From Quantum Many-Body Systems to Ideal Fluids

Abstract

We give a rigorous, quantitative derivation of the incompressible Euler equation from the many-body problem for N bosons on T d with binary Coulomb interactions in the semiclassical regime. The coupling constant of the repulsive interaction potential is 1/(ε 2 N ), where ε ≪ 1 and N ≫ 1, so that by choosing ε = N −λ , for appropriate λ > 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 → ∞ and → 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 ε on N is essentially optimal, while in dimension 3, heuristic considerations suggest our scaling is optimal. To the best of our knowledge, our result is a new connection between quantum many-body systems and ideal hydrodynamics, complementing the previously known connection to compressible fluids. 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 [GP21] on the derivation of the pressureless Euler-Poisson equation in the classical and mean-field limits and of Han-Kwan and Iacobelli [HKI21] and the author [Ros21] 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 Schrödinger-Poisson equation in the limit as + ε → 0, corresponding to a combined classical and quasineutral limit.

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

“…One of the interesting features of the method of proof of [7] is the observation that Serfaty's inequality (which was originally applied in the context of a classical mean field limit) can be adapted to the semi-classical regime as well. This observation has also been utilized in the recent work [16], which proves a semi-classical combined mean field quasineutral limit, which is a semi-classical version of theorem 1.1 in [10]. Our new contribution-stated precisely in theorem 2.7 of the next section-is a derivation of the incompressible Euler equation (1.5) from the von Neumann equation (1.6), thereby complementing both of the abovementioned works [7], [10].…”

“…This fact is essentially already contained in the literature (see e.g. [16], section 3.4), and is included mainly for the sake of clarity of exposition. In the expense of a bit more technique, one can show that in fact ρ N :1,ǫ, → ω in some appropriate negative Sobolev space -see [16], section 3.4 for a guidance on how this is to be done.…”

“…[16], section 3.4), and is included mainly for the sake of clarity of exposition. In the expense of a bit more technique, one can show that in fact ρ N :1,ǫ, → ω in some appropriate negative Sobolev space -see [16], section 3.4 for a guidance on how this is to be done. A key ingredient to establish this weak convergence is the following coercivity estimate for the modulated energy There are constants C = C(α) > 0, λ = λ(α) > 0 such that for any ϕ ∈ C 0,α (R 2 ), any probability density µ ∈ L ∞ (R 2 ) and any X N with ∀i = j :…”

“…The calculations which lead to this estimate are more tedious in comparison to the classical setting, partially due to the fact that quantization gives rise to commutators of a differential operator with a multiplication operator-which contributes non zero terms of course. As in [7], [10], [16] this estimate heavily relies on Serfaty's inequality. Section 4 aims to explain how weak convergence is implied from this estimate-which is a simple consequence of a different (yet intimately related) inequality of Serfaty.…”

The derivation of the compressible Euler equation from quantum many-body dynamics

“…As the first breakthrough, Golse and Paul [33], with the help of Serfaty's inequality [54,Corollary 3.4], used the modulated energy method in the quantum N-body setting to justify the validity of the joint mean-field and classical limit of the quantum N-body dynamics leading to the pressureless Euler-Poisson with repulsive Coulomb potential. Subsequently, Rosenzweig complemented [33] in [52] by combining meanfield, semiclassical and quasi-neutral limits to reach a derivation of an incompressible Euler equation on T d with binary Coulomb interactions.…”