Diffusion of impurities in a dense fluid near the glass transition

The diffusion of impurities of low concentration in a surrounding dense medium is studied near the liquid-glass transition point. Within a simplified model we obtain an analytical solution in the long-time or small-frequency regime for a generalized diffusion constant, which is expressed in terms of Bessel functions. At the glass transition of the medium the impurities can either become localized or still be delocalized. In the former case, there exists an Einstein-Stokes relation between the diffusion constant and the viscosity of the medium. In the latter case, the situation is similar to a superionic glass and reduces essentially to the Lorentz model. Explicit results for the generalized diffusion constant in the various regimes are obtained from asymptotic expansions of the Bessel functions.