Viscosity and effective temperature of an active dense system of self-propelled particles

We obtain a nonequilibrium theory for a simple model of a generic class of active dense systems consisting of self-propelled particles with a self-propulsion force, $f_0$, and persistence time, $\tau_p$, of their motion. We consider two models of activity and find the system is characterized by an evolving effective temperature $T_{eff}(\tau)$, defined through a generalized fluctuation-dissipation theorem. $T_{eff}(\tau)$ is equal to the equilibrium temperature at very short time $\tau$ and saturates to $T_{eff}=T_{eff}(\tau\to\infty)$ at long times; The transition time $t_{trans}$ when $T_{eff}(\tau)$ goes to the long-time limit depends on $\tau_p$ alone and $t_{trans}\sim \tau_p^{0.85}$ for both models. $f_0$ reduces the viscosity with increasing activity, $\tau_p$ on the other hand, may increase or decrease viscosity depending on the details of how the activity is included. However, as a function of $T_{eff}$, viscosity shows the same behavior for different models of activity and $\eta\sim (T_{eff}-T)^{-\gamma}$ with $\gamma=1.74$. Our theory gives reasonable agreement when compared with experimental data and is consistent with several experiments on diverse systems.