A cold tracer in a hot bath: in and out of equilibrium

We study the dynamics of a zero-temperature overdamped tracer in a bath of Brownian particles at positive temperature. Surprisingly, although none of the constituent particles are self-propelled, we find that the cold tracer generically displays all the signatures of an active particle, including boundary accumulation, sustained currents in ratchet potentials, and density rectification by asymmetric obstacles. However, in the limit where the tracer interacts with many hot particles at once, it transitions to a bona-fide equilibrium regime. To account for this, we construct the generalized Langevin equation of the tracer under the assumption of linear coupling to the bath and show convergence to equilibrium in the limit of large bath density. We then develop a perturbation theory to characterize the departure from equilibrium at large but finite bath densities, revealing an intermediate time-reversible but non-Boltzmann regime, followed by a fully irreversible one. Finally, we show that when the bath particles are connected as a lattice, mimicking a gel, the cold tracer drives the entire bath out of equilibrium, leading to a long-ranged suppression of bath fluctuations. We discuss implications for experiments and for the many-body physics of multi-temperature particle systems.