On the Comparison between Phenomenological and Kinetic Theories of Gas Mixtures with Applications to Flocking

The paper’s Theorem 5.2 proves the same flocking estimates for the KB-CS model (4.19) under Type A and Type B topologies, using the symmetric pairwise dissipation identity for V and E and a time-dependent lower bound on the minimal interaction strength to integrate decay rates; see the model definitions (4.17)–(4.19) and constraints (4.21) together with the dissipation steps (5.10), (5.13) and the lower bound argument (5.14)–(5.16) culminating in Theorem 5.2’s statements for V, X, E and Cλ(t) (definitions in (5.4), (5.8)) . The candidate solution follows the same structure: (i) conservation of the means from (4.21), (ii) exact dissipation for V and E via symmetry, (iii) for Type A, uniform positivity of min aαβ gives exponential decay and bounded X, and (iv) for Type B, a lower bound a*(t) ≥ Λ0(1+t)^{-2λ} based on interparticle distance growth yields the same explicit decay rates for V and E and the same bound for X via X' ≤ V. Small differences are cosmetic (e.g., a slightly different derivation of the interparticle distance bound), but the proofs are essentially the same.