We propose a microscopic description for the polarization from the first principle through the spin-orbit coupling in particle collisions. The model is different from previous ones based on local equilibrium assumptions for the spin degree of freedom.It is based on scatterings of particles as wave packets, an effective method to deal with particle scatterings at specified impact parameters. The polarization is then the consequence of particle collisions in a non-equilibrium state of spins. The spinvorticity coupling naturally emerges from the spin-orbit one encoded in polarized scattering amplitudes of collisional integrals when one assumes local equilibrium in momentum but not in spin.

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Many studies for nonequilibrium systems, e.g., the pre-equilibration puzzle in heavy-ion collisions, require solving the relativistic Boltzmann equation (BE) with the full collisional kernel to high precision. It is challenging to solve relativistic BE due to its high dimensional phase-space integrals and limited computing resources. We have developed a numerical framework for a full solution of the relativistic Boltzmann equations for the quark-gluon matter using the multiple graphics processing units (GPUs) on distributed clusters. Including all the 2 → 2 scattering processes of 3-flavor quarks and gluons, we compute the time evolution of distribution functions in both coordinate and momentum spaces for the cases of pure gluons, quarks, and the mixture of quarks and gluons. By introducing a symmetrical sampling method on GPUs which ensures the particle number conservation, our framework is able to perform the space-time evolution of quark-gluon system toward thermal equilibrium with high performance. We also observe that the gluons naturally accumulate in the soft region at the early time, which may indicate the gluon condensation.

Based on the Wigner function in local equilibrium, we derive hydrodynamical quantities for a system of polarized spin-1/2 particles: the particle number current density, the energy-momentum tensor, the spin tensor, and the dipole moment tensor. Compared with ideal hydrodynamics without spin, additional terms at the first and second orders in the Knudsen number Kn and the average spin polarization χs have been derived. The Wigner function can be expressed in terms of matrix-valued distributions, whose equilibrium forms are characterized by thermodynamical parameters in quantum statistics. The equations of motion for these parameters are derived by conservation laws at the leading and next-to-leading order Kn and χs .

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