## Abstract

We show that a measurable function g: S^{d−1} → ℝ, with d ≥ 3, satisfies the functional relation g(ω) + g(ω_{∗}) = g(ω^{′}) + g(ω^{′}∗), for all admissible ω, ω_{∗}, ω^{′}, ω^{′}∗ ∈ S^{d−1} in the sense that if and only if it can be written as ω + ω_{∗} = ω^{′} + ω^{′}∗, g(ω) = A + B · ω, for some constants A ∈ ℝ and B ∈ 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.

Original language | English (US) |
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Pages (from-to) | 229-239 |

Number of pages | 11 |

Journal | Communications in Mathematics |

Volume | 32 |

Issue number | 3 |

DOIs | |

State | Published - 2024 |

## Keywords

- Boltzmann–Fermi–Dirac equation
- Cauchy’s functional equation
- collision invariants
- hydrodynamic limits
- kinetic theory

## ASJC Scopus subject areas

- General Mathematics