Compressible solved-volatility stochastic fluid thermodynamics: source-consistent energy, finite-correlation reservoirs, entropy admissibility and boundary conditions
A variable-density thermodynamic extension is developed for the solved-volatility stochastic-fluid formulation of arXiv:2607.25536. Source-inclusive stochastic transport separates mass, momentum and total-energy conservation into time-evolution partial differential equations and martingale compatibility constraints. Density and temperature are the primitive thermodynamic fields: mass conservation determines density, internal energy determines temperature, and the equation of state determines pressure evolution along stochastic particle paths. The resolved kinetic-energy identity is combined with a finite-correlation reservoir, Green--Kubo calibration, an equilibrium counterterm and adjoint resolved-unresolved exchange. A stochastic Gibbs identity and Gaussian relative entropy yield a conditional entropy-admissibility result for a Hencky-reservoir formulation. Equation-of-state pressure fluctuations are distinguished from mechanical stress impulses; regular finite-Mach fluctuations produce no independent white-noise bulk pressure impulse, while fast mechanical pressure is represented by a causal finite-correlation carrier. Conservative boundary conditions and a calorically perfect ideal-gas specialization are given. In the zero-volatility limit, the classical compressible Navier--Stokes--Fourier equations are recovered. A frozen descriptor analysis identifies a mixed hyperbolic--parabolic drift subsystem coupled to algebraic martingale constraints, with closure-dependent elliptic blocks and a singular low-Mach pressure limit. Canonical calculations verify the pressure carrier, acoustic dispersion, viscous-thermal energy balance and low-Mach scaling. Nonlinear well-posedness, shock admissibility and developed turbulence are not claimed.