Relaxation time, viscosity and scaling at densities below jamming

We simulate soft-core bidisperse frictionless disks in two dimensions with overdamped dynamics at zero temperature and densities below jamming. We first prepare configurations by shearing at several constant shear rates $\dot\gamma$. These configurations are then used as starting points for simulations \emph{without} shearing that relax the system to zero energy. From these simulations we determine both the relaxation time, $\tau$, and the average path length traversed by the particles to reach the zero energy state. We find that $\tau$ diverges algebraically as a function of density, $\tau \sim (\phi_J-\phi)^{-\beta}$, if $\dot\gamma$ in the preparatory simulations is sufficiently small. We further find that the shear viscosity $\eta$ can be formally related to $\tau$, and that this gives a way to understand the origin of corrections to scaling in the scaling analysis of $\eta$[1]. The presence of the exponent $\beta+y$, where $y\approx 1.1$, in the scaling of the deviations from the $\dot\gamma\to0$ limit, $\eta(\phi,\dot\gamma)/ \eta(\phi,\dot\gamma\to0) = f((\phi_J-\phi)^{-(\beta+y)}\dot\gamma)$ [1], is also given an intuitive interpretation.\newline [1] P. Olsson and S. Teitel, Phys.\ Rev.\ E \textbf{83}, 030302(R), 2011.