Benjamin Anwasia ; Diogo Arsénio - Quantized collision invariants on the sphere

cm:12766 - Communications in Mathematics, April 25, 2024, Volume 32 (2024), Issue 3 (Special issue: Portuguese Mathematics) - https://doi.org/10.46298/cm.12766
Quantized collision invariants on the sphereArticle

Authors: Benjamin Anwasia ; Diogo Arsénio

    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(\omega)+g(\omega_*)=g(\omega')+g(\omega_*'), \end{equation*} for all admissible $\omega,\omega_*,\omega',\omega_*'\in\mathbb{S}^{d-1}$ in the sense that \begin{equation*} \omega+\omega_*=\omega'+\omega_*', \end{equation*} if and only if it can be written as \begin{equation*} g(\omega)=A+B\cdot\omega, \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.


    Volume: Volume 32 (2024), Issue 3 (Special issue: Portuguese Mathematics)
    Published on: April 25, 2024
    Accepted on: March 7, 2024
    Submitted on: January 2, 2024
    Keywords: Mathematics - Classical Analysis and ODEs,Mathematics - Analysis of PDEs

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