Nonequilibrium in Thermodynamic Formalism: the Second Law, gases and Information Geometry

The paper explicitly states and proves that DKL(µ1, µ2) = DKL(L*_{log J}(µ1), L*_{log J}(µ2)) for a Hölder Jacobian J and suitable probabilities µ1, µ2 with continuous IRN (Theorem 23, eq. (31)), using the explicit IRN transformation J3(z) = J1(σz) J(z) / J(σz) (Proposition 24, eq. (32)) and a direct cancellation argument . The candidate solution proves the same identity by: (i) showing σ_* L*_{log J}(µ) = µ and that the conditional first symbol given the tail under the lifted measure equals J; (ii) identifying dµ/dL*_{log J}(µ) = J_µ/J via the IRN identity L*_{log J_µ}(µ)=µ; and (iii) decomposing each IRN as a conditional part times the pushforward density, leading again to cancellation. These steps rely on standard identities in the paper (sum_a J(a x)=1 for Jacobians, definition of DKL via IRNs, and Proposition 10: L*_{log J_µ}(µ)=µ) . The proofs are logically consistent and complete under the stated assumptions, but differ in technique: the paper uses the explicit transformed IRN (32), while the model uses disintegration and Radon–Nikodym comparisons.