Derivation of the compressible Euler equations from the dynamics of interacting Bose gas in the hard-core limit regime

TL;DR

This paper rigorously derives the compressible Euler equations from the many-body dynamics of interacting Bose gases, revealing new insights into the energy structure and coupling constants in the hard-core limit regime.

Contribution

It provides the first rigorous derivation of the Euler equations from Bose gas dynamics in the hard-core limit, highlighting the kinetic energy's role in internal energy and identifying the coupling constant as the electrostatic capacity.

Findings

Internal energy in the hard-core regime is proportional to the square of density.

The coupling constant equals the electrostatic capacity of the interaction potential.

The limiting dynamics can be described by an eikonal system in certain regimes.

Abstract

We investigate the dynamics of short-range interacting Bose gases with varying degrees of diluteness and interaction strength. By applying a combined mean-field and semiclassical space-time rescaling to the dynamics in both the Gross--Pitaevskii and hard-core limit regimes, we prove that the local one-particle mass, momentum, and energy densities of the many-body system can be quantitatively approximated by solutions to the compressible Euler system in the strong sense, up to the first blow-up time of the fluid description, as the number of particles tends to infinity. In the hard-core limit regime, two novel results are presented. First, we rigorously prove, for the first time, that the internal energy of the fluid takes the form $4\pi \mathfrak{c}_{0}\rho^{2}$ (equivalently, pressure $P=2\pi \mathfrak{c}_{0}\rho^{2}$), arising solely from the kinetic energy density of the many-body system, rather than the interaction energy density, marking a fundamental difference from the Gross--Pitaevskii and other mean-field regimes. Second, the newly discovered coupling constant $\mathfrak{c}_{0}$ is the electrostatic capacity of the interaction potential, corresponding to the scattering length of the hard-core potential. Furthermore, in other limiting regimes, including those beyond the Gross--Pitaevskii regime, we find that the limiting equation is described by an eikonal system, offering a rigorous first-principle justification for using the ``geometric optics approximation'' to describe the dynamics of ultracold Bose gases.