Recently, there have been significant experimental progresses in observing and/or controlling spin-dependent bulk quantities in broad areas in physics, e.g., relativistic heavy-ion collisions, spintronics. Although hydrodynamics is one of the most powerful theoretical frameworks to describe bulk quantities, its extension to a spinful fluid has not been developed well, especially for relativistic systems.
In this presentation, we formulate relativistic spin hydrodynamics with first-order dissipative corrections for the first time . Our formulation is based on the phenomenological entropy-current analysis, in which the first and second laws of thermodynamics are utilized to constrain constitutive relations. We also clarify that spin should be a non-hydrodynamic (diffusive) mode by explicitly solving the obtained hydrodynamic equations within the linear-mode analysis on top of a global thermal equilibrium configuration. This diffusive behavior is a consequence of the mutual convertibility between spin and orbital angular momentum.
 K. Hattori, M. Hongo, X.-G. Huang, M. Matsuo, H. Taya, "Fate of spin polarization in a relativistic fluid: An entropy-current analysis," Phys. Lett. B 795, 100 (2019)