  In an equilibrium thermal environment, random elastic collisions between
background particles and a tracer establish the picture of Brownian motion
fulfilling the celebrated Einstein relation between diffusivity and mobility.
In nature, environments often comprise collections of autonomously moving
objects that exhibit fascinating non-equilibrium phenomena and are nowadays
termed active matter. We investigate experimentally the impact of an active
background on a passive tracer using vibrationally excited active particles.
They produce multiple correlated tapping collisions with the tracer, for which
a persistent memory emerges in the dynamics. The system is described by a
generalized active Einstein relation that constrains fluctuations, dissipation,
and effective activity, by taking the emerging tracer memory into account.
Since the resulting persistence can largely be tuned by the environmental
density and motility, our findings can be useful to engineer properties of
various active systems in biomedical applications, microfluidics, chemical
engineering, or swarm robotics.

Statistical Mechanics;Soft Condensed Matter