Levi-Civita identities, it is easy to see that $i[\textbf{V}^{(a)}, \textbf{F}^{(a)}] $ is the spin-$1$ analog of the corresponding spin-$\frac{1}{2}$ term which acts as an effective magnetic field for the spins. But the last term $i[\textbf{V}^{(s)}, \textbf{F}^{(s)}]$ shows that fluctuations of the nematicity can change the spin dynamics.

Spin waves in a two-component gas
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As a starting point for understanding the spin-$1$ gas, we review the spin-$\frac{1}{2}$ case. As explained in [@oktel], spin dynamics in the cold collision regime are governed by momentum exchange collisions with a characteristic timescale $\tau^{-1}_{ex} \sim \frac{4\pi a_{12}n}{m}$, where $a_{12}$ is the two-body scattering length. When $k_{B}T \ll \frac{1}{m a^{2}_{12}}$, or alternatively, the thermal deBroglie wavelength is large compared to the scattering length $\Lambda_{T} = \sqrt{\frac{2\pi}{m k_{B}T}} \ge a_{12}$, $\tau_{ex} \ll \tau_{el}$, and for times shorter than $\tau_{el}$, particles exchange momentum several times without significantly altering the energy distribution. Therefore excitations with wavelength longer than $v_{T}\tau_{ex}$ are a collective effect.

For typical densities ($\sim 10^{14}$cm$^{-3}$) and scattering lengths, $n^{1/3}a \sim 0.01$ the condensation temperature ($T_{c}$) for the interacting system may be approximated to that of an ideal Bose gas [@baym]. Thus in order to see collective spin phenomena, we require $T_{c} \sim \langle \omega \rangle 