Quantitative Derivation of the Euler–Poisson Equation from Quantum Many-Body Dynamics

Abstract

We study the three dimensional quantum many-body dynamics with repulsive Coulomb interaction in the mean-field setting. The Euler–Poisson equation is its limit as the particle number tends to infinity and Planck’s constant tends to zero. By a new scheme combining the hierarchy method and the modulated energy method, we establish strong and quantitative microscopic to macroscopic convergence of mass and momentum densities as well as kinetic and potential energies before the 1st blow up time of the limiting Euler–Poisson equation.