Steady state self-diffusion at low density

We prove that the motion of a test particle in a hard sphere fluid in thermal equilibrium converges, in the Boltzmann-Grad limit, to the stochastic process governed by the linear Boltzmann equation. The convergence is in the sense of weak convergence of the path measures. We use this result to study the steady state of a binary mixture of hard spheres of different colors (but equal masses and diameters) induced by color-changing boundary conditions. In the Boltzmann-Grad limit the steady state is determined by the stationary solution of the linear Boltzmann equation under appropriate boundary conditions.