Global equilibrium at order $\mathcal{O}(\hbar^2)$: pseudo-gauge ambiguity, Maxwell relations, and thermodynamic consistency

We study massive spin-$1/2$ Boltzmann particles in global thermodynamic equilibrium with rotation and acceleration, consistently including quantum corrections up to order $\mathcal{O}(\hbar^2)$. For such a system, the fluid velocity becomes an independent thermodynamic variable and fundamental therm