the self- and cross- interactions between the three sublevels produce diagonal and off-diagonal contributions to the interaction potential, the spin-relaxation or population exchange interactions only gives rise to coherences, and are absent for a single component gas.

Qualitative Features
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Here we elaborate on our argument for why one expects richer spin physics in a spin-$1$ system versus a spin-$\frac{1}{2}$ one. We assume a uniform, collisionless gas with $\nabla_{R}c_{0}n/m v_{T} \ll c_{0}n$, such that all the physics is governed by the commutator in (\[eq:2\]).

The pseudo-spin $\frac{1}{2}$ case may be understood starting from the fact that that a $2 \times 2$ matrix ($A$) can be decomposed into $\textbf{A} = A_{0}\textbf{I} + A_{\mu}\sigma_{\mu}$, where $A_{0} = \text{Tr}(\textbf{A})$, and $\vec{\sigma}$ are the Pauli matrices. Expressing the density matrix and interaction potential in this way, the equations of motion are $\frac{\partial}{\partial t}F_{0} + \frac{\vec{p}}{m}\cdotp\vec{\nabla}_{R}F_{0} = 0$ and $\frac{\partial}{\partial t}{F_{\mu}} + \frac{\vec{p}}{m}\cdotp\vec{\nabla}_{R}F_{\mu} =  - F_{\rho}V_{\nu}\epsilon_{\rho\nu\mu}$, where $\epsilon_{\rho\nu\mu}$ is the completely antisymmetric unit tensor. The second equation, which is responsible for much of the spin wave physics in a spin-$\frac{1}{2}$ system simply says that the interactions act as an effective magnetic field about which the spins precess.

To analyze the spin-$1$ case, we follow Ohmi 