Second-order fast-slow dynamics of non-ergodic Hamiltonian systems: Thermodynamic interpretation and simulation

The paper’s Theorem 5.2 asserts that, under non-resonance Assumptions 1–2, the scaled residuals (φε2, θε2, yε2, pε2) split into explicit rapidly oscillatory correctors [φ2]ε, [θ2]ε, [y2]ε, [p2]ε and slowly varying components (φ̄2, θ̄2, ȳ2, p̄2) solving a linear inhomogeneous system with initial data (22). The stated system matches equations (21a)–(21d), and the construction of correctors is given in Definition 5.1. The candidate solution reproduces the same structure: it introduces the same scaled residuals/correctors, repeatedly integrates by parts, uses the same non-resonance to control small denominators, and identifies exactly the same surviving drifts (e.g., dφ̄λ2/dt = ⟨Dωλ(y0), ȳ2⟩ + θλ∗|DLλ0|^2/8 − (DtLλ0)^2/(8ωλ(y0)); compare 21a). It also mirrors the paper’s weak-* arguments for oscillatory terms (via the same kind of lemmas about oscillatory integrals and stationary phase) and concludes uniqueness by linear ODE theory, in line with the paper’s subsequence-extraction-then-uniqueness step. Minor differences are expository (e.g., a moment where an O(ε) rate for θε1 − [θ1]ε is implicitly used; the paper only needs uniform o(1) with weak-* control), but these do not affect the limit system or convergence statements. Overall, both proofs use substantially the same approach and arrive at the same result. Citations: Theorem 5.2 statement and system (21a–d) and (16)–(20) are explicit in the paper, as are Assumptions 1–2 and the correctors in Definition 5.1 . The proof strategy using integration by parts and oscillatory lemmas is summarized around Lemmas 5.4–5.12 , with details for θ̄2 (including the explicit dθ̄λ2/dt) and weak-* treatment of oscillations . The integrated form in Theorem 6.1 confirms θ̄λ2(t) = θλ∗⟨p0, Dωλ(y0)⟩^2/(8ωλ(y0)^4) + const., consistent with (21b) upon differentiation .