线性模分析和自旋弛豫
The linear mode analysis and spin relaxation
摘要：
本文对线性化的Hermite碰撞算符的简正模进行了详细的分析，它是由\cite{Weickgenannt:2021cuo}中提出的忽略跃迁率非对角部分的线性化的大质量费米子的自旋Boltzmann方程导出的，并用一个截断算符近似得到的。在总角动量守恒的假设下，证明了碰撞项能很好地描述平衡态，并对碰撞不变量给出了合理的解释，从而对局域自旋极化的研究具有重要意义.按照量子力学中常见的方法，我们把简并模的求解问题看作简并微扰问题，并计算了11个有趣的零模的色散关系，它们与所有的碰撞不变量一一对应.我们发现在普通流体力学中出现的无旋模的结果与教科书中已有的结论是一致的。对于自旋相关模，我们得到了波矢中的二阶频率，并将它们与自旋密度涨落的耗散联系起来。此外，自旋和动量两个弛豫时间尺度的比值是约化质量的函数，这表明在现有的框架下，对于夸克-胶子等离子体（QGP）中的奇异夸克自旋，自旋平衡几乎和动量平衡一样慢.
英文摘要：
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.
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