we turn to the spin-$1$ gas. Starting from a microscopic Hamiltonian, we obtain a linearized Boltzmann equation about the ferromagnetic ($m_{F} = 1$) and polar ($m_{F} = 0$) states, and calculate the spin wave dispersion in each case. We find that a polar gas with ferromagnetic interactions is dynamically unstable towards spin-mixing collisions for small enough Zeeman fields, analogous to the condensed case. By explicit calculation of the dispersion relation, we show that for strong enough anti-ferromagnetic interactions, an instability occurs in the polar state [@mueller2].

Following these analytic calculations, we perform numerical simulations of a trapped gas using an effective $1$D Boltzmann equation. We explore the evolution of transversely polarized spins, and investigate dynamical instabilities.

Our work complements prior work on the kinetics of a normal spin-$1$ Bose gas by Endo and Nikuni [@nikuni]. While their focus is on the effect of collisions in the damping of collective modes of a trapped gas, with particular emphasis on dipole modes, we focus here on the collisionless, Knudsen regime.

Basic Setup
===========

Kinetic Equations
-----------------

As described by Ho [@jason] and Ohmi and Machida [@Ohmi], the second-quantized Hamiltonian for a spin-$1$ gas, expressed in a frame where each spin component is 