Nonequilibrium mode-coupling theory for dense active systems of self-propelled particles

The physics of active systems of self-propelled particles, in the regime of a dense liquid state, is an open puzzle of great current interest, both for statistical physics and because such systems appear in many biological contexts. We develop a nonequilibrium mode-coupling theory (MCT) for such systems, where activity is included as a colored noise with the particles having a self-propulsion foce $f_0$ and persistence time $\tau_p$. Using the extended MCT and a generalized fluctuation-dissipation theorem, we calculate the effective temperature $T_{eff}$ of the active fluid. The nonequilibrium nature of the systems is manifested through a time-dependent $T_{eff}$ that approaches a constant in the long-time limit, which depends on the activity parameters $f_0$ and $\tau_p$. We find, phenomenologically, that this long-time limit is captured by the potential energy of a single, trapped active particle (STAP). Through a scaling analysis close to the MCT glass transition point, we show that $\tau_\alpha$, the $\alpha$-relaxation time, behaves as $\tau_\alpha\sim f_0^{-2\gamma}$, where $\gamma=1.74$ is the MCT exponent for the passive system. $\tau_\alpha$ may increase or decrease as a function of $\tau_p$ depending on the type of active force correlations, but the behavior is always governed by the same value of the exponent $\gamma$. Comparison with numerical solution of the nonequilibrium MCT as well as simulation results give excellent agreement with the scaling analysis.