to the usual mean-field regime. For approximately monokinetic initial states with nearly uniform density, we show that the density of the first marginal converges to 1 as $N\rightarrow\infty$ and $\hbar\rightarrow 0$, while the current of the first marginal converges to a solution $u$ of the incompressible Euler equation on an interval for which the equation admits a classical solution. In dimension 2, the dependence of $\varepsilon$ on $N$ is essentially optimal, while in dimension 3, heuristic considerations suggest our scaling is optimal. Our proof is based on a Gronwall relation for a quantum modulated energy with an appropriate corrector and is inspired by recent work of Golse and Paul arXiv:1912.06750 on the derivation of the pressureless Euler-Poisson equation in the classical and mean-field limits and of Han-Kwan and Iacobelli arXiv:2006.14924 and the author arXiv:2104.11723 on the derivation of the incompressible Euler equation from Newton's second law in the supercritical mean-field limit. As a byproduct of our analysis, we also derive the incompressible Euler equation from the Schr\"odinger-Poisson equation in the limit as $\hbar+\varepsilon\rightarrow 0$, corresponding to a combined classical and quasineutral limit.