is the collision integral and $\textbf V$ is the interaction potential. This form of the Boltzmann equation (\[eq:2\]) is completely general, and holds for any non-condensed spinor gas. The role of spin enters in determining the dimension of the density matrix, and the exact form of the interaction potential $\textbf{V}$.

The first two terms on the right hand side of the Boltzmann equation (\[eq:2\]) arise from forward and backward scattering collisions. While the former type merely alter the mean field seen by the atoms, backward scattering collisions allow the colliding atoms to exchange momentum.

The last term in the Boltzmann equation is the collision integral, responsible for energy relaxation. While a detailed derivation of the collision integral is non trivial [@nikuni], the qualitative properties are well described within a simple relaxation time approximation $I_{c} = -(f - f_{0})/\tau$. The relaxation time ($\tau$) is proportional to the elastic scattering rate $\tau^{-1}_{el} \sim 8\pi a^{2}_{0}v_{T}n$, where $v_{T} = \sqrt{\frac{2 k_{B}T}{m}}$ is the thermal velocity and $n$ is the density.

The precise expression for $\bf{V}$ determined by including the interactions within the Hartree-Fock description is$$\label{eq:3}
 \textbf{V} = \left(U + c_{0}\text{Tr}(\textbf n)\right)\textbf 1 + c_{0}\textbf n + c_{2}\vec{\textbf{S}}\textbf n \cdotp \vec{\textbf{S}} + c_{2} \vec{M}\cdotp\vec{\textbf{S}}$$ 