Effective Temperatures in Athermal Systems Sheared at Fixed Normal Load

We perform molecular dynamics simulations of repulsive athermal systems sheared at fixed normal load to study the effective temperature $T_L$ defined from time-dependent fluctuation-dissipation relations for density. We show that these systems possess two distinct regimes as a function of the ratio $T_S/V$ of the granular temperature to the potential energy per particle. At small $T_S/V$, these systems are pressure-controlled and $T_L$ is set by the normal load. In contrast, they behave as quasi-equilibrium systems with $T_L \approx T_S$ that increases with shear rate at large $T_S/V$. These results point out several problems with using $T_L$ in thermodynamic descriptions of slowly sheared athermal systems.