A thermodynamically consistent, fully compressible, fluctuating hydrodynamics model of a Cahn Hilliard binary fluid mixture

We derive, from first principles, a thermodynamically consistent set of fully compressible fluctuating hydrodynamic equations for a Cahn-Hilliard binary fluid mixture, in isothermal and non-isothermal form and for symmetric and non-symmetric mixtures. The derivation uses the GENERIC formalism, with the friction operator constructed from flux fluctuations following Jelic and Ottinger, so that deterministic and stochastic fluxes derive from the same operator and correlation matrix and the fluctuation-dissipation theorem holds by construction. We show that the momentum equation must use the total mass density, that the Cahn-Hilliard variable is the mass fraction, and that the diffusion flux must be driven by the specific, not the volumetric, exchange potential. We derive a generalized Korteweg tensor for square-gradient energies in density and composition, and show that the reversible pressure tensor of the non-isothermal model is fixed by the degeneracy condition, producing a thermal capillary stress when the interfacial free energy has an entropic part. Thermodynamic consistency is proved in full: Jacobi identity, positivity and degeneracy, local energy conservation with interstitial-working fluxes, a local entropy balance, Soret-Dufour reciprocity, and Einstein statistics of the linearized equations. The final equations are written in conservative form for finite-volume discretization, and closed-form static and dynamic structure factors are given for all models as analytic benchmarks. An appendix shows that the same models follow from classical irreversible thermodynamics with the Helmholtz or Gibbs free energy; this route is unique for isothermal flows and has a gauge freedom in the non-isothermal case that GENERIC removes.