Derivation of the compressible Euler equations from the dynamics of interacting Bose gas in the hard-core limit regime
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arXiv
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| Natura: | Preprint |
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2024
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| _version_ | 1866916644834508800 |
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| author | Chong, Jacky Shen, Shunlin Zhang, Zhifei |
| author_facet | Chong, Jacky Shen, Shunlin Zhang, Zhifei |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_14812 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Derivation of the compressible Euler equations from the dynamics of interacting Bose gas in the hard-core limit regime Chong, Jacky Shen, Shunlin Zhang, Zhifei Analysis of PDEs Mathematical Physics 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. |
| title | Derivation of the compressible Euler equations from the dynamics of interacting Bose gas in the hard-core limit regime |
| topic | Analysis of PDEs Mathematical Physics |
| url | https://arxiv.org/abs/2409.14812 |