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We investigate the relation between mobility and diffusivity for Brownianparticles under steady shear near the glass transition, using mode couplingapproximations. For the two directions perpendicular to the shear direction,the particle motion is diffusive at long times and the mobility reaches afinite constant. Nevertheless, the Einstein relation holds only for theshort-time in-cage motion and is violated for long times. In order to get therelation between diffusivity and mobility, we perform the limit of smallwavevector for the relations derived previously [Phys. Rev. Lett. 102 (2009),135701], without further approximation. We find good agreement to simulationresults. Furthermore, we split the extra term in the mobility in an exact wayinto three terms. Two of them are expressed in terms of mean squareddisplacements. The third is given in terms of the (less handy) force-forcecorrelation function.

Non-Equilibrium relation between mobility and diffusivity of interacting Brownian particles under shear