# Boltzmann equation derivation for $H=v\sigma \cdot p$ hamiltonian

I am trying to write the Boltzmann equation for $$H=v_{F}\vec{\sigma}\cdot(\vec{p}-e\vec{A}).$$ This is a free charged particles gas. The velocity for this hamiltonian is $$\vec{v}=v_{F} \vec{\sigma}.$$ The Lorentz force can be derived $$\vec{F}=q(\vec{v}\times \vec{B})$$. I am considering to start from $$\frac {df}{dt}=\frac {\partial f}{\partial t}+\vec{F}\cdot\vec{\nabla}_{p}f(\vec{p},\vec{\sigma},\epsilon).$$ I am trying to rigorously reach to boltzmann equation from this hamiltonian.