Ratio of effective temperature to pressure controls the dynamics of sheared hard spheres

Using molecular dynamics simulations, we calculate the effective temperature, $T_{\rm eff}$, and the pressure, $p$, of steadily sheared mixtures of hard spheres of mass $m$ and diameters $\sigma$ and $1.4 \sigma$ in contact with a thermal reservoir at temperature $T$. We vary the packing fraction, $\phi$, and the shear stress, $\Sigma$. We define $T_{\rm eff}$ from the ratio of correlations to response and show that different correlation-response relations yield a consistent numerical value $T_{\rm eff}\ge T$ that reduces to $T_{\rm eff}=T$ when $\Sigma=0$. We show that the effective temperature represents the limiting value of the effective temperature for soft spheres in the limit $p\sigma^3/\epsilon\rightarrow 0$, where $\epsilon$ is the repulsive energy scale. We find that the dimensionless ratio $T_{\rm eff}/p \sigma^3$ controls the dynamic jamming transition that occurs with decreasing shear stress and increasing packing fraction. In particular, we find that the dependence of the dimensionless relaxation time, $\tau \sqrt{p \sigma/m}$, on $T_{\rm eff}/p \sigma^3$ as shear stress is varied is quantitatively similar to the dependence of $\tau \sqrt{p \sigma/m}$ on $T/p \sigma^3$ in equilibrium.