We investigate the relation between mobility and diffusivity for Brownian
particles under steady shear near the glass transition, using mode coupling
approximations. For the two directions perpendicular to the shear direction,
the particle motion is diffusive at long times and the mobility reaches a
finite constant. Nevertheless, the Einstein relation holds only for the
short-time in-cage motion and is violated for long times. In order to get the
relation between diffusivity and mobility, we perform the limit of small
wavevector for the relations derived previously [Phys. Rev. Lett. 102 (2009),
135701], without further approximation. We find good agreement to simulation
results. Furthermore, we split the extra term in the mobility in an exact way
into three terms. Two of them are expressed in terms of mean squared
displacements. The third is given in terms of the (less handy) force-force
correlation function.Comment: 14 pages, 4 figures, accepted for Prog. Theor. Phys. Suppl., issue
for the workshop "Frontiers in Nonequilibrium Physics", Kyoto, 200
