Cross effects in spin hydrodynamics: Entropy analysis and statistical operator

We revisit the construction of first-order spin hydrodynamics and find that constitution relations receive corrections from cross effects resulting spin-orbit coupling. Starting a routine entropy analysis, we show how to identify new transport coefficients second law thermodynamics. Interestingly, conventional coefficient heat conductivity $\kappa$ is bounded below by product coefficients, which means threshold conduction changed. With recourse Zubarev's non-equilibrium statistical operator, reproduce identification in more rigorous way. By seeking dispersion normal modes, these suppress attenuation sound modes mode appearing also have impacts on damping non-hydrodynamic modes.