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 with binary Coulomb interactions in the semiclassical regime. The coupling constant of the repulsive interaction potential is 1/(εN), where ε ≪ 1 and N ≫ 1, so that by choosing ε = N, 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.