  As the core ingredient for spin polarization, the local equilibrium spin
distribution function is derived from the detailed balance principle. The
kinetic theory for interacting fermionic systems is applied to the
Nambu--Jona-Lasinio model at quark level. Under the semi-classical expansion
with respect to $\hbar$ and non-perturbative expansion with respect to $N_c$,
the kinetic equations for the vector and axial-vector distribution functions
are derived with collision terms. It is found that, for an initially
unpolarized system, non-zero spin polarization can be generated at the order of
$\hbar$ from the coupling between the vector and axial-vector charges. The
local equilibrium spin polarization is derived from the requirement of detailed
balance. It arises from the thermal vorticity and is orthogonal to the particle
momentum.
