  Characterizing the properties of an extended system driven by active
reservoirs is a question of increasing importance. Here we address this
question in two steps. We start by investigating the dynamics of a probe
particle connected to an `active Rubin bath' -- a linear chain of overdamped
run-and-tumble particles. We derive exact analytical expressions for the
effective noise and dissipation kernels, acting on the probe, and show that the
active nature of the bath leads to a modified fluctuation-dissipation relation.
In the next step, we study the properties of an activity-driven system, modeled
by a chain of harmonic oscillators connected to two such active reservoirs at
the two ends. We show that the system reaches a nonequilibrium stationary state
(NESS), remarkably different from that generated due to a thermal gradient. We
characterize this NESS by computing the kinetic temperature profile, spatial
and temporal velocity correlations of the oscillators, and the average energy
current flowing through the system. It turns out that, the activity drive leads
to the emergence of two characteristic length scales, proportional to the
activities of the reservoirs. Strong signatures of activity are also manifest
in the anomalous short-time decay of the velocity autocorrelations. Finally, we
find that the energy current shows a non-monotonic dependence on the activity
drive and reversal in direction, corroborating previous findings.
