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We derive the collision term in the Boltzmann equation using the equation of motion for the Wigner function of massive spin-1/2 particles. To next-to-lowest order in h, it contains a nonlocal contribution, which is responsible for the conversion of orbital into spin angular momentum. In a proper choice of pseudogauge, the antisymmetric part of the energy-momentum tensor arises solely from this nonlocal contribution. We show that the collision term vanishes in global equilibrium and that the spin potential is, then, equal to the thermal vorticity. In the nonrelativistic limit, the equations of motion for the energy-momentum and spin tensors reduce to the well-known form for hydrodynamics for micropolar fluids.
We study the Wigner function for massive spin-1/2 fermions in electromagnetic fields. The Wigner function is analytically solved in five cases when electromagnetic fields are constants. For a general space-time dependent field configuration, we use the method of semi-classical expansion and solved the Wigner function at linear order in the Planck's constant. At the same order, we obtained a generalized Boltzmann equation for particle distribution, and a generalized BMT equation for spin polarization. Using the Wigner function, we calculated some physical quantities in a thermal equilibrium system.