A Light impurity in an Equilibrium Gas

We investigate the evolution of a light impurity particle in a Lorentz gas where the background atoms are in thermal equilibrium. As in the standard Lorentz gas, we assume that the particle is negligibly light in comparison with the background atoms. The thermal motion of atoms causes the average particle speed to grow. In the case of the hard-sphere particle-atom interaction, the temporal growth is ballistic, while generally it is sub-linear. For the particle-atom potential that diverges as r^{-\lambda} in the small separation limit, the average particle speed grows as t^{\lambda /(2(d-1)+ \lambda)} in d dimensions. The particle displacement exhibits a universal growth, linear in time and the average (thermal) speed of the atoms. Surprisingly, the asymptotic growth is independent on the gas density and the particle-atom interaction. The velocity and position distributions approach universal scaling forms which are non-Gaussian. We determine the velocity distribution in arbitrary dimension and for arbitrary interaction exponent \lambda. For the hard-sphere particle-atom interaction, we compute the position distribution and the joint velocity-position distribution.