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

Using molecular dynamics simulation, we calculate fluctuations and response for steadily sheared hard spheres over a wide range of packing fractions $ϕ$ and shear strain rates $γ$, using two different methods to dissipate energy. To a good approximation, shear stress and density fluctuations are related to their associated response functions by a single effective temperature $T_{eff}$ that is equal to or larger than the kinetic temperature $T_{kin}$. We find a crossover in the relationship between the relaxation time $τ$ and the the nondimensionalized effective temperature $T_{eff}/pσ^3$, where $p$ is the pressure and $σ$ is the sphere diameter. In the solid response regime, the behavior at fixed packing fraction satisfies $τγ\propto \exp(-cpσ^3/T_{eff})$, where $c$ depends weakly on $ϕ$, suggesting that the average local yield strain is controlled by the effective temperature in a way that is consistent with shear transformation zone theory. In the fluid response regime, the relaxation time depends on $T_{eff}/pσ^3$ as it depends on $T_{kin}/pσ^3$ in equilibrium. This regime includes both near-equilibrium conditions where $T_{eff} ~ T_{kin}$ and far-from-equilibrium conditions where $T_{eff} \ne T_{kin}$. We discuss the implications of our results for systems with soft repulsive interactions.