Effective temperature and activity induced memory effects of a passive particle in an active bath

We present an overview of the properties of a system of a passive particle immersed in a bath of active particles with translational inertia. In this overview, we not only classify the behavior of the passive particle, but also that of the active bath particles. Previous studies have shown analytically that the temperature of an overdamped active particle scales quadratically with its active velocity in the dilute limit. We analytically extend this theory to show that this scaling law also applies to active particles with translational inertia, though the scaling is now between the active particle temperature and its active force. Through computer simulations, we furthermore verify that this law holds in dense systems. We additionally demonstrate through computer simulations that the temperature of a passive particle immersed in an active bath also scales quadratically with the active force of the bath particles. However, the coefficient of the quadratic term is not the same and, consequently, a passive particle in an active bath will not equilibrate to the same temperature as the active bath particles. In contrast to models proposed in some previous studies, we show that the memory kernel of the passive particle can neither be approximated as Markovian nor as decaying exponentially on one time scale. Instead, at high active forces, the memory kernel develops a negative portion, a possible indication of backflow, which can also be seen in the long time tail of its velocity autocorrelation function.