Transient spin modes from relaxational axial kinetic theory

We study the dynamics of spin mode by solving the axial kinetic equations under the relaxation time approximation in the presence of dissipative sources. We find transient spin modes in response to the electric field with spacetime inhomogeneity, fluid acceleration and shear. To the lowest order in spatial momentum $k$, we find the responses to electric field and acceleration can be interpreted as a retarded response to temporal variations of the magnetic field and vorticity respectively. The response to shear occurs at $O({k}^{2})$ and can be reduced to a retarded response to the spatial variation of vorticity. Beyond lowest order, the responses to all three sources are nonlocal with branch cut in the dispersions. We argue that the nonlocality is a consequence of the quasiparticle picture underlying the kinetic description. We also analyze the spin transport equation taking into account the spin response to temporal and spatial variations of vorticity. We find the corrections turn the original first order spin transport equation into a third order one (or a second order one in the homogeneous limit). The change in order of the transport equation is a consequence of the nonlocal nature of the responses, suggesting a possible breakdown of gradient expansion in spin hydrodynamics for microscopic theories with quasiparticles.