  We study self-diffusion within a simple hopping model for glassy materials.
(The model is Bouchaud's model of glasses [J.-P. Bouchaud, J. Physique I 2,
1705 (1992)], as extended to describe rheological properties [P. Sollich, F.
Lequeux, P. Hebraud and M.E. Cates, Phys. Rev. Lett. 78, 2020 (1997)].) We
investigate the breakdown, near the glass transition, of the (generalized)
Stokes-Einstein relation between self-diffusion of a tracer particle and the
(frequency-dependent) viscosity of the system as a whole. This stems from the
presence of a broad distribution of relaxation times of which different moments
control diffusion and rheology. We also investigate the effect of flow
(oscillatory shear) on self-diffusion and show that this causes a finite
diffusivity in the temperature regime below the glass transition (where this
was previously zero). At higher temperatures the diffusivity is enhanced by a
power law frequency dependence that also characterises the rheological
response. The relevance of these findings to soft glassy materials (foams,
emulsions etc.) as well as to conventional glass-forming liquids is discussed.
