Quantized collision invariants on the sphere
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866911978472079360 |
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| author | Anwasia, Benjamin Arsénio, Diogo |
| author_facet | Anwasia, Benjamin Arsénio, Diogo |
| contents | We show that a measurable function $g:\mathbb{S}^{d-1}\to\mathbb{R}$, with $d\geq 3$, satisfies the functional relation \begin{equation*} g(ω)+g(ω_*)=g(ω')+g(ω_*'), \end{equation*} for all admissible $ω,ω_*,ω',ω_*'\in\mathbb{S}^{d-1}$ in the sense that \begin{equation*} ω+ω_*=ω'+ω_*', \end{equation*} if and only if it can be written as \begin{equation*} g(ω)=A+B\cdotω, \end{equation*} for some constants $A\in \mathbb{R}$ and $B\in\mathbb{R}^d$.
Such functions form a family of quantized collision invariants which play a fundamental role in the study of hydrodynamic regimes of the Boltzmann--Fermi--Dirac equation near Fermionic condensates, i.e., at low temperatures. In particular, they characterize the elastic collisional dynamics of Fermions near a statistical equilibrium where quantum effects are predominant. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_00433 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Quantized collision invariants on the sphere Anwasia, Benjamin Arsénio, Diogo Classical Analysis and ODEs Analysis of PDEs We show that a measurable function $g:\mathbb{S}^{d-1}\to\mathbb{R}$, with $d\geq 3$, satisfies the functional relation \begin{equation*} g(ω)+g(ω_*)=g(ω')+g(ω_*'), \end{equation*} for all admissible $ω,ω_*,ω',ω_*'\in\mathbb{S}^{d-1}$ in the sense that \begin{equation*} ω+ω_*=ω'+ω_*', \end{equation*} if and only if it can be written as \begin{equation*} g(ω)=A+B\cdotω, \end{equation*} for some constants $A\in \mathbb{R}$ and $B\in\mathbb{R}^d$. Such functions form a family of quantized collision invariants which play a fundamental role in the study of hydrodynamic regimes of the Boltzmann--Fermi--Dirac equation near Fermionic condensates, i.e., at low temperatures. In particular, they characterize the elastic collisional dynamics of Fermions near a statistical equilibrium where quantum effects are predominant. |
| title | Quantized collision invariants on the sphere |
| topic | Classical Analysis and ODEs Analysis of PDEs |
| url | https://arxiv.org/abs/2401.00433 |