Quantized collision invariants on the sphere

Benjamin Anwasia, Diogo Arsénio

Research output: Contribution to journalArticlepeer-review


We show that a measurable function g: Sd−1 → ℝ, with d ≥ 3, satisfies the functional relation g(ω) + g(ω) = g(ω) + g(ω∗), for all admissible ω, ω, ω, ω∗ ∈ Sd−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 ∈ ℝd. 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 languageEnglish (US)
Pages (from-to)229-239
Number of pages11
JournalCommunications in Mathematics
Issue number3
StatePublished - 2024


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

ASJC Scopus subject areas

  • General Mathematics


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