  This paper gives the first full proof of the justification of the linear
Boltzmann equation from an underlying long range particle evolution. We suppose
that a tagged particle is interacting with a background via a two body
potential that is decaying faster than $ C\exp\left(-C|x|^{\frac{3}{2}}\right)
$, and that the background is initially distributed according to a function in
$L^1((\mathbb{R}^3,(1+|v|^2)\mathrm{d}v)$ in velocity and uniformly in space.
Under finite mass and energy assumptions on the initial density, the tagged
particle density converges weak-$\star$ in $L^\infty$ to a solution of the
linear Boltzmann equation.
  The proof uses estimates on two body scattering and on the relationship
between long range dynamics and dynamics with a truncated interaction potential
to explicitly estimate the error between densities for long and short range
dynamics. To compare the difference between the short range dynamics and the
linear Boltzmann equation, we use a tree based structure to encode the
collisional history of the tagged particle.
