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

TL;DR

This paper explores how mobility and diffusivity relate in sheared Brownian particles near the glass transition, revealing violations of the Einstein relation at long times and providing a detailed analysis of the underlying terms.

Contribution

It derives a precise relation between diffusivity and mobility under shear without additional approximations, extending previous work and confirming results with simulations.

Findings

Mobility reaches a finite constant in directions perpendicular to shear.

The Einstein relation holds only at short times, not long times.

Good agreement with simulation results.

Abstract

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.