Conditional extension of the macroscopic limit to higher dimensions

Conjecture that, if stable equilibria of the angular collision operator C(f) exist in dimensions n ≥ 4, then the macroscopic limit derived in Theorem (the density–direction system for ρ and Ω) extends to dimensions n ≥ 4.

Background

The macroscopic derivation uses stable equilibria of C to identify the local angular equilibria and close the moment system via the method of Generalized Collision Invariants. This structure underpins the derivation of the macroscopic equations for ρ and Ω.

The authors point out that, provided stable equilibria for C can be established for n ≥ 4, the same derivation framework should apply and yield an extension of their main macroscopic result beyond n ∈ {2,3}.