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

Jacky Chong, Shunlin Shen, Zhifei Zhang

TL;DR

This work rigorously derives macroscopic fluid limits for interacting Bose gases by combining a quantum many-body Bogoliubov analysis with semiclassical scalings. It shows that, in the GP and hard-core regimes, local mass, momentum, and energy densities converge strongly to the compressible Euler system up to the first blow-up time, with the hard-core case yielding an internal energy form tied to the kinetic density and a pressure P=2πc0ρ^2 controlled by the electrostatic capacity c0 of the interaction. The coupling constant c0 arises as the capacity of the potential, linking scattering theory to hydrodynamic behavior, and the analysis clarifies how the limiting dynamics reduce to eikonal equations in beyond-GP regimes. The paper also develops a comprehensive Bogoliubov-corrected mean-field framework, providing precise fluctuation bounds and quantifying convergence rates in terms of N and the semiclassical parameter ε, with N large and ε→0. Overall, it offers a first-principles, multi-regime justification for using compressible Euler and geometric optics descriptions in ultracold Bose gases and connects microscopic scattering data to macroscopic fluid properties, including a novel internal-energy structure in the hard-core limit.

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 $4π\mathfrak{c}_{0}ρ^{2}$ (equivalently, pressure $P=2π\mathfrak{c}_{0}ρ^{2}$), 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.

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

TL;DR

This work rigorously derives macroscopic fluid limits for interacting Bose gases by combining a quantum many-body Bogoliubov analysis with semiclassical scalings. It shows that, in the GP and hard-core regimes, local mass, momentum, and energy densities converge strongly to the compressible Euler system up to the first blow-up time, with the hard-core case yielding an internal energy form tied to the kinetic density and a pressure P=2πc0ρ^2 controlled by the electrostatic capacity c0 of the interaction. The coupling constant c0 arises as the capacity of the potential, linking scattering theory to hydrodynamic behavior, and the analysis clarifies how the limiting dynamics reduce to eikonal equations in beyond-GP regimes. The paper also develops a comprehensive Bogoliubov-corrected mean-field framework, providing precise fluctuation bounds and quantifying convergence rates in terms of N and the semiclassical parameter ε, with N large and ε→0. Overall, it offers a first-principles, multi-regime justification for using compressible Euler and geometric optics descriptions in ultracold Bose gases and connects microscopic scattering data to macroscopic fluid properties, including a novel internal-energy structure in the hard-core limit.

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 (equivalently, pressure ), 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 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.
Paper Structure (53 sections, 44 theorems, 407 equations, 1 figure)

This paper contains 53 sections, 44 theorems, 407 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Scalings and models
  3. One-particle conservation laws
  4. The compressible Euler equations and eikonal system
  5. Physical models and motivations
  6. Hard-core limit regime
  7. The hyper-ultradilute limit regime: $\beta>1$ Case
  8. Beyond Gross--Pitaevskii regime
  9. Bosonic Fock space
  10. Literature overview
  11. Effective description of interacting Bose gas
  12. Semiclassical limit of the Gross--Pitaevskii equation
  13. Many-body dynamics to Euler-type equations
  14. Main results
  15. Outline, novelties, and notations: a guide to the paper
  16. ...and 38 more sections

Key Result

Theorem 1.1

Fix $\beta=\kappa=1$ and $\lambda(N)=(\ln N)^\upalpha$ for $\upalpha \in [0, 1)$ fixed. Let $v$ be a bounded nonnegative radially symmetric function supported in $\overline{B_{R_0}(0)}=\overline{\{\left\lvert x \right\rvert< R_0\}}$ and is strictly positive in $B_{R_0}(0)$. Suppose $\Psi_{N, t}$ is Let $(\rho^\mathrm{in}, \boldsymbol{\mathbf{u}}^\mathrm{in})\in H^5(\mathbb R^3)\times H^4(\mathbb

Figures (1)

  • Figure 1:

Theorems & Definitions (83)

  • Theorem 1.1: GP/HC limit regime
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Theorem 1.2: BGP/SGP limit regimes
  • Remark 1.7
  • Proposition 2.1
  • ...and 73 more