The linear mode analysis and spin relaxation

TL;DR

This paper analyzes the normal modes of the linearized Hermite collision operator derived from the spin Boltzmann equation, revealing insights into spin relaxation and its relation to momentum relaxation in fermionic systems.

Contribution

It provides a detailed analysis of spin-related normal modes, deriving dispersion relations and relaxation times, extending understanding of spin dynamics in quantum kinetic theory.

Findings

Spinless modes align with classical hydrodynamics results.

Spin modes' frequencies relate to spin density fluctuation dissipation.

Spin relaxation is nearly as slow as momentum relaxation in quark-gluon plasma.

Abstract

In this paper, a detailed analysis on normal modes of the linearized Hermite collision operator is presented, which follows from linearizing spin Boltzmann equation for massive fermions proposed in \cite{Weickgenannt:2021cuo} with the non-diagonal part of the transition rate neglected and approximating what we got with a mutilated operator. With the assumption of total angular momentum conservation, the collision term is proved to well describe the equilibrium state and gives proper interpretation for collisional invariants, thus is relevant for the research on local spin polarization. Following the familiar fashion as used in quantum mechanics, we treat the problem of solving normal modes as a degenerate perturbation problem and calculate the dispersion relations for intriguing eleven zero modes, which form one-to-one correspondence to all collisional invariants. We find that the results of spinless modes appearing in ordinary hydrodynamics are consistent with available conclusions in textbooks. As for spin-related modes, we obtain the frequencies up to second order in wave vector and relate them with the dissipation of spin density fluctuation. In addition, the ratio of two relaxation time scales for spin and momentum is shown as a function of reduced mass, which reads that based on present framework spin equilibration is almost as slow as momentum equilibration as far as the strange quark spins in quark gluon plasma (QGP) are concerned.