Trajectory Class Fluctuation Theorem

The paper states the class decomposition Σ = ΘC + ψC (its Eq. (16)), and proves the Nominal and Exponential Class Fluctuation Theorems: ⟨Σ⟩C = ΘC + ΨC (Eq. (19)) and ⟨e^{−Σ}⟩C = e^{−ΘC} = R(C)/P(C) (Eq. (20)) using class-conditioned densities and the identity ⟨e^{−ψC}⟩C = 1; see the definitions and proofs around Theorem 1 and Theorem 2 in the TCFT section . The candidate solution reproduces exactly these results with a measure-theoretic Radon–Nikodym argument. One minor caveat: in the change-of-measure step it implicitly uses dR/dP (or dR(·|C)/dP(·|C)), which requires mutual absolute continuity that was not stated; the paper avoids this by working with pointwise densities and the ratio R(·|C)/P(·|C). Appendix A of the paper also derives ⟨e^{−Σ}⟩C = R(C)/P(C) directly as a path-ensemble identity . Aside from that minor technical assumption, the approaches are the same in substance and yield the same statements.