The Boundary Structure of Zero-temperature Driven Hard Spheres

We study the fundamental problem of two gas species whose molecules collide as hard spheres in the presence of a flat boundary and with dependence on only one space dimension. More specifically the steady linear problem considered is the one arising when the second gas dominates as a flow moving towards the boundary with constant microscopic velocity (and hence zero temperature). The boundary condition adopted consists of prescribing the outgoing velocity distribution of the first gas at the boundary. It is discovered that the presence of the boundary under general assumptions on the outgoing distribution ensures the convergence of a series of path integrals resulting in a convenient representation for the distribution of the velocities of the molecules returning at the boundary.