We derive relativistic hydrodynamic equations with a dynamical spin degree of freedom on the basis of an entropy-current analysis. The first and second laws of local thermodynamics constrain possible structures of the constitutive relations including a spin current and the antisymmetric part of the (canonical) energy-momentum tensor. Solving the obtained hydrodynamic equations within the linear-mode analysis, we find spin-diffusion modes, indicating that spin density is damped out after a characteristic time scale controlled by transport coefficients introduced in the antisymmetric part of the energy-momentum tensor in the entropy-current analysis. This is a consequence of mutual convertibility between spin and orbital angular momentum.

We compute the momentum diffusion coefficients of heavy quarks, κ and κ ⊥ , in a strong magnetic field B along the directions parallel and perpendicular to B, respectively, at the leading order in QCD coupling constant α s . We consider a regime relevant for the relativistic heavy ion collisions, α s eB T 2 eB, so that thermal excitations of light quarks are restricted to the lowest Landau level (LLL) states. In the vanishing light-quark mass limit, we find κ LO ⊥ ∝ α 2 s T eB in the leading order that arises from screened Coulomb scatterings with (1+1)-dimensional LLL quarks, while κ gets no contribution from the scatterings with LLL quarks due to kinematic restrictions. We show that the first non-zero leading order contributions to κ LO come from the two separate effects: 1) the screened Coulomb scatterings with thermal gluons, and 2) a finite light-quark mass m q . The former leads to κ LO, gluon ∝ α 2 s T 3 and the latter to κ LO, massive ∝ α s (α s eB) 1/2 m 2 q . Based on our results, we propose a new scenario for the large value of heavy-quark elliptic flow observed in RHIC and LHC. Namely, when κ ⊥ κ , an anisotropy in drag forces gives rise to a sizable amount of the heavy-quark elliptic flow even if heavy quarks do not fully belong to an ellipsoidally expanding background fluid. *