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In this paper, a detailed analysis on normal modes of the linearized Hermitecollision operator is presented, which follows from linearizing spin Boltzmannequation for massive fermions proposed in \cite{Weickgenannt:2021cuo} with thenon-diagonal part of the transition rate neglected and approximating what wegot with a mutilated operator. With the assumption of total angular momentumconservation, the collision term is proved to well describe the equilibriumstate and gives proper interpretation for collisional invariants, thus isrelevant for the research on local spin polarization. Following the familiarfashion as used in quantum mechanics, we treat the problem of solving normalmodes as a degenerate perturbation problem and calculate the dispersionrelations for intriguing eleven zero modes, which form one-to-onecorrespondence to all collisional invariants. We find that the results ofspinless modes appearing in ordinary hydrodynamics are consistent withavailable conclusions in textbooks. As for spin-related modes, we obtain thefrequencies up to second order in wave vector and relate them with thedissipation of spin density fluctuation. In addition, the ratio of tworelaxation time scales for spin and momentum is shown as a function of reducedmass, which reads that based on present framework spin equilibration is almostas slow as momentum equilibration as far as the strange quark spins in quarkgluon plasma (QGP) are concerned.

The linear mode analysis and spin relaxation